UniversitextEditorial Board

(North America):

S. AxlerF.W. Gehring

K.A. Ribet

SpringerNew YorkBerlinHeidelbergBarcelonaBudapestHong KongLondonMilanParisSingaporeTokyo

Universitext

Editors (North America): S. Axler, F.W. Gehring, and K.A. Ribet

Aksoy/Khamsi: Nonstandard Methods in Fixed Point TheoryAndersson: Topics in Complex AnalysisAupetit: A Primer on Spectral TheoryBerberian: Fundamentals of Real AnalysisBooss/Bleecker: Topology and AnalysisBorkar: Probability Theory: An Advanced CourseCarlcson/Gamelin: Complex DynamicsCecil: Lie Sphere Geometry: With Applications to SubmanifoldsChae: Lebesgue Integration (2nd ed.)Charlap: Bieberbach Groups and Flat ManifoldsChern: Complex Manifolds Without Potential TheoryCohn: A Classical Invitation to Algebraic Numbers and Class FieldsCurtis: Abstract Linear AlgebraCurtis: Matrix GroupsDiBenedetto: Degenerate Parabolic EquationsDimca: Singularities and Topology of HypersurfacesEdwards: A Formal Background to Mathematics I a/bEdwards: A Formal Background to Mathematics II a/bFoulds: Graph Theory ApplicationsFriedman: Algebraic Surfaces and Holomorphic Vector BundlesFuhrmann: A Polynomial Approach to Linear AlgebraGardiner: A First Course in Group TheoryGarding/Tambour: Algebra for Computer ScienceGoldblatt: Orthogonality and Spacetime GeometryGustafson/Rao: Numerical Range: The Field of Values of Linear Operators

and MatricesHahn: Quadratic Algebras, Clifford Algebras, and Arithmetic Witt GroupsHolmgren: A First Course in Discrete Dynamical SystemsHowe/Tan: Non-Abelian Harmonic Analysis: Applications of SL(2, R)Howes: Modern Analysis and TopologyHumi/Miller: Second Course in Ordinary Differential EquationsHurwitz/Kritikos: Lectures on Number TheoryJennings: Modern Geometry with ApplicationsJones/Morris/Pearson: Abstract Algebra and Famous ImpossibilitiesKannan/Krueger: Advanced AnalysisKelly/Matthews: The Non-Euclidean Hyperbolic PlaneKostrikin: Introduction to AlgebraLuecking/Rubel: Complex Analysis: A Functional Analysis ApproachMacLane/Moerdijk: Sheaves in Geometry and LogicMarcus: Number FieldsMcCarthy: Introduction to Arithmetical FunctionsMeyer: Essential Mathematics for Applied FieldsMines/Richman/Ruitenburg: A Course in Constructive AlgebraMoise: Introductory Problems Course in Analysis and TopologyMorris: Introduction to Game TheoryPolster: A Geometrical Picture BookPorter/Woods: Extensions and Absolutes of Hausdorff SpacesRamsay/Richtmyer: Introduction to Hyperbolic GeometryReisel: Elementary Theory of Metric SpacesRickart: Natural Function Algebras

(continued after index)

M.W. Wong

Weyl Transforms

Springer

M.W. WongDepartment of Mathematics and StatisticsYork UniversityToronto, Ontario M3J 1P3Canada

Editorial Board(North America):

S. Axler F.W. GehringMathematics Department Mathematics DepartmentSan Francisco State University East HallSan Francisco, CA 94132 University of MichiganUSA Ann Arbor, MI 48109

USA

K.A. RibetDepartment of MathematicsUniversity of California at BerkeleyBerkeley, CA 94720-3840USA

Mathematics Subject Classification (1991): 44A15, 42-01, 43-01

Library of Congress Cataloging-in-Publication DataWong, M.W.

Weyl transforms / M.W. Wong.p. cm. — (Universitext)

Includes bibliographical references and indexes.ISBN 0-387-98414-3 (hardcover : alk. paper)l.Pseudodifferential operators. 2. Fourier analysis. I. Title.

Qa329.7.W66 1998515'.7242—dc21 98-13042

© 1998 Springer-Verlag New York, Inc.All rights reserved. This work may not be translated or copied in whole or in part without thewritten permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York,NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Usein connection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use of general descriptive names, trade names, trademarks, etc., in this publication, even if theformer are not especially identified, is not to be taken as a sign that such names, as understood bythe Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

ISBN 0-387-98414-3 Springer-Verlag New York Berlin Heidelberg SPIN 10663135

Preface

This book is an outgrowth of courses given by me for graduate students at YorkUniversity in the past ten years. The actual writing of the book in this form wascarried out at York University, Peking University, the Academia Sinica in Beijing,the University of California at Irvine, Osaka University, and the University ofDelaware. The idea of writing this book was first conceived in the summer of1989, and the protracted period of gestation was due to my daily duties as aprofessor at York University. I would like to thank Professor K.C. Chang, of PekingUniversity; Professor Shujie Li, of the Academia Sinica in Beijing; ProfessorMartin Schechter, of the University of California at Irvine; Professor MichihiroNagase, of Osaka University; and Professor M.Z. Nashed, of the University ofDelaware, for providing me with stimulating environments for the exchange ofideas and the actual writing of the book.

We study in this book the properties of pseudo-differential operators arising inquantum mechanics, first envisaged in [33] by Hermann Weyl, as bounded linearoperators on L2(Rn). Thus, it is natural to call the operators treated in this bookWeyl transforms.

To be specific, my original plan was to supplement the standard graduate coursein pseudo-differential operators at York University by writing a set of lecturenotes on the derivation of a formula from first principles for the product of twoWeyl transforms. This was achieved in the summer of 1990 when I was visitingPeking University and the Academia Sinica in Beijing. Chapters 2–6 of the book,which appeared then, albeit in embryonic form, already contained the formula forthe product of two Weyl transforms obtained by Pool in [20]. Chapters 8 and 9were written in the summer of 1993 at York University in order to get anotherformula for the product of two Weyl transforms using relatively new ideas, e.g.,

vi Preface

the Heisenberg group and the twisted convolution, in noncommutative harmonicanalysis developed by Folland in [6] and Stein in [26], among others. The result wasan account, given in Chapter 9, of a formula for the product of two Weyl transformsin the paper [10] by Grossmann, Loupias, and Stein. A preliminary version of thederivations of the two formulas was written up for private circulation in the secondquarter of 1994–95 at the University of California at Irvine.

In the summer of 1994, I gave a course in special topics in pseudo-differentialoperators tailored to the needs of my Ph.D. students at York University. I chose tostudy the criteria in terms of the symbols for the boundedness and compactness ofthe Weyl transforms. Two sets of results were presented. The first set was aboutthe compactness of a Weyl transform with symbol in Lr (R2n), 1 ≤ r ≤ ∞, andthe second set, inspired by the book [29] by Thangavelu, was concerned withthe criteria for the boundedness and compactness of Weyl transforms in terms ofsymbols evaluated at Wigner transforms of Hermite functions. The two sets ofresults can be found in, respectively, Chapters 11–14 and Chapters 24–27. Chapter28 is devoted to the study of the eigenvalues and eigenfunctions of a Weyl transformof which the symbol is a Dirac delta on a disk in R

2.The preliminary version of the formulas for the product of two Weyl transforms

and the lecture notes of the topics course given in the summer of 1994 were thenput together, simplified, polished, and supplemented with background materialsat Osaka University and the University of Delaware in the winter of 1997. To thisend, I found it instructive to add new chapters, i.e., Chapters 15–17, on localizationoperators initiated by Daubechies in [3, 4] and Daubechies and Paul in [5], andthe closely related theory of square-integrable group representations studied byGrossmann, Morlet, and Paul in [11, 12]. The final two chapters were added inan attempt to make explicit the role of the symplectic group in the study of Weyltransforms.

The connections of the Weyl transforms with quantization in physics, high-lighted in this book, can be found in the references [6, 10, 20, 26, 33] alreadycited, the book [2] by Berezin and Shubin, the paper [18] by Iancu and Wong, andthe papers [37, 38] by Wong.

All the topics in this book should be accessible to a first-year graduate student.The book is a natural sequel to a first course in pseudo-differential operators, butno familiarity with even the basics of pseudo-differential operators is required fora good understanding of the entire book. The only essential prerequisites are theelementary properties of the Fourier transform and tempered distributions givenin the beginning chapters of, say, the book [8] by Goldberg, the book [27] by Steinand Weiss, and the book [36] by Wong, and these are collected in Chapter 1. Ofcourse, a nodding acquaintance with basic functional analysis is necessary for anintelligent reading of this book.

Finally, it must be emphasized that this book is far from being a definitive treatiseon Weyl transforms. Thus, the choice of topics in this book was guided by personalpredilections, and the references at the end of the book are limited to those thathave been instrumental in my understanding of Weyl transforms.

Contents

Preface v

1 Prerequisite Topics in Fourier Analysis 1

2 The Fourier–Wigner Transform 9

3 The Wigner Transform 13

4 The Weyl Transform 19

5 Hilbert–Schmidt Operators on L2(Rn) 25

6 The Tensor Product in L2(Rn) 29

7 H ∗-Algebras and the Weyl Calculus 33

8 The Heisenberg Group 37

9 The Twisted Convolution 43

10 The Riesz–Thorin Theorem 47

11 Weyl Transforms with Symbols in Lr (R2n), 1 ≤ r ≤ 2 55

12 Weyl Transforms with Symbols in L∞(R2n) 59

viii Contents

13 Weyl Transforms with Symbols in Lr (R2n), 2 < r < ∞ 63

14 Compact Weyl Transforms 71

15 Localization Operators 75

16 A Fourier Transform 79

17 Compact Localization Operators 83

18 Hermite Polynomials 87

19 Hermite Functions 93

20 Laguerre Polynomials 95

21 Hermite Functions on C 101

22 Vector Fields on C 103

23 Laguerre Formulas for Hermite Functions on C 107

24 Weyl Transforms on L2(R) with Radial Symbols 113

25 Another Fourier Transform 119

26 A Class of Compact Weyl Transforms on L2(R) 123

27 A Class of Bounded Weyl Transforms on L2(R) 127

28 A Weyl Transform with Symbol in S ′(R2) 131

29 The Symplectic Group 135

30 Symplectic Invariance of Weyl Transforms 145

Notation Index 155

Index 157

1Prerequisite Topics in Fourier Analysis

The basic topics in Fourier analysis that we need for a good understanding of thebook are collected in this chapter. In view of the fact that these topics can be foundin many books on Fourier analysis, e.g., [8] by Goldberg, [27] by Stein and Weiss,and [36] by Wong, among others, we provide only the proofs of the key results inthe study of the Weyl transform. Another important role played by this chapter isto fix the notation used throughout the book.

Let Rn � {(x1, x2, . . . , xn) : xj real numbers}. Points in R

n are denoted byx, y, ξ, η, etc. Let x � (x1, x2, . . . , xn) and y � (y1, y2, . . . , yn) be in R

n. Theinner product x · y of x and y is defined by

x · y �n∑

j�1xjyj ,

and the norm |x| of x is defined by

|x| �(

n∑j�1

x2j

) 12

.

We denote the differential operators ∂∂x1

, ∂∂x2

, . . . , ∂∂xn

on Rn by ∂1, ∂2, . . . , ∂n,

respectively, and the differential operators −i∂1, −i∂2, . . . , −i∂n on Rn by

D1, D2, . . . , Dn, respectively, where i2 � −1. A reason for using the factor of−i is to make some formulas, e.g., Proposition 1.8, look better, but the mainjustification for its appearance lies in the fact that the quantum-mechanical mo-mentum observable in the direction of the j th coordinate is represented by Dj ifwe choose to work with units that give the value 1 to Planck’s constant. More de-

2 1. Prerequisite Topics in Fourier Analysis

tails are given in the discussion at the end of Chapter 4. A linear partial differentialoperator P (x, D) on R

n is given by

P (x, D) �∑

|α|≤m

aα(x)Dα, x ∈ Rn,

where α � (α1, α2, . . . , αn) is a multi-index, i.e., an n-tuple of nonnegative in-tegers; |α| � ∑n

j�1 αj is the length of α; Dα � Dα1Dα2 · · · Dαn , and aα isa measurable complex-valued function on R

n for |α| ≤ m. The symbol of thedifferential operator P (x, D) is the function on R

2n defined by

P (x, ξ ) �∑

|α|≤m

aα(x)ξα, x, ξ ∈ Rn,

where ξα � ξα11 ξ

α22 · · · ξαn

n .The differential operator ∂α , for any multi-index α, defined by

∂α � ∂α11 ∂

α22 · · · ∂αn

n ,

will also be used frequently in the book. We write ∂αx (or ∂α

ξ ) for ∂α , and Dαx

(or Dαξ ) for Dα , when we need to specify the variable with respect to which we

differentiate.Let f and g be infinitely differentiable functions on R

n. Then we have theLeibnitz formula

Dα(fg) �∑β≤α

(α

β

)(Dβf )(Dα−βg)

for all multi-indices α, and the more general Leibnitz formula

P (D)(fg) �∑

|µ|≤m

(P (µ)(D)f )(Dµg)

for any linear partial differential operator P (D) � ∑|α|≤m aαDα with constant

coefficients, where β ≤ α means βj ≤ αj , j � 1, 2, · · · , n,(α

β

) � (α1β1

)(α2β2

). . .(αn

βn

),

µ! � µ1!µ2! · · · µn!, and P (µ)(D) is the linear partial differential operator withsymbol P (µ) on R

n given by

P (µ)(ξ ) � (∂µP )(ξ ), ξ ∈ Rn.

Now we let C∞0 (Rn) be the set of all infinitely differentiable functions on R

n

with compact supports, and we let S(Rn) be the set of all infinitely differentiablefunctions on R

n such that

supx∈Rn

|xα(∂βϕ)(x)| < ∞

for all multi-indices α and β.

Theorem 1.1. C∞0 (Rn) and S(Rn) are dense in Lr (Rn), 1 ≤ r < ∞.

Theorem 1.1 can be proved using the convolution of functions on Rn.

1. Prerequisite Topics in Fourier Analysis 3

Theorem 1.2. (Young’s Inequality) Let f ∈ L1(Rn) and g ∈ Lr (Rn), 1 ≤ r ≤ ∞.Then the integral ∫

Rn

f (x − y)g(y)dy

exists for almost all x in Rn. If we denote the value of the integral by (f ∗ g)(x),

then f ∗ g ∈ Lr (Rn) and

‖f ∗ g‖Lr (Rn) ≤ ‖f ‖L1(Rn)‖g‖Lr (Rn).

Remark 1.3. The function f ∗ g in Theorem 1.2 is usually called the convolutionof f and g. The formulation of Young’s inequality for the convolution of sequences{ak}∞k�−∞ and {bk}∞k�−∞ of complex numbers is left as an exercise.

It is useful to have the following result.

Proposition 1.4. Let f and g be in S(Rn). Then f ∗ g ∈ S(Rn).

Our next result is a technique of regularization to be used in the proofs ofTheorems 1.11, 3.1, and 16.1.

Theorem 1.5. Let ϕ ∈ L1(Rn) be such that∫

Rn ϕ(x)dx � a. For any positivenumber ε, we define the function ϕε on R

n by

ϕε(x) � ε−nϕ(x

ε

), x ∈ R

n.

Then, for any bounded function f on Rn that is continuous on an open subset V

of Rn, f ∗ ϕε → af uniformly on compact subsets of V as ε → 0.

Proof. Without loss of generality, we can assume that ϕ(x) � 0 for almost all x

in Rn. Since ∫

Rn

ϕε(x)dx � a

for any positive number ε, it follows that

(f ∗ ϕε)(x) − af (x) �∫

Rn

{f (x − εy) − f (x)}ϕ(y)dy (1.1)

for all x in Rn. Let K be a compact subset of V and δ be a positive number. Let

K1 be a compact subset of V such that K is properly contained in K1. Then thereexists a positive number δ1 such that

|f (x) − f (y)| <δ

2‖ϕ‖−1

L1(Rn) (1.2)

for all x and y in K1 with |x − y| < δ1. Let U be an open subset of Rn such that

K ⊂ U ⊂ K1. Let δ2 be the distance between K and the complement of U in Rn

and let δ0 � min(δ1, δ2). Then

K ⊂⋃x∈K

B(x, δ0) ⊂ U, (1.3)

4 1. Prerequisite Topics in Fourier Analysis

where B(x, δ0) is the open ball with center x and radius δ0. Since ϕ ∈ L1(Rn),there exists a positive number R such that∫

|y|≥R

|ϕ(y)|dy <δ

4M, (1.4)

where M � supx∈Rn

|f (x)|. Hence, for any positive number ε, we get, by (1.1) and

(1.4),

|(f ∗ ϕε)(x) − af (x)|≤∫

|y|≥R

|f (x − εy) − f (x)‖ϕ(y)|dy +∫

|y|<R

|f (x − εy) − f (x)‖ϕ(y)|dy

<δ

2+∫

|y|<R

|f (x − εy) − f (x)‖ϕ(y)|dy, x ∈ Rn. (1.5)

For any x in K and any y in Rn with |y| < R, we have

x − εy ∈ B(x0, δ), (1.6)

provided that ε < δ0R

. Thus, by (1.2), (1.3), (1.5), and (1.6),

|(f ∗ ϕε)(x) − af (x)| < δ, x ∈ K,

whenever ε < δ0R

, and the proof is complete. �

Of fundamental importance in this book is the Fourier transform, which we nowdefine. The Fourier transform of a function in L1(Rn) is the function f , sometimesdenoted by Ff , on R

n defined by

f (ξ ) � (2π )−n/2∫

Rn

e−ix·ξ f (x)dx, ξ ∈ Rn.

We give in the following proposition some useful, albeit simple, properties of theFourier transform.

Proposition 1.6. (The Riemann–Lebesgue Lemma) Let f ∈ L1(Rn). Then f is acontinuous function on R

n such that lim|ξ |→∞ f (ξ ) � 0.

Proposition 1.7. Let f and g be in L1(Rn). Then

(f ∗ g)(ξ ) � (2π )n/2f (ξ )g(ξ ), ξ ∈ Rn.

Proposition 1.8. Let ϕ ∈ S(Rn). Then, for all multi-indices α,

(i) (Dαϕ )(ξ ) � ξαϕ(ξ ), ξ ∈ Rn,

(ii) (Dαϕ)(ξ ) � ((−x)αϕ )(ξ ), ξ ∈ Rn.

Proposition 1.9. Let ϕ ∈ L1(Rn). Then

(i) (Tyf )(ξ ) � (Myf )(ξ ), ξ ∈ Rn,

(ii) (Myf )(ξ ) � (T−yf )(ξ ), ξ ∈ Rn,

(iii) (Daf )(ξ ) � |a|−n(D 1af )(ξ ), ξ ∈ R

n,

1. Prerequisite Topics in Fourier Analysis 5

where

(Tyf )(x) � f (x + y), x ∈ Rn,

(Myf )(x) � eix·yf (x), x ∈ Rn,

and

(Daf )(x) � f (ax), x ∈ Rn,

for all y in Rn and all nonzero real numbers a.

Ty , My , and Da in Proposition 1.9 are respectively the translation operator, themodulation operator, and the dilation operator on R

n.

Proposition 1.10. (The Adjoint Formula) Let f and g be in L1(Rn). Then∫Rn

f (x)g(x)dx �∫

Rn

f (x)g(x)dx.

Important and less superficial properties of the Fourier transform are given inthe following two theorems.

Theorem 1.11. (The Fourier Inversion Formula) The Fourier transform is a oneto one and onto mapping from S(Rn) into S(Rn). Moreover,

(f ) � f, f ∈ S(Rn),

where

g(x) � (2π )−n/2∫

Rn

eix·ξ g(ξ )dξ, x ∈ Rn,

for all g in S(Rn).

Proof. Let f ∈ S(Rn). Then, for any positive number ε, we define the functionIε on R

n by

Iε(x) � (2π )−n/2∫

Rn

eix·ξ− ε2 |ξ |22 f (ξ )dξ, x ∈ R

n. (1.7)

Then, by (1.7), Propositions 1.9, 1.10, and the well-known fact that the Fouriertransform ϕ of the function ϕ on R

n given by

ϕ(x) � e− |x|22 , x ∈ R

n, (1.8)

is equal to ϕ, we get

Iε(x) � (2π )−n/2(f ∗ ϕε)(x), x ∈ Rn, (1.9)

where ϕε(x) � ε−nϕ(

xε

)for all x in R

n. Since f ∈ S(Rn), it follows from (1.8),(1.9), and Theorem 1.5 that

Iε → (2π )−n/2(∫

Rn

e− |x|22 dx

)f � f (1.10)

6 1. Prerequisite Topics in Fourier Analysis

uniformly on compact subsets of Rn as ε → 0. By (1.7) and the Lebesgue

dominated convergence theorem,

Iε(x) → (2π )−n/2∫

Rn

eix·ξ f (ξ )dξ (1.11)

for all x in Rn. Hence, by (1.10) and (1.11), (f ) � f . That the Fourier transform

from S(Rn) into S(Rn) is one-to-one and onto is then an easy consequence. �

Remark 1.12. The function g in Theorem 1.11 is called the inverse Fouriertransform of g and is sometimes denoted by F−1g.

Theorem 1.13. (The Plancherel Theorem) The mapping F : S(Rn) → S(Rn)can be extended uniquely to a unitary operator on L2(Rn).

Proof. By Theorems 1.1 and 1.11, it is sufficient to prove that

‖ϕ‖L2(Rn) � ‖ϕ‖L2(Rn), ϕ ∈ S(Rn). (1.12)

To this end, let ϕ ∈ S(Rn) and let ψ be the function on Rn defined by

ψ(x) � ϕ(−x), x ∈ Rn. (1.13)

Then ψ ∈ S(Rn), and an easy computation gives

ψ(ξ ) � ϕ(ξ ), ξ ∈ Rn. (1.14)

Thus, by (1.13),

‖ϕ‖2L2(Rn) �

∫Rn

ϕ(x)ϕ(x)dx �∫

Rn

ϕ(x)ψ(−x)dx � (ϕ ∗ ψ)(0). (1.15)

By Proposition 1.4, ϕ∗ψ ∈ S(Rn). Hence, by (1.14), Proposition 1.7, and Theorem1.11,

(ϕ ∗ ψ)(0) � (2π )−n/2∫

Rn

(ϕ ∗ ψ )(ξ )dξ �∫

Rn

ϕ(ξ )ψ(ξ )dξ

�∫

Rn

ϕ(ξ )ϕ(ξ )dξ � ‖ϕ‖2L2(Rn). (1.16)

Hence, by (1.15) and (1.16), (1.12) follows. �

Remark 1.14. In view of the Plancherel theorem, we can define the Fourier trans-form of a function f in L2(Rn), again denoted by f or Ff . The inverse Fouriertransform of a function f in L2(Rn) is denoted by f or F−1f .

Let us now review the very basic notions of tempered distributions used in thisbook.

Let {ϕj }∞j�1 be a sequence of functions in S(Rn) such that for all multi-indicesα and β,

supx∈Rn

|xα(∂βϕj )(x)| → 0

1. Prerequisite Topics in Fourier Analysis 7

as j → ∞. Then we say that ϕj → 0 in S(Rn) as j → ∞. A linear functional T

on S(Rn) is called a tempered distribution on Rn if

T (ϕj ) → 0

as j → ∞ for every sequence {ϕj }∞j�1 of functions in S(Rn) such that ϕj → 0 inS(Rn) as j → ∞. The collection of all tempered distributions on R

n is denotedby S ′(Rn), and a sequence {Tj }∞j�1 of tempererd distributions in S ′(Rn) is said toconverge to zero in S ′(Rn), denoted by Tj → 0 in S ′(Rn), as j → ∞ if

Tj (ϕ) → 0, ϕ ∈ S(Rn),

as j → ∞. The most important tempered distributions to us are given by temperedfunctions on R

n. Let us recall that a measurable function f on Rn is said to be

tempered if ∫Rn

|f (x)|(1 + |x|)N dx < ∞

for some positive integer N . All functions in Lr (Rn), 1 ≤ r ≤ ∞, are thereforetempered.

Proposition 1.15. Let f be a tempered function on Rn. Then the linear functional

Tf on S(Rn) defined by

Tf (ϕ) �∫

Rn

f (x)ϕ(x)dx, ϕ ∈ S(Rn),

is a tempered distribution.

Remark 1.16. It is customary to identify the tempered distribution Tf with thefunction f .

We leave it as an exercise to prove the following proposition, which will be usedin Chapter 28.

Proposition 1.17. Let δ : S(R2) → C be the linear mapping defined by

δ(ϕ) �∫ 2π

0ϕ(ρeie)ρdθ, ρ > 0.

Then δ ∈ S ′(R2).

Another exercise is to prove the following proposition.

Proposition 1.18. S(Rn) is dense in S ′(Rn).

We can now introduce the class of pseudo-differential operators studied in thebook [36] by Wong. They will be used in Chapter 4 to motivate the definition ofthe Weyl transform. Familiarity of pseudo-differential operators is desirable, butnot necessary, for a good understanding of the materials in this book.

Let m be any real number. Then we define Sm to be the set of all infinitelydifferentiable functions on R

2n such that for all multi-indices α and β, there is a

8 1. Prerequisite Topics in Fourier Analysis

positive constant Cα,β , depending on α and β only, for which

|(Dαx D

β

ξ σ )(x, ξ )| ≤ Cα,β(1 + |ξ |)m−|β|, x, ξ ∈ Rn.

We call any σ ∈ ⋃m∈R

Sm a symbol. Let σ be a symbol. Then we define thepseudo-differential operator Tσ corresponding to the symbol σ by

(Tσϕ)(x) � (2π )−n/2∫

Rn

eix·ξ σ (x, ξ )ϕ(ξ )dξ, x ∈ Rn,

for all ϕ in S(Rn). Using Proposition 1.8 and the Fourier inversion formula, it canbe shown that a linear partial differential operator

∑|α|≤m

aα(x)Dα on Rn, where aα

is an infinitely differentiable function on Rn such that

supx∈R

|(Dβaα)(x)| < ∞, |α| ≤ m,

for all multi-indices β, is a pseudo-differential operator corresponding to thesymbol σ in Sm given by

σ (x, ξ ) �∑

|α|≤m

aα(x)ξα, x, ξ ∈ Rn.

Proposition 1.19. Let σ be a symbol. Then the pseudo-differential operator Tσ

maps S(Rn) continuously into S(Rn).

2The Fourier–Wigner Transform

A basic tool we use in the study of the Weyl transform is the Wigner transform.We find it convenient to introduce first a related transform, which we call theFourier–Wigner transform.

Let q and p be in Rn, and let f be a measurable function on R

n. We define thefunction ρ(q, p)f on R

n by

(ρ(q, p)f )(x) � eiq·x+ 12 iq·pf (x + p), x ∈ R

n. (2.1)

Proposition 2.1. ρ(q, p) : L2(Rn) → L2(Rn) is a unitary operator for all q andp in R

n.

Proof. We only need to prove that

‖ρ(q, p)f ‖L2(Rn) � ‖f ‖L2(Rn), f ∈ L2(Rn),

and ρ(q, p) is onto for all q and p in Rn. But, by (2.1),

‖ρ(q, p)f ‖2L2(Rn) �

∫Rn

|eiq·x+ 12 iq·pf (x + p)|2dx

�∫

Rn

|f (x + p)|2dx

�∫

Rn

|f (x)|2dx

� ‖f ‖2L2(Rn), f ∈ L2(Rn),

10 2. The Fourier–Wigner Transform

for all q and p in Rn. To prove that ρ(q, p) is onto, we let g ∈ L2(Rn) and define

the function f on Rn by

f (x) � e−iq·x+ 12 iq·pg(x − p), x ∈ R

n. (2.2)

Then f is obviously in L2(Rn), and by (2.1) and (2.2),

(ρ(q, p)f )(x) � eiq·x+ 12 iq·pf (x + p)

� eiq·x+ 12 iq·pe−iq·(x+p)+ 1

2 iq·pg(x)� g(x), x ∈ R

n. �

Remark 2.2. It is clear from the proof of Proposition 2.1 that ρ(q, p)−1 �ρ(−q, −p), q, p ∈ R

n. In fact, ρ is a projective representation, i.e., a unitaryrepresentation up to phase factors, of the phase space R

2n on L2(Rn), and it isclosely related to the Schrodinger representation R1 of the Heisenberg group Hn

on L2(Rn) to be studied in Chapter 8. The elucidation of the connection betweenρ and R1 is given in Remark 8.8.

Let f and g be in S(Rn). Then we define the function V (f, g) on R2n by

V (f, g)(q, p) � (2π )−n/2〈ρ(q, p)f, g〉, q, p ∈ Rn, (2.3)

where 〈 , 〉 is the inner product in L2(Rn). We call V (f, g) the Fourier–Wignertransform of f and g.

The notation 〈 , 〉 will also be used to denote the inner product in L2(R2n).

Proposition 2.3. Let f and g be in S(Rn). Then

V (f, g)(q, p) � (2π )−n/2∫

Rn

eiq·yf(y + p

2

)g(y − p

2

)dy (2.4)

for all q and p in Rn.

Proof. By (2.1) and (2.3),

V (f, g)(q, p) � (2π )−n/2〈ρ(q, p)f, g〉� (2π )−n/2

∫Rn

eiq·x+ 12 iq·pf (x + p)g(x)dx (2.5)

for all q and p in Rn. Let x � y− p

2in (2.5). Then we get (2.4) immediately. �

Proposition 2.4. V : S(Rn) × S(Rn) → S(R2n) is a bilinear mapping.

Recall that ifX andY are complex vector spaces, then a mappingf : X×X → Y

is said to be bilinear if for all α1 and α2 in C and all x1 and x2 in X, we have

f (α1x1 + α2x2, x) � α1f (x1, x) + α2f (x2, x)

and

f (x, α1x1 + α2x2) � α1f (x, x1) + α2f (x, x2)

for all x in X.

2. The Fourier–Wigner Transform 11

The bilinearity in Proposition 2.4 is easy to check. To prove that V maps S(Rn)×S(Rn) into S(R2n), we need a lemma.

Lemma 2.5. Let ϕ ∈ S(R2n). Then the function � on R2n defined by

�(q, p) �∫

Rn

eiq·yϕ(y, p)dy, q, p ∈ Rn, (2.6)

is also in S(R2n).

We assume Lemma 2.5 for a moment and use it to complete the proof of Propo-sition 2.4. To do this, we note that for all f and g in S(Rn), the function ϕ onS(R2n) defined by

ϕ(y, p) � f (y)g(p), y, p ∈ Rn,

is obviously in S(R2n). Hence the function ψ on R2n defined by

ψ(y, p) � f(y + p

2

)g(y − p

2

), y, p ∈ R

n, (2.7)

is also in S(R2n). Therefore, by (2.4), (2.7), and Lemma 2.5, V (f, g) ∈ S(R2n).The proof that ψ is in S(R2n) is left as an exercise.

Proof of Lemma 2.5. Let α, β, γ , and δ be multi-indices. Then, by (2.6),

qαpβ(∂γq ∂δ

p�)(q, p) � qαpβ

∫Rn

(iy)γ eiq·y(∂δpϕ)(y, p)dy

�∫

Rn

1i|α| i

|γ |(∂αy eiq·y)yγ pβ(∂δ

pϕ)(y, p)dy

� (−1)|α|i|γ |−|α|∫

Rn

eiq·y∂αy {yγ pβ(∂δ

pϕ)(y, p)}dy (2.8)

for all q and p in Rn. Now, there exists a positive constant Cαβγ δ , depending on

α, β, γ , and δ only, such that

|∂αy {yγ pβ(∂δ

pϕ)(y, p)}| ≤ Cαβγ δ(1 + |y|2)−N, y ∈ Rn, (2.9)

where N is some positive integer greater thann

2. Hence, by (2.8) and (2.9),

supq,p∈Rn

|qαpβ(∂γq ∂δ

p�)(q, p)| ≤ Cαβγ δ

∫Rn

(1 + |y|2)−Ndy,

and the proof is complete. �

3The Wigner Transform

In this chapter, we introduce the Wigner transform and study some of its very basicproperties. The Wigner transform W (f ) of a function f in L2(Rn), introducedby Wigner in [35], is a tool for the study of the nonexisting joint probabilitydistribution of position and momentum in the state f . In order to study the Weyltransform, it is necessary to have the notion of the Wigner transform of two arbitraryfunctions in L2(Rn). To do this, we begin by computing the Fourier transform ofthe Fourier–Wigner transform.

Theorem 3.1. Let f and g be in S(Rn). Then

V (f, g)(x, ξ ) � (2π )−n/2∫

Rn

e−iξ ·pf(x + p

2

)g(x − p

2

)dp, x, ξ ∈ R

n.

(3.1)

Proof. For any positive number ε, we define the function Iε on R2n by

Iε(x, ξ ) �∫

Rn

∫Rn

e− ε2 |q|22 e−ix·q−iξ ·pV (f, g)(q, p)dq dp, x, ξ ∈ R

n. (3.2)

Then, using Fubini’s theorem and the fact that the Fourier transform of the functionϕ given by

ϕ(x) � e− |x|22 , x ∈ R

n, (3.3)

14 3. The Wigner Transform

is equal to ϕ, we get, by (3.2),

Iε(x, ξ )

� (2π )−n/2∫

Rn

∫Rn

e− ε2 |q|22 e−ix·q−iξ ·p

×{∫

Rn

eiq·yf(y + p

2

)g(y − p

2

)dy

}dq dp

� (2π )−n/2∫

Rn

e−iξ ·p

×{∫

Rn

(∫Rn

e−i(x−y)·qe− ε2 |q|22 dq

)f(y + p

2

)g(y − p

2

)dy

}dp

�∫

Rn

e−iξ ·p{∫

Rn

ε−ne− |x−y|2

2ε2 f(y + p

2

)g(y − p

2

)dy

}dp,

x, ξ ∈ Rn. (3.4)

Now, for each p in Rn, we define the function Fp on R

n by

Fp(y) � f(y + p

2

)g(y − p

2

), y ∈ R

n. (3.5)

Then, by (3.4) and (3.5),

Iε(x, ξ ) �∫

Rn

e−iξ ·p(Fp ∗ ϕε)(x)dp, x, ξ ∈ Rn, (3.6)

where

ϕε(x) � ε−nϕ(x

ε

), x ∈ R

n. (3.7)

Note that, for each fixed p in Rn, by (3.3) and (3.5),

Fp ∗ ϕε →(∫

Rn

ϕ(x)dx

)Fp � (2π )

n2 Fp (3.8)

uniformly on compact subsets of Rn as ε → 0. Let N be any positive integer.

Then, by (3.3), (3.5), and (3.7), there exists a positive constant CN such that

|(Fp ∗ ϕε)(x)| ≤ ‖Fp‖L∞(Rn)‖ϕε‖L1(Rn)

� ‖Fp‖L∞(Rn)‖ϕ‖L1(Rn)

≤ (2π )n/2 supy∈Rn

∣∣∣f (y + p

2

)g(y − p

2

)∣∣∣≤ CN (1 + |p|2)−N, x, p ∈ R

n, (3.9)

for all positive numbers ε. So, by (3.6), (3.8), (3.9), and the Lebesgue dominatedconvergence theorem,

limε→0

Iε(x, ξ ) � (2π )n/2∫

Rn

e−iξ ·pf(x + p

2

)g(x − p

2

)dp, x, ξ ∈ R

n.

(3.10)

3. The Wigner Transform 15

But, using (3.2) and again the Lebesgue dominated convergence theorem,

limε→0

Iε(x, ξ ) �∫

Rn

∫Rn

e−ix·q−iξ ·pV (f, g)(q, p)dq dp

� (2π )nV (f, g)(x, ξ ), x, ξ ∈ Rn. (3.11)

So, by (3.10) and (3.11), (3.1) is valid. �

We can now define the Wigner transform of two functions in S(Rn). To do this,let f and g be in S(Rn). Then the function W (f, g) on R

2n, defined by

W (f, g)(x, ξ ) � (2π )−n/2∫

Rn

e−iξ ·pf(x + p

2

)g(x − p

2

)dp, x, ξ ∈ R

n,

(3.12)is called the Wigner transform of f and g, and can be used, as in the paper [19] byMoyal, to interpret quantum mechanics as a form of nondeterministic statisticaldynamics.

Some of the most basic properties of the Wigner transform are given in thischapter.

Theorem 3.2. (The Moyal Identity) For all f1, g1, f2, and g2 in S(Rn), we have

〈W (f1, g1), W (f2, g2)〉 � 〈f1, f2〉〈g1, g2〉. (3.13)

Proof. We define W : S(R2n) → S(R2n) by

(WF )(x, ξ ) � (2π )−n/2∫

Rn

e−iξ ·pF(x + p

2, x − p

2

)dp, x, ξ ∈ R

n, (3.14)

for all F in S(R2n). Then, by (3.14) and the Plancherel theorem,

〈WF1, WF2〉 �∫

Rn

∫Rn

(WF1)(x, ξ )(WF2)(x, ξ )dx dξ

�∫

Rn

{∫Rn

(WF1)(x, ξ )(WF2)(x, ξ )dξ

}dx

�∫

Rn

{∫Rn

F1

(x + p

2, x − p

2

)F2

(x + p

2, x − p

2

)dp

}dx

�∫

Rn

∫Rn

F1

(x + p

2, x − p

2

)F2

(x + p

2, x − p

2

)dp dx (3.15)

for all F1 and F2 in S(R2n). Let u � x + p

2and v � x − p

2. Then, by (3.15), we

get

〈WF1, WF2〉 �∫

Rn

∫Rn

F1(u, v)F2(u, v)du dv

� 〈F1, F2〉, F1, F2 ∈ S(R2n). (3.16)

Now, let f1, g1, f2, and g2 be in S(Rn). Let F1 and F2 be functions on R2n defined

by

F1(u, v) � f1(u)g1(v), u, v ∈ Rn, (3.17)

16 3. The Wigner Transform

and

F2(u, v) � f2(u)g2(v), u, v ∈ Rn. (3.18)

Then, by (3.12), (3.14), and (3.16)–(3.18),

〈W (f1, g1), W (f2, g2)〉 � 〈WF1, WF2〉 � 〈F1, F2〉�∫

Rn

∫Rn

F1(u, v)F2(u, v)du dv

�∫

Rn

∫Rn

f1(u)g1(v)f2(u)g2(v)du dv

�(∫

Rn

f1(u)f2(u)du

)(∫Rn

g1(v)g2(v)dv

)� 〈f1, f2〉〈g1, g2〉.

�

Corollary 3.3. The Moyal identity is also true for the Fourier–Wigner transformV .

Corollary 3.4. W : S(Rn) × S(Rn) → S(R2n) can be extended uniquely to abilinear operator

W : L2(Rn) × L2(Rn) → L2(R2n)

such that

‖W (f, g)‖L2(R2n) � ‖f ‖L2(Rn)‖g‖L2(Rn)

for all f and g in L2(Rn).

Corollary 3.5. The preceding corollary is also true for the Fourier–Wignertransform V .

Proposition 3.6.

(i) Let t ∈ R − {0} and let f be any measurable function on Rn. Let f t be the

function on Rn defined by

f t (x) � |t | n2 f (tx), x ∈ R

n.

Then, for all f and g in S(Rn),

W (f t , gt )(x, ξ ) � W (f, g)(tx, t−1ξ ), x, ξ ∈ Rn.

(ii) Let a, b, c, and d be in Rn and let f and g be in S(Rn). Then

W (ρ(a, b)f, ρ(c, d)g)(x, ξ )

� ei{(a−c)·x+(b−d)·ξ}e12 i(a·d−b·c)W (f, g)

(x + b + d

2, ξ − a + c

2

)

for all x and ξ in Rn.

(iii) W (g, f ) � W (f, g), f, g ∈ S(Rn).

3. The Wigner Transform 17

Corollary 3.7. Let W (f ) � W (f, f ), f ∈ L2(Rn). Then

(i) W (ρ(a, b)f )(x, ξ ) � W (f )(x + b, ξ − a), a, b, x, ξ ∈ Rn,

(ii) W (f ) is real-valued.

The proofs of Corollaries 3.3, 3.4, 3.5, Proposition 3.6, and Corollary 3.7 areleft as exercises.

Proposition 3.8. Let f ∈ L2(Rn) and G ∈ L2(R2n). Then

(W (f ) ∗ G)(x, ξ ) � 〈W (ρ(ξ, −x)f ), G〉, x, ξ ∈ Rn, (3.19)

where f (x) � f (−x), x ∈ Rn.

Proof. We begin by noting that

(W (f ) ∗ G)(x, ξ ) �∫

Rn

∫Rn

W (f )(x − y, ξ − η)G(y, η)dy dη, x, ξ ∈ Rn.

(3.20)But, by part (i) of Proposition 3.6 and part (i) of Corollary 3.7,

W (f )(x − y, ξ − η) � W (f )(y − x, η − ξ )� W (ρ(ξ, −x)f )(y, η), x, y, ξ, η ∈ R

n. (3.21)

So, by (3.20) and (3.21), we get

(W (f ) ∗ G)(x, ξ ) �∫

Rn

∫Rn

W (ρ(ξ, −x)f )(y, η)G(y, η)dy dη

� 〈W (ρ(ξ, −x)f ), G〉for all x and ξ in R

n. �

Corollary 3.9. Let f and g be in S(Rn). Then

(W (f ) ∗ W (g))(x, ξ ) � (2π )n|V (f , g)(ξ, −x)|2, x, ξ ∈ Rn.

Proof. Let G � W (g). Then, by part (ii) of Corollary 3.6, G is real-valued.Thus, by (3.13) and (3.19),

(W (f ) ∗ W (g))(x, ξ ) � 〈W (ρ(ξ, −x)f ), Wg〉� 〈ρ(ξ, −x)f , g〉〈ρ(ξ, −x)f , g〉� |〈ρ(ξ, −x)f , g〉|2, x, ξ ∈ R

n. (3.22)

Therefore, by (2.3) and (3.22), we complete the proof. �

4The Weyl Transform

We can now introduce the Weyl transform and explicate its beautiful connectionwith the Wigner transform. It is instructive, though absolutely unnecessary, to moti-vate the definition of the Weyl transform by means of pseudo-differential operatorsbriefly described in Chapter 1. The role of the Weyl transform in quantization isgiven at the end of the chapter as an impetus for further development of the subject.

Let σ ∈ Sm, m ∈ R. Then, for any function ϕ in S(Rn), the function Tσϕ on Rn

can be written as

(Tσϕ)(x) � (2π )−n

∫Rn

∫Rn

ei(x−y)·ξ σ (x, ξ )ϕ(y)dy dξ, x ∈ Rn, (4.1)

where the integral in (4.1) is understood to be an iterated integral.There is another way of associating a linear operator from S(Rn) into S(Rn)

with a symbol σ in Sm. Note that for all ϕ in S(Rn), we can define another functionWσϕ on R

n by

(Wσϕ)(x) � (2π )−n

∫Rn

∫Rn

ei(x−y)·ξ σ(

x + y

2, ξ

)ϕ(y)dy dξ, x ∈ R

n, (4.2)

where, again, the integral in (4.2) is understood to be an iterated integral. In orderto obtain another representation of Wσϕ, we let θ be any function in C∞

0 (Rn) suchthat θ (0) � 1. Then we get the following result.

Lemma 4.1. The limit

limε→0+

(2π )−n

∫Rn

∫Rn

θ (εξ )ei(x−y)·ξ σ(

x + y

2, ξ

)ϕ(y)dy dξ

20 4. The Weyl Transform

exists and is independent of the choice of the function θ . Moreover, the convergenceis uniform with respect to x on R

n.

Proof. Note that for any positive integer N ,

(1 − �y)N {ei(x−y)·ξ } � (1 + |ξ |2)Nei(x−y)·ξ , x, y, ξ ∈ Rn. (4.3)

So, by (4.3) and integration by parts, we get

(2π )−n

∫Rn

∫Rn

θ (εξ )ei(x−y)·ξ σ(x + y

2, ξ

)ϕ(y)dy dξ

� (2π )−n

∫Rn

∫Rn

θ (εξ )(1 + |ξ |2)−N ((1 − �y)N {ei(x−y)·ξ })σ(x + y

2, ξ

)ϕ(y)dy dξ

� (2π )−n

∫Rn

∫Rn

θ (εξ )(1 + |ξ |2)−Nei(x−y)·ξ (1 − �y)N{σ

(x + y

2, ξ

)ϕ(y)

}dy dξ

(4.4)

for all x in Rn and all positive numbers ε. Let P (D) � (1 − �)N . Then, by (4.4)

and Leibnitz’s formula,

(2π )−n

∫Rn

∫Rn

θ (εξ )ei(x−y)·ξ σ(

x + y

2, ξ

)ϕ(y)dy dξ

�∑

µ

1µ!

(2π )−n

∫Rn

∫Rn

θ (εξ )(1 + |ξ |2)−Nei(x−y)·ξ

× 12|µ| (D

µσ )(

x + y

2, ξ

)(P (µ)(D)ϕ)(y)dy dξ (4.4)

for all x in Rn and all positive numbers ε, where P (µ)(D) is the partial differential

operator with symbol P (µ), where P (µ)(ξ ) � (∂µP )(ξ ), ξ ∈ Rn. Now, for each

fixed x in Rn,

θ (εξ )(1 + |ξ |2)−Nei(x−y)·ξ (Dµσ )(

x + y

2, ξ

)(P (µ)(D)ϕ)(y)

−→ (1 + |ξ |2)−Nei(x−y)·ξ (Dµσ )(

x + y

2, ξ

)(P (µ)(D)ϕ)(y) (4.5)

for all y and ξ in Rn as ε → 0+. Furthermore, there exists a positive constant C

such that ∣∣∣∣θ (εξ )(1 + |ξ |2)−Nei(x−y)·ξ (Dµσ )(

x + y

2, ξ

)(P (µ)(D)ϕ)(y)

∣∣∣∣≤ C(1 + |ξ |2)−N (1 + |ξ |)m|(P (µ)(D)ϕ)(y)|, y, ξ ∈ R

n. (4.6)

Since

(1 + |ξ |2)−N (1 + |ξ |)m|(P (µ)(D)ϕ)(y)|is in L1(R2n) as a function of (y, ξ ) on R

2n if 2N − m > n, i.e., N > m+n2 , it

follows from (4.4), (4.5), (4.6), and the Lebesgue dominated convergence theorem

4. The Weyl Transform 21

that

limε→0

(2π )−n

∫Rn

∫Rn

θ (εξ )ei(x−y)·ξ σ(

x + y

2, ξ

)ϕ(y)dy dξ

exists and is independent of the choice of θ . It can also be checked that theconvergence is uniform with respect to x on R

n. �

From the proof of Lemma 4.1, we can get another formula for Wσϕ when ϕ isin S(Rn). Indeed,

(Wσϕ)(x) � (2π )−n

∫Rn

∫Rn

(1 + |ξ |2)−Nei(x−y)·ξ (1 − �y)N

×{σ

(x + y

2, ξ

)ϕ(y)

}dy dξ

for all x in Rn, where N is any positive integer greater than m+n

2 .The proof of the following proposition is left as an exercise.

Proposition 4.2. Let σ ∈ Sm, m ∈ R. Then Wσ : S(Rn) → S(Rn) is continuous.

We call the linear operator Wσ the Weyl transform associated with the symbolσ . The following result illuminates the relationship between the Weyl transformand the Wigner transform and plays a major role in the development of the theoryof the Weyl transform in this book.

Theorem 4.3. Let σ ∈ Sm, m ∈ R. Then

〈Wσf, g〉 � (2π )−n/2∫

Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ, f, g ∈ S(Rn).

Proof. Let θ be any function in C∞0 (Rn) such that θ (0) � 1. Then, by (3.12),

Lemma 4.1, the Lebesgue dominated convergence theorem, and Fubini’s theorem,∫Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ

� limε→0+

∫Rn

∫Rn

θ (εξ )σ (x, ξ )W (f, g)(x, ξ )dx dξ

� limε→0+

(2π )−n/2∫

Rn

∫Rn

θ (εξ )σ (x, ξ )

×{∫

Rn

e−iξ ·pf(x + p

2

)g(x − p

2

)dp

}dx dξ

� limε→0+

(2π )−n/2∫

Rn

θ (εξ )

×{∫

Rn

∫Rn

σ (x, ξ )e−iξ ·pf(x + p

2

)g(x − p

2

)dp dx

}dξ.

(4.7)

22 4. The Weyl Transform

Let u � x+ p

2 and v � x− p

2 in the last term in (4.7). Then, by Lemma 4.1, Fubini’stheorem, and the Lebesgue dominated convergence theorem, (4.7) becomes∫

Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ

� limε→0+

(2π )−n/2∫

Rn

θ (εξ ){∫

Rn

∫Rn

σ

(u + v

2, ξ

)ei(v−u)·ξ f (u)g(v)du dv

}dξ

� limε→0+

(2π )−n/2∫

Rn

g(v){∫

Rn

∫Rn

θ (εξ )σ(

u + v

2, ξ

)ei(v−u)·ξ f (u)du dξ

}dv

� (2π )−n/2∫

Rn

g(v)(Wσf )(v)dv � (2π )n/2〈Wσf, g〉. �

We denote the C∗-algebra of all bounded linear operators from L2(Rn) intoL2(Rn) by B(L2(Rn)).

Theorem 4.4. There exists a unique bounded linear operator Q : L2(R2n) →B(L2(Rn)) such that

〈(Qσ )f, g〉 � (2π )−n/2∫

Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ (4.8)

and

‖Qσ‖∗ ≤ (2π )−n/2‖σ‖L2(R2n) (4.9)

for all f and g in L2(Rn) and σ in L2(R2n), where ‖ ‖∗ denotes the norm inB(L2(Rn)).

Proof. Let σ ∈ S(R2n). Then, for any f in S(Rn), we define (Qσ )f by

(Qσ )f � Wσf. (4.10)

Then, by Theorem 4.3 and (4.10),

〈(Qσ )f, g〉 � 〈Wσf, g〉� (2π )−n/2

∫Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ (4.11)

for all f and g in S(Rn). Therefore, by Theorem 3.2 and (4.11),

|〈(Qσ )f, g〉| ≤ (2π )−n/2‖σ‖L2(R2n)‖W (f, g)‖L2(R2n)

� (2π )−n/2‖σ‖L2(R2n)‖f ‖L2(Rn)‖g‖L2(Rn) (4.12)

for all f and g in S(Rn). Hence, by (4.12),

‖(Qσ )f ‖L2(Rn) ≤ (2π )−n/2‖σ‖L2(R2n)‖f ‖L2(Rn) (4.13)

for all f in S(Rn). Therefore, by (4.13),

‖Qσ‖∗ ≤ (2π )−n/2‖σ‖L2(R2n), σ ∈ S(R2n). (4.14)

Now, let σ ∈ L2(R2n). Let {σk}∞k�1 be a sequence of functions in S(R2n) such thatσk → σ in L2(R2n) as k → ∞. Then, by (4.14),

‖Qσk − Qσl‖∗ ≤ (2π )−n/2‖σk − σl‖L2(R2n) → 0

4. The Weyl Transform 23

as k, l → ∞. Thus, {Qσk}∞k�1 is a Cauchy sequence in B(L2(Rn)). We defineQσ to be the limit in B(L2(Rn)) of the sequence {Qσk}∞k�1. This definition isindependent of the choice of the sequence {σk}∞k�1. Indeed, let {τk}∞k�1 be anothersequence of functions in S(R2n) such that τk → σ in L2(R2n) as k → ∞. Then,again, by (4.14),

‖Qσk − Qτk‖∗ ≤ (2π )−n/2‖σk − τk‖L2(R2n) → 0

as k → ∞. Thus, the limits in B(L2(Rn)) of {Qσk}∞k�1 and {Qτk}∞k�1 are equal.Next, let σ ∈ L2(R2n), and let {σk}∞k�1 be a sequence of functions in S(R2n) suchthat σk → σ in L2(R2n) as k → ∞. Then, by (4.14),

‖Qσ‖∗ � limk→∞

‖Qσk‖∗ ≤ (2π )−n/2 limk→∞

‖σk‖L2(R2n) � (2π )−n/2‖σ‖L2(R2n),

and (4.9) is proved. Now, if f and g are in S(Rn), then, by (4.1),

〈(Qσ )f, g〉 � limk→∞

〈(Qσk)f, g〉

� limk→∞

(2π )−n/2∫

Rn

∫Rn

σk(x, ξ )W (f, g)(x, ξ )dx dξ

� (2π )−n/2∫

Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ.

Finally, let f and g be in L2(Rn). Then we pick sequences {fk}∞k�1 and {gk}∞k�1 inS(Rn) such that fk → f in L2(Rn) and gk → g in L2(Rn) as k → ∞. We have

〈(Qσ )f, g〉 � limk→∞

〈(Qσ )fk, gk〉

� limk→∞

(2π )−n/2∫

Rn

∫Rn

σ (x, ξ )W (fk, gk)(x, ξ )dx dξ

� (2π )−n/2∫

Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ.

It is obvious that Q : L2(R2n) → B(L2(Rn)) is the only bounded linear operatorsatisfying (4.8) for all f and g in L2(Rn) and σ in L2(R2n). �

Remark 4.5. From now on, we shall denote Qσ by Wσ for any function σ inL2(R2n).

In classical mechanics, the phase space used to describe the motion of a particlemoving in R

n is given by

R2n � {(x, ξ ) : x, ξ ∈ R

n},where the variables x and ξ are used to denote the position and momentum ofthe particle, respectively. The observables of the motion are given by real-valuedtempered distributions on R

2n. The rules of quantization, with Planck’s constantadjusted to 1, say that a quantum-mechanical model of the motion can be set upusing the Hilbert space L2(Rn) for the phase space, the multiplication operator onL2(Rn) by the function xj for the position variable xj , and the differential operatorDj for the momentum variable ξj . Thus, the quantum-mechanical analogue of

24 4. The Weyl Transform

the classical mechanical observable σ (x, ξ ) should be the linear operator σ (x, D)obtained by direct substitution, provided that D is understood to be the “vector”(D1, D2, . . . , Dn). The mathematical problem then is to define σ (x, D) for, atleast, a good class of functions σ on R

2n. Let σ ∈ Sm, m ≤ 0, say. Can we use the(bounded) pseudo-differential operator Tσ on L2(Rn) for σ (x, D)? To answer thisquestion, let us note that in quantum mechanics, observables must be representedby self-adjoint operators. Unfortunately, the pseudo-differential operator Tσ failsin general to be self-adjoint, despite the fact that σ is real-valued. Hence, Tσ is notthe correct definition for σ (x, D), and our immediate task is to develop a functionalcalculus for the Weyl transform. The resulting Weyl calculus, given in Theorem7.5, shows that σ (x, D) should be defined to be Wσ .

Good accounts of the mathematical foundations of quantum mechanics basedon self-adjoint operators on Hilbert spaces, pioneered by von Neumann in [32] andadopted in this book, can be found in, e.g., the book [2], by Berezin and Shubin;the book [21], by Prugovecki; and the book [22], by Schechter.

5Hilbert–Schmidt Operators on L2(Rn)

We give in this chapter a self-contained treatment of Hilbert–Schmidt operatorson L2(Rn), which play a central role in the development of a functional calculusfor the Weyl transform.

Let h ∈ L2(R2n). Then we define the integral operator Sh : L2(Rn) → L2(Rn)by

(Shf )(x) �∫

Rn

h(x, y)f (y)dy, x ∈ Rn, (5.1)

for all f in L2(Rn). To check that Shf is in L2(Rn), we note that by (5.1),

{∫Rn

|(Shf )(x)|2dx

} 12

�{∫

Rn

∣∣∣∣∫

Rn

h(x, y)f (y)dy

∣∣∣∣2

dx

} 12

≤∫

Rn

{∫Rn

|h(x, y)f (y)|2 dx

} 12

dy

�∫

Rn

|f (y)|{∫

Rn

|h(x, y)|2dx

} 12

dy

≤{∫

Rn

|f (y)|2dy

} 12{∫

Rn

∫Rn

|h(x, y)|2dx dy

} 12

� ‖f ‖L2(Rn)‖h‖L2(R2n). (5.2)

We call Sh the Hilbert–Schmidt operator corresponding to the kernel h.

26 5. Hilbert–Schmidt Operators on L2(Rn)

Let g and h be in L2(R2n). Then we define the function g ◦ h on R2n by

(g ◦ h)(x, y) �∫

Rn

g(x, z)h(z, y)dz, x, y ∈ Rn. (5.3)

Lemma 5.1. Let g and h be in L2(R2n). Then g ◦ h ∈ L2(R2n).

Proof. By (5.3),

|(g ◦ h)(x, y)|2 ≤{∫

Rn

|g(x, z)|2dz

}{∫Rn

|h(z, y)|2dz

}(5.4)

for all x and y in Rn. By (5.4),∫

Rn

∫Rn

|(g ◦ h)(x, y)|2dx dy

≤∫

Rn

∫Rn

{∫Rn

|g(x, z)|2dz

}{∫Rn

|h(z, y)|2dz

}dx dy

�∫

Rn

{∫Rn

|h(z, y)|2dz

}(∫Rn

{∫Rn

|g(x, z)|2dz

}dx

)dy

�∫

Rn

{∫Rn

|h(z, y)|2dz

}dy‖g‖2

L2(R2n) � ‖g‖2L2(R2n)‖h‖2

L2(R2n).�

Theorem 5.2. Let Sg and Sh be the Hilbert–Schmidt operators corresponding tokernels g and h respectively. Then

(i) SgSh � Sg◦h;(ii) S∗

h � Sh∗ ,

where S∗h is the adjoint of Sh and h∗ is the function on R

2n defined by

h∗(x, y) � h(y, x), x, y ∈ Rn. (5.5)

Proof. By (5.3),

((SgSh)f )(x) � (Sg(Shf ))(x)

�∫

Rn

g(x, z)(Shf )(z)dz

�∫

Rn

g(x, z){∫

Rn

h(z, y)f (y)dy

}dz

�∫

Rn

{∫Rn

g(x, z)h(z, y)dz

}f (y)dy

�∫

Rn

(g ◦ h)(x, y)f (y)dy � (Sg◦hf )(x)

for all x in Rn and f in L2(Rn), provided that we can justify the interchange of

the order of integration. To do this, we let

I (x) �∫

Rn

∫Rn

|g(x, z)‖h(z, y)‖f (y)|dy dz, x ∈ Rn, (5.6)

5. Hilbert–Schmidt Operators on L2(Rn) 27

and we only need to show that I (x) < ∞ for almost all x in Rn. But, by Lemma

5.1 and (5.6),

I (x) �∫

Rn

(∫Rn

|g(x, z)‖h(z, y)|dz

)|f (y)|dy

≤ ‖f ‖L2(Rn)

{∫Rn

(∫Rn

|g(x, z)‖h(z, y)|dz

)2

dy

} 12

� ‖f ‖L2(Rn)‖(|g| ◦ |h|)(x, ·)‖L2(Rn) < ∞for almost all x in R

n. Next, let f and g be in L2(Rn). Then

〈Shf, g〉 �∫

Rn

(Shf )(x)g(x)dx

�∫

Rn

{∫Rn

h(x, y)f (y)dy

}g(x)dx

�∫

Rn

f (y)

(∫Rn

h(x, y)g(x)dx

)dy

�∫

Rn

f (x)

(∫Rn

h(y, x)g(y)dy

)dx � 〈f, Sh∗g〉,

provided that we can interchange the order of integration. But,∫Rn

∫Rn

|h(x, y)‖f (y)‖g(x)|dx dy

�∫

Rn

|g(x)|{∫

Rn

|h(x, y)‖f (y)|dy

}dx

≤ ‖g‖L2(Rn)

{∫Rn

(∫Rn

|h(x, y)‖f (y)|dy

)2

dx

}1/2

� ‖g‖L2(Rn)‖S|h||f |‖L2(Rn) < ∞,

and hence the order of integration can be interchanged. �

6The Tensor Product in L2(Rn)

In this chapter, we use some basic facts about tensor products of functions inL2(Rn) to prove that the set of Weyl transforms with symbols in L2(R2n) is equalto the set of Hilbert–Schmidt operators on L2(Rn).

Let f and g be in L2(Rn). Then we define the function f ⊗ g on R2n by

(f ⊗ g)(x, y) � f (x)g(y), x, y ∈ Rn. (6.1)

We call f ⊗ g the tensor product of f and g.

Proposition 6.1. Let f and g be in L2(Rn). Then f ⊗ g ∈ L2(R2n), and

‖f ⊗ g‖L2(R2n) � ‖f ‖L2(Rn)‖g‖L2(Rn). (6.2)

Proof. By (6.1),∫Rn

∫Rn

|(f ⊗ g)(x, y)|2dx dy �∫

Rn

∫Rn

|f (x)g(y)|2dx dy

�(∫

Rn

|f (x)|2dx

)(∫Rn

|g(y)|2dy

)� ‖f ‖2

L2(Rn)‖g‖2L2(Rn). (6.3)

Therefore, by (6.3), f ⊗ g ∈ L2(R2n), and (6.2) follows. �

Proposition 6.2. The set of all finite linear combinations of the formm∑

k�1akfk ⊗ gk, fk, gk ∈ L2(Rn), ak ∈ C,

is dense in L2(R2n).

30 6. The Tensor Product in L2(Rn)

The proof of Proposition 6.2 is left as an exercise.Let f be in L2(R2n). Then∫

Rn

∫Rn

|f (x, y)|2dx dy < ∞. (6.4)

Therefore, by (6.4), f (·, y) is in L2(Rn) for almost all y in Rn. Similarly, f (x, ·) is

in L2(Rn) for almost all x in Rn. We denote by F1f and F2f the Fourier transforms

of f with respect to the “first” and “second” variables, respectively. The inverseFourier transforms of f with respect to the “first” and “second” variables aredenoted by F−1

1 f and F−12 f , respectively.

Proposition 6.3. F1 and F2 are unitary operators on L2(R2n).

Proposition 6.4. Let f and g be in L2(Rn). Then

(i) F1(f ⊗ g) � (Ff ) ⊗ g,(ii) F2(f ⊗ g) � f ⊗ (Fg).

The proofs of Propositions 6.3 and 6.4 are left as exercises.

Remark 6.5. We can obtain analogues of Propositions 6.3 and 6.4 if F1, F2, andF are replaced by F−1

1 , F−12 , and F−1, respectively.

We define the linear operator T : L2(R2n) → L2(R2n) by

(Tf )(x, y) � f(x + y

2, x − y

2

), x, y ∈ R

n, (6.5)

for all f in L2(R2n). We call T the twisting operator on L2(R2n).

Proposition 6.6. The twisting operator T : L2(R2n) → L2(R2n) is unitary, and

(T −1f )(x, y) � f

(x + y

2, x − y

), x, y ∈ R

n, (6.6)

for all f in L2(R2n).

The proof of Proposition 6.6 is also left as an exercise.Now we define the linear operator K : L2(R2n) → L2(R2n) by

(Kf )(x, y) � (T −1F2f )(y, x), x, y ∈ Rn, (6.7)

for all f in L2(R2n).

Proposition 6.7. The linear operator K on L2(R2n) defined by (6.7) has thefollowing properties:

(i) K : L2(R2n) → L2(R2n) is a unitary operator.(ii) K � T −1F−1

2 .(iii) Kf � (Kf )∗, f ∈ L2(R2n).(iv) W (f, g) � K−1(f ⊗ g), f, g ∈ L2(Rn).

6. The Tensor Product in L2(Rn) 31

Proof. Part (i) is obvious. For part (ii), we note that in view of Proposition 6.2,it is enough to prove that

K(f ⊗ g) � T −1F−12 (f ⊗ g), f, g ∈ L2(Rn). (6.8)

But for all f and g in L2(Rn), by (6.7) and Propositions 6.4 and 6.6,

K(f ⊗ g)(x, y) � T −1F2(f ⊗ g)(y, x)� T −1(f ⊗ Fg)(y, x)

� f

(y + x

2

)(Fg)(y − x), x, y ∈ R

n, (6.9)

and

T −1F−12 (f ⊗ g)(x, y) � T −1(f ⊗ F−1g)(x, y)

� f

(x + y

2

)(F−1g)(x − y)

� f

(x + y

2

)(Fg)(y − x), x, y ∈ R

n. (6.10)

Therefore, by (6.9) and (6.10), (6.8) is true, and part (ii) is proved. To prove part (iii),again, in view of Proposition 6.2, it is enough to prove the formula for functionsof the form f ⊗ g, f, g ∈ L2(Rn). But for all x and y in R

n, we get, by (6.7) andPropositions 6.4 and 6.6,

K(f ⊗ g)(x, y) � K(f ⊗ g)(x, y)� T −1F2(f ⊗ g)(y, x)� T −1(f ⊗ Fg)(y, x)

� f

(y + x

2

)(Fg)(y − x), x, y ∈ R

n. (6.11)

Next, for all x and y in Rn, we get, by (5.5), part (ii), and (6.6),

(K(f ⊗ g))∗(x, y) � K(f ⊗ g)(y, x) � f

(x + y

2

)(Fg)(y − x). (6.12)

Thus, by (6.11) and (6.12), the proof of part (iii) is complete. Finally, by (3.12),(6.1), and (6.5),

W (f, g)(x, ξ ) � (2π )−n/2∫

Rn

e−iξ ·pf(x + p

2

)g(x − p

2

)dp

� (2π )−n/2∫

Rn

e−iξ ·pT (f ⊗ g)(x, p)dp

� F2T (f ⊗ g)(x, ξ ), x, ξ ∈ Rn, (6.13)

for all f and g in S(Rn). So, by Corollary 3.4 and (6.13),

W (f, g) � F2T (f ⊗ g), f, g ∈ L2(Rn). (6.14)

32 6. The Tensor Product in L2(Rn)

But by part (ii) and (6.14),

K � T −1F−12 ⇒ K−1 � F2T ⇒ W (f, g) � K−1(f ⊗ g)

for all f and g in L2(Rn), and hence, by Proposition 6.2, part (iv) is proved. �

We can now give the following important property of the Weyl transform.

Theorem 6.8. Let σ ∈ L2(R2n). Then Wσ : L2(Rn) → L2(Rn) is a Hilbert–Schmidt operator with kernel (2π )−n/2Kσ .

Proof. Let f and g be in L2(Rn). Then, by (4.8), Remark 4.5 and parts (i), (iii),and (iv) of Proposition 6.7, we get

〈Wσf, g〉 � (2π )−n/2∫

Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ

� (2π )−n/2〈W (f, g), σ 〉 � (2π )−n/2〈K−1(f ⊗ g), σ 〉� (2π )−n/2〈f ⊗ g, (Kσ )∗〉. (6.15)

Therefore, by (5.1), (5.5), (6.1), and (6.15),

〈Wσf, g〉 � (2π )−n/2∫

Rn

∫Rn

f (x)g(y)(Kσ )(y, x)dx dy

� (2π )−n/2∫

Rn

{∫Rn

(Kσ )(y, x)f (x)dx

}g(y)dy

� 〈Skf, g〉, f, g ∈ L2(Rn),

where k is the function on R2n defined by

k(x, y) � (2π )−n/2(Kσ )(x, y), x, y ∈ Rn,

and the proof is complete. �

We now know that if σ ∈ L2(R2n), then Wσ is a Hilbert–Schmidt operator withkernel (2π )−n/2Kσ . Suppose that A is an arbitrary Hilbert–Schmidt operator. Is itnecessarily of the form Wσ for some σ in L2(R2n)?

The answer to the question is yes. Indeed, if

A � Sh (6.16)

for some h in L2(R2n), then we let σ be the function on R2n given by

σ � (2π )n/2K−1h. (6.17)

Then, by (6.17) and Theorem 6.8,

Wσ � Sk, (6.18)

where

k � (2π )−n/2Kσ � h. (6.19)

Therefore, by (6.16), (6.18), and (6.19),

Wσ � Sh � A.

7H ∗-Algebras and the Weyl Calculus

This chapter is an account, based on the paper [20] by Pool, of a functional calculusfor the Weyl transform with symbol in L2(R2n). In addition to the identificationof Weyl transforms with symbols in L2(R2n) with Hilbert–Schmidt operators onL2(Rn) proved in the preceding chapter, we need the notion of an H ∗-algebrastudied by Ambrose in [1].

Let H be a complex and separable Hilbert space in which the norm and innerproduct are denoted by ‖ ‖ and 〈 , 〉, respectively. Suppose that we are given twooperations in H ,

a, b �→ ab,

a �→ a∗,

satisfying the following conditions.

(i) a(b + c) � ab + ac, (a + b)c � ac + bc,(ii) λ(ab) � (λa)b � a(λb),(iii) a(bc) � (ab)c,(iv) a∗∗ � a,(v) (a + b)∗ � a∗ + b∗,(vi) (ab)∗ � b∗a∗,(vii) (λa)∗ � λa∗,(viii) ‖a∗‖ � ‖a‖,(ix) ‖ab‖ ≤ ‖a‖ ‖b‖,(x) 〈ab, c〉 � 〈b, a∗c〉,for all a, b, and c in H , and λ in C. Then we call H an H ∗-algebra with respect tothe given operations.

34 7. H ∗-Algebras and the Weyl Calculus

It is easy to prove the following proposition.

Proposition 7.1. The Hilbert space L2(R2n) equipped with the two operations

f, g �→ f ◦ g,

f �→ f ∗,

defined by (5.3) and (5.5), respectively, is an H ∗-algebra.

The H ∗-algebra so obtained is denoted by (L2(R2n), ◦, ∗).We denote by HS(L2(Rn)) the set of all Hilbert–Schmidt operators on L2(Rn).

Let A and B be in HS(L2(Rn)). Then we define (A, B)HS by

(A, B)HS �∞∑

k�1〈Aϕk, Bϕk〉, (7.1)

where {ϕk : k � 1, 2, . . .} is an orthonormal basis for L2(Rn). It can be shownthat the definition of (A, B)HS given by (7.1) is independent of the choice of theorthonormal basis {ϕk : k � 1, 2, . . .}. It can also be shown that ( , )HS givenby (7.1) is an inner product in HS(L2(Rn)). It is easy to prove the followingproposition.

Proposition 7.2. The space HS(L2(Rn)) equipped with the inner product ( , )HS

given by (7.1) and the operations of taking the usual products and adjoints is anH ∗-algebra.

The H ∗-algebra in Propostion 7.2 is again denoted by HS(L2(Rn)).The following proposition will be used later.

Proposition 7.3. The H ∗-algebra (L2(R2n), ◦, ∗) is isometrically ∗-isomorphicto the H ∗-algebra HS(L2(Rn)) under the mapping h �→ Sh, where Sh is given by(5.1).

The proofs of Propositions 7.1, 7.2, and 7.3 are left as exercises.Let f and g be in L2(R2n). Then we define the function f × g on R

2n by

f × g � K−1(Kf ◦ Kg), (7.2)

where K : L2(R2n) → L2(R2n) is given by (6.7). The following propositionprovides us with another H ∗-algebra structure on L2(R2n).

Proposition 7.4. The Hilbert space L2(R2n) equipped with the two operations

f, g �→ f × g,

f �→ f ,

is an H ∗-algebra. Moreover, the linear operator K : L2(R2n) → L2(R2n) definedby (6.7) is an isometric ∗-isomorphism of the H ∗-algebra (L2(R2n), ×, −) ontothe H ∗-algebra (L2(R2n), ◦, ∗).

Proposition 7.4 is an immediate consequence of (7.2) and parts (i) and (ii) ofProposition 6.7. The task of providing the details is left as an exercise.

7. H ∗-Algebras and the Weyl Calculus 35

A functional calculus for the Weyl transform is provided by the followingtheorem.

Theorem 7.5. The mapping

L2(R2n) � σ �→ (2π )n2 Wσ ∈ HS(L2(Rn))

is an isometric ∗-isomorphism of the H ∗-algebra (L2(R2n), ×, −) onto the H ∗-algebra HS(L2(R2n)). Consequently, we have

(i) W ∗σ � Wσ ,

(ii) WσWτ � W(2π )−n/2σ×τ ,(iii) Wσ + Wτ � Wσ+τ ,(iv) λWσ � Wλσ ,(v) ‖Wσ‖∗ ≤ (2π )−n/2‖σ‖L2(R2n) � ‖Wσ‖HS ,

where ‖ ‖HS is the norm in HS(L2(Rn)) induced by the inner product ( , )HS givenby (7.1).

Proof. We have

(L2(R2n), ×, −) (L2(R2n), ◦, ∗)

HS(L2(Rn))Fig. 1

�K

�

S

�

where S is the mapping defined by

Sh � Sh, h ∈ L2(R2n),

and Sh is given by (5.1). Then it is clear from Fig. 1 that

L2(R2n) � σ �→ SKσ ∈ HS(L2(Rn))

is an isometric ∗-isomorphism from (L2(R2n), ×, −) onto HS(L2(Rn)). ByTheorem 6.8, we get

SKσ � (2π )n/2Wσ,

and hence complete the proof of the theorem. �

By part (i) of Theorem 7.5, we see that Wσ is self-adjoint if and only if σ

is real-valued. Thus, a good model for the quantization σ (x, D) of the classicalmechanical observable σ (x, ξ ), discussed at the end of Chapter 4, is Wσ insteadof Tσ .

8The Heisenberg Group

Although the formula provided by part (ii) of Theorem 7.5 is very useful for the ac-tual computation of the symbol of the product of two Weyl transforms with symbolsin L2(R2n), it is inadequate in revealing its connections with some fundamentalnotions of quantum mechanics.

In order to obtain a conceptual formula for the symbol of the product of twoWeyl transforms Wσ and Wτ , where σ and τ are in L2(R2n), we use the Heisenberggroup and the twisted convolution studied, respectively, in this and the next chapter.

We identify any point (q, p) in R2n with the point z � q + ip in C

n. We definethe symplectic form [ , ] on C

n by

[z, w] � 2Im(z · w), z, w ∈ Cn, (8.1)

where

z � (z1, z2, . . . , zn),w � (w1, w2, . . . , wn),

and

z · w �n∑

j�1zj wj .

In the following proposition, we give some easy properties of [ , ].

Proposition 8.1. Let ζ, z, and w be in Cn, and let λ and µ be in R. Then

(i) [λζ + µz, w] � λ[ζ, w] + µ[z, w],(ii) [ζ, λz + µw] � λ[ζ, z] + µ[ζ, w],

38 8. The Heisenberg Group

(iii) [z, w] � −[w, z],(iv) [z, z] � 0.

The proof of Proposition 8.1 is easy and left as an exercise.We define the multiplication · on C

n × R by

(z, t) · (w, s) � (z + w, t + s + [z, w]) (8.2)

for all (z, t) and (w, s) in Cn × R, where [ , ] is the symplectic form on C

n definedby (8.1). It is obvious that · is a binary operation on C

n × R.

Proposition 8.2. Cn × R is a group with respect to the multiplication defined by

(8.2).

Proof. Let (ζ, u), (z, t), and (w, s) be in Cn × R. Then, by (8.2),

((ζ, u) · (z, t)) · (w, s) � (ζ + z, u + t + [ζ, z]) · (w, s)� (ζ + z + w, u + t + [ζ, z] + s + [ζ + z, w])� (ζ + z + w, u + t + s + [ζ, z] + [ζ, w] + [z, w]). (8.3)

Also, by (8.2),

(ζ, u) · ((z, t) · (w, s)) � (ζ, u) · (z + w, t + s + [z, w])� (ζ + z + w, u + t + s + [z, w] + [ζ, z + w])� (ζ + z + w, u + t + s + [z, w] + [ζ, z] + [ζ, w]). (8.4)

Therefore, by (8.3) and (8.4), the associative law is valid. Now, for all (z, t) inC

n × R, we have, by (8.2),

(z, t) · (0, 0) � (z, t + [z, 0]) � (z, t),

and

(0, 0) · (z, t) � (z, t + [0, z]) � (z, t).

Therefore, (0, 0) is the identity element in Cn ×R. Finally, for all (z, t) in C

n ×R,we get, by (8.2), parts (ii) and (iv) of Proposition 8.1,

(z, t) · (−z, −t) � (0, 0 + [z, −z]) � (0, 0),

and, by (8.2), parts (i) and (iv) of Proposition 8.1,

(−z, −t) · (z, t) � (0, 0 + [−z, z]) � (0, 0).

Therefore, the inverse of any element (z, t) in Cn × R is equal to (−z, −t). �

The group Cn × R with respect to the multiplication defined by (8.2) is de-

noted by Hn and is called the Heisenberg group. This terminology stems from thefact that the structure equations of the Lie algebra of Hn, i.e., the vector spaceof all left-invariant vector fields on Hn equipped with the bracket operation oftwo left-invariant vector fields given by their commutator, satisfy the canonicalcommutation relations, due to Heisenberg, in quantum mechanics.

8. The Heisenberg Group 39

For each real number λ, we define the mapping Rλ from Hn into the group G

of all unitary operators on L2(Rn) by

(Rλ(z, t)f )(x) � eiλ(q·x+ 12 q·p+ 1

4 t)f (x + p), x ∈ Rn, (8.5)

for all (z, t) in Hn and f in L2(Rn), where z � q + ip. It is easy to see that Rλ(z, t)is indeed a unitary operator on L2(Rn) for all (z, t) in Hn.

Proposition 8.3. Let λ be a fixed real number. Then

(i) Rλ : Hn → G is a group homomorphism,(ii) Rλ(z, t)f → f in L2(Rn) as (z, t) → (0, 0).

Remark 8.4. For any fixed real number λ, Rλ is a unitary representation of Hn onL2(Rn). We call it the Schrodinger representation of Hn on L2(Rn) with parameterλ.

Proof of Proposition 8.3. Let (z, t) � (q + ip, t) and (z′, t ′) � (q ′ + ip′, t ′) bein Hn, and let f be in L2(Rn). Then, by (8.5),

(Rλ(z, t)Rλ(z′, t ′)f )(x) � eiλ(q·x+ 12 q·p+ 1

4 t)(Rλ(z′, t ′)f )(x + p)

� eiλ(q·x+ 12 q·p+ 1

4 t)eiλ(q ′ ·(x+p)+ 12 q ′ ·p′+ 1

4 t ′)f (x + p + p′)(8.6)

for all x in Rn. We also have, by (8.2) and (8.5),

(Rλ((z, t) · (z′, t ′))f )(x) � (Rλ(z + z′, t + t ′ + [z, z′])f )(x)

� eiλ((q+q ′)·x+ 12 (q+q ′)·(p+p′)+ 1

4 (t+t ′+[z,z′]))f (x + p + p′)(8.7)

for all x in Rn. But, by (8.1),

[z, z′] � 2Im(q + ip) · (q ′ − ip′) � 2(q ′ · p − q · p′). (8.8)

Therefore, by (8.7) and (8.8),

(Rλ((z, t) · (z′, t ′))f )(x)

� eiλ((q+q ′)·x+ 12 q·p+ 1

2 q·p′+ 12 q ′ ·p+ 1

2 q ′ ·p′+ 14 (t+t ′)+ 1

2 (q ′ ·p−q·p′))f (x + p + p′) (8.9)

for all x in Rn. Hence, by (8.6) and (8.9),

Rλ(z, t)Rλ(z′, t ′) � Rλ((z, t) · (z′, t ′)),

and part (i) is proved. To prove part (ii), let f be in L2(Rn). Then, for all (z, t) �(q + ip, t) in Hn, we get, by (8.5),

‖Rλ(z, t)f − f ‖2L2(Rn) �

∫Rn

|(Rλ(z, t)f )(x) − f (x)|2dx

�∫

Rn

|eiλ(q·x+ 12 q·p+ 1

4 t)f (x + p) − f (x)|2dx

40 8. The Heisenberg Group

�∫

Rn

|eiλ(q·x+ 12 q·p+ 1

4 t){f (x + p) − f (x)}

+ eiλ(q·x+ 12 q·p+ 1

4 t)f (x) − f (x)|2dx

≤ 2∫

Rn

|f (x + p) − f (x)|2dx

+ 2∫

Rn

|eiλ(q·x+ 12 q·p+ 1

4 t)f (x) − f (x)|2dx. (8.10)

By the L2-continuity of translations,∫Rn

|f (x + p) − f (x)|2dx → 0 (8.11)

as p → 0. For almost all x in Rn,

|eiλ(q·x+ 12 q·p+ 1

4 t)f (x) − f (x)|2 → 0 (8.12)

as (z, t) → (0, 0), and

|eiλ(q·x+ 12 q·p+ 1

4 t)f (x) − f (x)|2 ≤ 4|f (x)|2. (8.13)

Hence, by (8.12), (8.13), and the Lebesgue dominated convergence theorem,∫Rn

|eiλ(q·x+ 12 q·p+ 1

4 t)f (x) − f (x)|2dx → 0 (8.14)

as (z, t) → (0, 0). Therefore, by (8.10), (8.11), and (8.14),

‖Rλ(z, t)f − f ‖L2(Rn) → 0

as (z, t) → (0, 0), and the proof is complete. �

Proposition 8.5. Let λ be a fixed real number. Then the unitary representationRλ of Hn on L2(Rn) is irreducible in the sense that the only closed subspaces ofL2(Rn) that are invariant under all the operators Rλ(z, t), (z, t) ∈ Hn, are {0}and L2(Rn).

To prove Proposition 8.5, it is enough to prove that any bounded linear operatoron L2(Rn) that commutes with Rλ(z, t) for all (z, t) in Hn is a scalar multiple ofthe identity operator on L2(Rn). For then, if M is a closed subspace of L2(Rn) thatis invariant under all the operators Rλ(z, t), (z, t) ∈ Hn, then so is the orthogonalcomplement M⊥ of M . Indeed, if f ∈ M⊥ and g ∈ M , then, using the fact thatRλ is a unitary representation of Hn on L2(Rn) and the fact that (−z, −t) is theinverse of (z, t) for all (z, t) in Hn, we get

〈Rλ(z, t)f, g〉 � 〈f, Rλ(−z, −t)g〉 � 0, (z, t) ∈ Hn.

Now let P be the orthogonal projection of L2(Rn) onto M . Then, for all (z, t) inHn and f in L2(Rn),

PRλ(z, t)f � PRλ(z, t)(f1 + f2), (8.15)

8. The Heisenberg Group 41

where f � f1 + f2, f1 ∈ M , and f2 ∈ M⊥. Thus, using (8.15) and the fact thatM and M⊥ are both invariant under all the operators Rλ(z, t), (z, t) ∈ Hn, we get

PRλ(z, t)f � PRλ(z, t)f1 � Rλ(z, t)f1, (z, t) ∈ Hn. (8.16)

Since

Rλ(z, t)Pf � Rλ(z, t)f1, (z, t) ∈ Hn, (8.17)

it follows from (8.16) and (8.17) that

PRλ(z, t) � Rλ(z, t)P, (z, t) ∈ Hn,

and hence P is a scalar multiple of the identity operator on L2(Rn). Thus, M �L2(Rn) or M � {0}. So, we now let A be a bounded linear operator on L2(Rn)that commutes with all the operators Rλ(z, t), (z, t) ∈ Hn, i.e.,

(Rλ(z, t)Af )(x) � (ARλ(z, t)f )(x), x ∈ Rn, (8.18)

for all (z, t) ∈ Hn. Therefore, by (8.5) and (8.18),

eiλ(q·x+ 12 q·p+ 1

4 t)(Af )(x + p) � (Aeiλ(q·?+ 12 q·p+ 1

4 t)f (? + p))(x) (8.19)

for all x in Rn and (z, t) in Hn. Let q � 0 and t � 0 in (8.19). Then

((TpA)f )(x) � ((ATp)f )(x), x, p ∈ Rn, (8.20)

where, for any measurable function g on Rn, Tpg is the function on R

n defined inProposition 1.9; i.e., A commutes with translations on R

n. Hence, by (8.20), thereexists a function τ in L∞(Rn) such that

(Af ) � τ f , f ∈ L2(Rn). (8.21)

Now, let p � 0 and t � 0 in (8.19). Then

((MλqA)f )(x) � ((AMλq)f )(x), x, q ∈ Rn, (8.22)

where for any measurable function g on Rn, Mλqg is the function on R

n definedin Proposition 1.9 by

(Mλqg)(x) � eiλq·xg(x), x ∈ Rn; (8.23)

i.e., A commutes with modulations on Rn. Thus, by (8.21), (8.22), and (8.23), we

get

((AMλq)f )(x) � (2π )−n/2∫

Rn

eix·ξ τ (ξ )(Mλqf )(ξ )dξ

� (2π )−n/2∫

Rn

eix·ξ τ (ξ )f (ξ − λq)dξ

� (2π )−n/2∫

Rn

eix·(η+λq)τ (η + λq)f (η)dη

� eix·λq(2π )−n/2∫

Rn

eix·ητ (η + λq)f (η)dη, x ∈ Rn, (8.24)

42 8. The Heisenberg Group

for all f in S(Rn). But by (8.21) and (8.23),

((MλqA)f )(x) � eix·λq(2π )−n/2∫

Rn

eix·ητ (η)f (η)dη, x ∈ Rn, (8.25)

for all f in S(Rn). Hence, by (8.24) and (8.25),

τ (η + λq) � τ (η) (8.26)

for almost all η and q in Rn. Therefore, by (8.26), τ is equal to a constant a.e.

on Rn. Consequently, by (8.21), A is a scalar multiple of the identity operator on

L2(Rn), and the proof that Rλ is irreducible is complete.Two irreducible unitary representations R1 and R2 of Hn on L2(Rn) are said to

be unitarily equivalent if there is a unitary operator U on L2(Rn) such that

UR1(z, t) � R2(z, t)U, (z, t) ∈ Hn. (8.27)

Proposition 8.6. Two unitary representations Rλ and Rµ of Hn on L2(Rn) areunitarily equivalent if and only if λ � µ.

Proof. It is enough to prove that if Rλ and Rµ are unitarily equivalent, thenλ � µ. By (8.27), there is a unitary operator on L2(Rn) such that

URλ(z, t) � Rµ(z, t)U, (z, t) ∈ Hn. (8.28)

Let z � 0 in (8.28). Then

URλ(0, t) � Rµ(0, t)U, t ∈ R. (8.29)

So, by (8.5) and (8.29),

Ue14 iλt � e

14 iµtU, t ∈ R,

and hence λ � µ. �

Remark 8.7. By a theorem of Stone and von Neumann, we know that up to unitaryequivalence, the only irreducible and unitary representations of Hn on L2(Rn) aregiven by {Rλ : −∞ < λ < ∞} and {Rα,β : α, β ∈ R

n}, where Rλ is given by(8.5) for all λ in R and

(Rα,β(z, t)f )(x) � ei(α·q+β·p)f (x)

for all α and β in Rn, x in R

n, f ∈ L2(Rn), and (z, t) � (q + ip, t) in Hn. Thus,the only nontrivial irreducible and unitary representations of Hn on L2(Rn) aregiven by {Rλ : −∞ < λ < ∞}.Remark 8.8. It is interesting to note that the unitary operator ρ(q, p), defined by(2.1) for all q and p in R

n, is equal to R1(z, 0), where z � q + ip. This observationwill be useful to us in the next chapter.

9The Twisted Convolution

The aim of this chapter is to express the symbol of the product of two Weyltransforms with symbols in L2(R2n) in terms of a twisted convolution, which wenow define.

Let λ be a fixed real number. Then we define the twisted convolution f ∗λ g oftwo measurable functions f and g on C

n by

(f ∗λ g)(z) �∫

Cn

f (z − w)g(w)eiλ[z,w]dw, z ∈ Cn, (9.1)

where [z, w] is the symplectic form of z and w defined by (8.1), dw is the Lebesguemeasure on C

n, and provided that the integral exists.

Proposition 9.1. Let f and g be measurable functions on Cn such that (f ∗λ g)(z)

exists at the point z in Cn. Then (g ∗−λ f )(z) exists, and

(f ∗λ g)(z) � (g ∗−λ f )(z).

Proof. In (9.1), we change the variable of integration from w to ζ by w � z−ζ .Then we get

(f ∗λ g)(z) �∫

Cn

g(z − ζ )f (ζ )eiλ[z,z−ζ ]dζ. (9.2)

By (9.1), (9.2), and parts (i) and (iv) of Proposition (8.1), we get

(f ∗λ g)(z) �∫

Cn

g(z − ζ )f (ζ )e−iλ[z,ζ ]dζ

� (g ∗−λ f )(z). �

44 9. The Twisted Convolution

Remark 9.2. It is clear from Proposition 9.1 that the twisted convolution is, ingeneral, noncommutative.

We can now give a formula for the product WσWτ of two Weyl transforms Wσ

and Wτ in terms of a twisted convolution of σ and τ . We begin with the case whenboth σ and τ are in S(R2n). To do this, let ϕ and ψ be in S(Rn). Then, by (2.3),Theorem 3.1, (3.12), Theorem 4.3, and the adjoint formula in the theory of theFourier transform,

〈Wσϕ, ψ〉 � (2π )−n

∫Rn

∫Rn

σ (q, p)〈ρ(q, p)ϕ, ψ〉dq dp. (9.3)

Therefore, by (9.3) and Fubini’s theorem,

〈Wσϕ, ψ〉 � (2π )−n

∫Rn

∫Rn

σ (q, p){∫

Rn

(ρ(q, p)ϕ)(x)ψ(x)dx

}dq dp

� (2π )−n

∫Rn

ψ(x){∫

Cn

σ (z)(ρ(z)ϕ)(x)dz

}dx,

and hence

(Wσϕ)(x) � (2π )−n

∫Cn

σ (z)(ρ(z)ϕ)(x)dz, x ∈ Rn, (9.4)

for all ϕ in S(Rn). But by (2.1) and (9.4),

(ρ(z)(Wτϕ))(x) � ei(q·x+ 12 q·p)(Wτϕ)(x + p)

� ei(q·x+ 12 q·p)(2π )−n

∫Cn

τ (w)(ρ(w)ϕ)(x + p)dw

� (2π )−n

∫Cn

τ (w)(ρ(z)ρ(w)ϕ)(x)dw, x ∈ Rn, (9.5)

for all ϕ in S(Rn). Thus, by (9.4) and (9.5),

(WσWτϕ)(x) � (2π )−2n

∫Cn

∫Cn

σ (z)τ (w)(ρ(z)ρ(w)ϕ)(x)dz dw, x ∈ Rn.

(9.6)Now, by (2.1), (8.2), (8.5), the fact that R1 is a unitary representation of Hn onL2(Rn), and Remark 8.8, we get

ρ(z)ρ(w) � R1(z, 0)R1(w, 0) � R1(z + w, [z, w])� ρ(z + w)e

14 i[z,w], z, w ∈ C

n. (9.7)

So, by (9.6) and (9.7),

(WσWτϕ)(x) � (2π )−2n

∫Cn

∫Cn

σ (z)τ (w)(ρ(z+w)ϕ)(x)e14 i[z,w]dz dw, x ∈ R

n,

(9.8)for all ϕ in S(Rn). Now, in (9.8), we change the variable z to ζ by z � ζ − w.Then, by (9.1) and parts (i) and (iv) of Proposition (8.1), we get

(WσWτϕ)(x) � (2π )−2n

∫Cn

∫Cn

σ (ζ − w)τ (w)(ρ(ζ )ϕ)(x)e14 i[ζ−w,w]dζ dw

9. The Twisted Convolution 45

� (2π )−2n

∫Cn

∫Cn

σ (ζ − w)τ (w)(ρ(ζ )ϕ)(x)e14 i[ζ,w]dζ dw

� (2π )−2n

∫Cn

{∫Cn

σ (ζ − w)τ (w)e14 i[ζ,w]dw

}(ρ(ζ )ϕ)(x)dζ

� (2π )−2n

∫Cn

(σ ∗ 14τ )(ζ )(ρ(ζ )ϕ)(x)dζ, x ∈ R

n. (9.9)

Hence, by (9.4) and (9.9),

WσWτ � Wω, (9.10)

where

ω � (2π )−n(σ ∗ 14τ ). (9.11)

Theorem 9.3. Let σ and τ be in L2(R2n). Then WσWτ � Wω, where

ω � (2π )−n(σ ∗ 14τ ).

Proof. Let {σk}∞k�1 and {τk}∞k�1 be sequences of functions in S(R2n) such that

σk → σ (9.12)

and

τk → τ (9.13)

in L2(R2n) as k → ∞. Then, by (9.10) and (9.11),

WσkWτk

� Wωk, (9.14)

where

ωk � (2π )−n(σk ∗ 14τk) (9.15)

for k � 1, 2, . . . . Now, by Remark 4.5, (4.9), (9.12), and (9.13),

Wσk→ Wσ (9.16)

and

Wτk→ Wτ (9.17)

in B(L2(Rn)) as k → ∞. So, by (9.16) and (9.17),

Wωk� Wσk

Wτk→ WσWτ (9.18)

in B(L2(Rn)) as k → ∞. By (9.1), (9.12), and (9.13),

(σk ∗ 14τk)(z) �

∫Cn

σk(z − w)τk(w)e14 i[z,w]dw

→∫

Cn

σ (z − w)τ (w)e14 i[z,w]dw

� (σ ∗ 14τ )(z) (9.19)

46 9. The Twisted Convolution

for almost all z in Cn as k → ∞. On the other hand, by (9.12)–(9.15) and part (v)

of Theorem 7.5, we get

‖σk ∗ 14τk − σj ∗ 1

4τj‖L2(R2n)

� ‖σk ∗ 14τk − σk ∗ 1

4τj + σk ∗ 1

4τj − σj ∗ 1

4τj‖L2(R2n)

≤ ‖σk ∗ 14

(τk − τj )‖L2(R2n) + ‖(σk − σj ) ∗ 14τj‖L2(R2n)

� (2π )3n2 ‖Wσk

Wτk−τj‖HS + (2π )

3n2 ‖Wσk−σj

Wτj‖HS

≤ (2π )3n2 ‖Wσk

‖HS‖Wτk−τj‖HS + (2π )

3n2 ‖Wσk−σj

‖HS‖Wτj‖HS

� (2π )n2 (‖σk‖L2(R2n)‖τk − τj‖L2(R2n) + ‖σk − σj‖L2(R2n)‖τj‖L2(R2n))

→ 0

as k, j → ∞. Hence, by the Plancherel theorem, there exists a function ω suchthat

σk ∗ 14τk → (2π )nω (9.20)

in L2(R2n) as k → ∞. Therefore, by (9.20), there exists a subsequence {σk′ ∗ 14

τk′ }∞k′�1 of {σk ∗ 14τk}∞k�1 such that

σk′ ∗ 14τk′ → (2π )nω (9.21)

a.e. on R2n as k′ → ∞. Thus, by (9.19) and (9.21),

(2π )nω � σ ∗ 14τ (9.22)

a.e. on R2n. By (9.15), (9.20), and the Plancherel theorem,

ωk → ω (9.23)

in L2(R2n) as k → ∞. Thus, by Remark 4.5, (4.9), and (9.23),

Wωk→ Wω (9.24)

in B(L2(Rn)) as k → ∞. So, by (9.18), (9.22), and (9.24), the proof of the theoremis complete. �

10The Riesz–Thorin Theorem

In Chapters 11–14, we shall define the Weyl transform Wσ corresponding to atempered distribution σ on R

2n and decide whether or not the resulting Weyltransform Wσ , for σ in Lr (R2n), 1 ≤ r ≤ ∞, is a bounded, or even compact,linear operator from L2(Rn) into L2(Rn). We need an interpolation theorem ofRiesz and Thorin for this task and also for the study of localization operators inChapter 15. The proof of the interpolation theorem is based on a fact in complexanalysis that we now present.

Theorem 10.1. (The Three Lines Theorem) Let f be a continuous and boundedfunction on the strip

B � {z ∈ C : α ≤ Rez ≤ β},where α and β are real numbers such that α < β. Suppose that f is analytic onthe interior of B and that there exist constants M1 and M2 such that

|f (α + iy)| ≤ M1, −∞ < y < ∞, (10.1)

and

|f (β + iy)| ≤ M2, −∞ < y < ∞. (10.2)

Then, for α ≤ x ≤ β, we have

|f (x + iy)| ≤ ML(x)1 M

1−L(x)2 , −∞ < y < ∞, (10.3)

where

L(x) � β − x

β − α, α ≤ x ≤ β. (10.4)

48 10. The Riesz–Thorin Theorem

Proof. Let F be the function on B defined by

F (z) � f (z)M

L(z)1 M

1−L(z)2

, z ∈ B. (10.5)

We first suppose that

f (x + iy) → 0 (10.6)

uniformly with respect to x on [α, β] as |y| → ∞. Note that by (10.4) and (10.5),

|F (z)| � |f (z)|M

L(x)1 M

1−L(x)2

≤ |f (z)|M

, z ∈ B, (10.7)

where M � min(M1, M2). Hence, by (10.6) and (10.7),

F (x + iy) → 0 (10.8)

uniformly with respect to x on [α, β] as |y| → ∞. Thus, by (10.8), there exists apositive number R such that

|F (x + iy)| ≤ 1 (10.9)

for α ≤ x ≤ β and |y| ≥ R. On the rectangle defined by α ≤ x ≤ β and |y| ≤ R,(10.9) is valid for |y| � R, and by (10.1), (10.2), and (10.5),

|F (α + iy)| ≤ 1, −∞ < y < ∞, (10.10)

and

|F (β + iy)| ≤ 1, −∞ < y < ∞. (10.11)

Hence, by (10.9), (10.10), (10.11), and the maximum modulus principle,

|F (z)| ≤ 1, z ∈ B. (10.12)

So, by (10.4), (10.5), and (10.12), we can conclude that (10.3) is true for α ≤ x ≤ β

and −∞ < y < ∞. If f (x + iy) does not go to zero uniformly with respect to x

on [α, β] as |y| → ∞, then for every positive integer k, we define the function fk

on B by

fk(z) � ez2k f (z), z ∈ B. (10.13)

Since f is a bounded function on B, and for k � 1, 2, . . . ,

|e z2k | � e

x2−y2k ≤ e

γ 2−y2k , −∞ < y < ∞, (10.14)

whereγ 2 � max(α2, β2), it follows from (10.13) and (10.14) that for k � 1, 2, . . . ,

|fk(x + iy)| → 0 (10.15)

uniformly with respect to x on [α, β] as |y| → ∞. Furthermore, by (10.1), (10.2),(10.13), and (10.14),

|fk(α + iy)| ≤ eα2k M1, −∞ < y < ∞, (10.16)

10. The Riesz–Thorin Theorem 49

and

|fk(β + iy)| ≤ eβ2k M2, −∞ < y < ∞. (10.17)

Thus, by (10.15), (10.16), (10.17), and what we have just shown, we get, fork � 1, 2, . . . ,

|e z2k f (x + iy)| ≤ e

α2k

L(x)ML(x)1 e

β2k

(1−L(x))M1−L(x)2 (10.18)

for α ≤ x ≤ β and −∞ < y < ∞. Therefore, by letting k → ∞ in (10.18), theproof of the theorem is complete. �

We can now formulate and prove the Riesz–Thorin theorem.

Theorem 10.2. (The Riesz–Thorin Theorem) Let (X, µ) be a measure space and(Y, ν) a σ -finite measure space. Let T be a linear transformation with domain Dconsisting of all µ-simple functions f on X such that

µ{s ∈ X : f (s) � 0} < ∞and such that the range of T is contained in the set of all ν-measurable functionson Y . Suppose that α1, α2, β1, and β2 are real numbers in [0, 1] and there existpositive constants M1 and M2 such that

‖Tf ‖L

1βj (Y )

≤ Mj‖f ‖L

1αj (X)

, f ∈ D, j � 1, 2. (10.19)

Then, for 0 < θ < 1,

α � (1 − θ )α1 + θα2, (10.20)

and

β � (1 − θ )β1 + θβ2, (10.21)

we have

‖Tf ‖L

1β (Y )

≤ M1−θ1 Mθ

2 ‖f ‖L

1α (X), f ∈ D.

Proof. For any complex number z, we let

α(z) � (1 − z)α1 + zα2 (10.22)

and

β(z) � (1 − z)β1 + zβ2. (10.23)

Let f be a µ-simple function on X and g a ν-simple function on Y such that

‖f ‖L

1α (X) � 1 (10.24)

and

‖g‖L

( 1β

)′ (Y )� 1, (10.25)

50 10. The Riesz–Thorin Theorem

where(

1β

)′is the conjugate index of 1

β. We first suppose that 1 ≤ 1

α< ∞ and

1 < 1β

≤ ∞. Let

f (s) � |f (s)|eiu(s), s ∈ X, (10.26)

and

g(t) � |g(t)|eiv(t), t ∈ Y. (10.27)

For any z in C, we write

F (s, z) �{

|f (s)| α(z)α eiu(s), f (s) � 0,

0, f (s) � 0,(10.28)

and

G(t, z) �{

|g(t)| 1−β(z)1−β eiv(t), g(t) � 0,

0, g(t) � 0,(10.29)

for all s in X and t in Y . Let

f (s) �m∑

j�1cjχSj

(s), s ∈ X, (10.30)

and

g(t) �n∑

k�1dkχTk

(t), t ∈ Y, (10.31)

where c1, c2, . . . , cm are distinct nonzero complex numbers; d1, d2, . . . , dn are dis-tinct nonzero complex numbers; S1, S2, . . . , Sm are pairwise disjoint measurablesubsets of X; T1, T2, . . . , Tn are pairwise disjoint measurable subsets of Y ; χ

Sjis

the characteristic function on Sj , j � 1, 2, . . . , m; χTk

is the characteristic functionon Tk , k � 1, 2, . . . , n. Let

cj � |cj |eiuj , j � 1, 2, . . . , m, (10.32)

and

dk � |dk|eivk , k � 1, 2, . . . , n. (10.33)

Then, by (10.26), (10.28), (10.30), and (10.32),

F (s, z) �m∑

j�1|cj |

α(z)α eiuj χ

Sj(s), s ∈ X, (10.34)

and, by (10.27), (10.29), (10.31), and (10.33),

G(t, z) �n∑

k�1|dk|

1−β(z)1−β eivkχ

Tk(t), t ∈ Y, (10.35)

for all z in C. Let � be the function on C defined by

�(z) �∫

Y

(T F (·, z))(t)G(t, z)dν(t), z ∈ C. (10.36)

10. The Riesz–Thorin Theorem 51

Then, by (10.34), (10.35), and (10.36),

�(z) �m∑

j�1

n∑k�1

ei(uj +vk )|cj |α(z)α |dk|

1−β(z)1−β

∫Y

(T χSj

)(t)χTk

(t)dν(t) (10.37)

for all z in C. Thus, � is entire. For 0 ≤ x ≤ 1 and −∞ < y < ∞, we get, by(10.22), (10.23), and (10.37),

|�(x + iy)| ≤m∑

j�1

n∑k�1

|cj |α(x)α |dk|

1−β(x)1−β

∣∣∣∣∫

Y

(T χSj

)(t)χTk

(t)dν(t)∣∣∣∣ . (10.38)

Hence, by (10.38), � is a bounded function on the strip {z ∈ C : 0 ≤ Rez ≤ 1}.For Rez � 0, we get by (10.20)–(10.23),

Reα(z) � α1 (10.39)

and

Reβ(z) � β1. (10.40)

Thus, by (10.28) and (10.39),

|F (s, iy)| �{

|f (s)| α1α , f (s) � 0,

0, f (s) � 0,(10.41)

and by (10.29) and (10.41),

|G(t, iy)| �{

|g(t)| 1−β11−β , g(t) � 0,

0, g(t) � 0,(10.42)

for all s in X and t in Y . Therefore, by (10.24) and (10.41),

‖F (·, iy)‖L

1α1 (X)

�{∫

X

|f (s)| 1α dµ(s)

}α1

� ‖f ‖α1α

L1α (X)

� 1, (10.43)

and by (10.25) and (10.42),

‖G(·, iy)‖L

( 1β1

)′(Y )

�{∫

Y

|g(t)| 11−β dν(t)

}1−β1

� ‖g‖1−β11−β

L( 1β

)′ (Y )� 1 (10.44)

for −∞ < y < ∞. So, by (10.19), (10.36), (10.43), and (10.44),

|�(iy)| �∣∣∣∣∫

Y

(T F (·, iy))(t)G(t, iy)dν(t)∣∣∣∣

≤ ‖T F (·, iy)‖L

1β1 (Y )

‖G(·, iy)‖L

( 1β1

)′(Y )

≤ M1 (10.45)

for −∞ < y < ∞. Similarly,

|�(1 + iy)| ≤ M2, −∞ < y < ∞. (10.46)

So, by (10.45), (10.46), Theorem 10.1, and the fact that � is entire,

|�(x + iy)| ≤ M1−x1 Mx

2 , 0 ≤ x ≤ 1, −∞ < y < ∞. (10.47)

52 10. The Riesz–Thorin Theorem

Thus, by (10.47),

|�(θ )| ≤ M1−θ1 Mθ

2 . (10.48)

Next, note that by (10.20), (10.22), (10.30), (10.32), and (10.34),

F (s, θ) � f (s), s ∈ X, (10.49)

and, by (10.21), (10.23), (10.31), (10.33), and (10.35),

G(t, θ) � g(t), t ∈ Y. (10.50)

Therefore, by (10.36) and (10.48)–(10.50),

|�(θ )| �∣∣∣∣∫

Y

(Tf )(t)g(t)dν(t)∣∣∣∣ ≤ M1−θ

1 Mθ2 . (10.51)

So, by (10.51),

‖Tf ‖L

1β (Y )

� sup∣∣∣∣∫

Y

(Tf )(t)g(t)dν(t)∣∣∣∣ ≤ M1−θ

1 Mθ2 ,

where the supremum is taken over all ν-simple functions g onY with ‖g‖L

( 1β

)′(Y )� 1, and this completes the proof of the theorem for 1 ≤ 1

α< ∞ and 1 < 1

β≤ ∞.

Now we have to look at case 1: α � 0 and β � 1; case 2: α � 0 and 0 < β < 1;case 3: α � 0 and β � 0; case 4: α � 1 and β � 1; and case 5: 0 < α < 1 andβ � 1. All cases are trivial except cases 2 and 5. For case 2, we take α(z)

α� 1

for all z in C, and the preceding proof goes through without any change. For case5, we take 1−β(z)

1−β� 1 for all z in C; then the preceding proof can be used, until

we get (10.51) for all ν-simple functions g on Y with ‖g‖L∞(Y ) � 1. Thus, for allν-simple functions g on Y ,∣∣∣∣

∫Y

(Tf )(t)g(t)dν(t)∣∣∣∣ ≤ M1−θ

1 Mθ2 ‖g‖L∞(Y ). (10.52)

Let S � {t ∈ Y : (Tf )(t) � 0} and let h be the function on Y defined by

h(t) �{ |(Tf )(t)|

(Tf )(t) , t ∈ S,

0, t ∈ S.(10.53)

Now, let {gl}∞l�1 be a sequence of ν-simple functions on Y such that

‖gl‖L∞(Y ) ≤ 1 (10.54)

for l � 1, 2, . . . , and gl → h a.e. on Y as l → ∞. Therefore,

(Tf )gl → (Tf )h (10.55)

a.e. on Y as l → ∞. By (10.53) and (10.54),

|(Tf )(t)gl(t)| ≤ |(Tf )(t)|, t ∈ Y. (10.56)

Since β � 1, it follows from (10.19) and (10.21) that∫Y

|(Tf )(t)|dν(t) < ∞. (10.57)

10. The Riesz–Thorin Theorem 53

Thus, using (10.53), (10.55)–(10.57), and the Lebesgue dominated convergencetheorem, we get∫

Y

|(Tf )(t)|dν(t) �∣∣∣∣∫

Y

(Tf )(t)h(t)dν(t)∣∣∣∣ � lim

l→∞

∣∣∣∣∫

Y

(Tf )(t)gl(t)dν(t)∣∣∣∣ .

(10.58)So, by (10.52), (10.54), and (10.58),∣∣∣∣

∫Y

(Tf )(t)dν(t)∣∣∣∣ ≤ M1−θ

1 Mθ2 ,

and the proof is therefore complete. �

A standard and beautiful application of the Riesz–Thorin theorem is theHausdorff–Young inequality for the Fourier transform of a function in Lr (Rn),1 ≤ r ≤ 2, which we state, prove, and use in Chapter 14. The Hausdorff–Younginequality is also attributed to Titchmarsh. See the paper [30] and the book [31]by Titchmarsh in this connection.

11Weyl Transforms with Symbols inLr(R2n), 1 ≤ r ≤ 2

The following result is an extension of Theorem 4.4 and is essential for the proofthat a Weyl transform with symbol in Lr (R2n), 1 ≤ r ≤ 2, is compact.

Theorem 11.1. For 1 ≤ r ≤ 2, there exists a unique bounded linear operatorQ : Lr (R2n) → B(L2(Rn)) such that (4.8) is valid for all σ in Lr (R2n) and f andg in S(Rn). Furthermore,

‖Qσ‖∗ ≤ 2−n

(2π

) nr

‖σ‖Lr (R2n), σ ∈ Lr (R2n). (11.1)

To prove Theorem 11.1, we begin by noting that for r � 2, Theorem 11.1 isjust Theorem 4.4. For r � 1, let σ ∈ S(R2n). Then, for any f in S(Rn), we define(Qσ )f by (4.10). Then, for all f and g in S(Rn), we get, by (4.10) and Theorem4.3,

|〈(Qσ )f, g〉| � (2π )−n/2∣∣∣∣∫

Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ

∣∣∣∣≤ (2π )−n/2‖σ‖L1(R2n)‖W (f, g)‖L∞(R2n). (11.2)

But by (3.12),

|W (f, g)(x, ξ )| � (2π )−n/2∣∣∣∣∫

Rn

e−iξ ·pf(x + p

2

)g(x − p

2

)dp

∣∣∣∣≤ (2π )−n/2

{∫Rn

∣∣∣f (x + p

2

)∣∣∣2 dp

} 12{∫

Rn

∣∣∣g (x − p

2

)∣∣∣2 dp

} 12

56 11. Weyl Transforms with Symbols in Lr (R2n), 1 ≤ r ≤ 2

� (2π )−n/22n

{∫Rn

|f (p)|2dp} 1

2{∫

Rn

|g(p)|2dp} 1

2

�(

2π

)n/2

‖f ‖L2(Rn)‖g‖L2(Rn), x, ξ ∈ Rn. (11.3)

Thus, by (11.2) and (11.3),

|〈(Qσ )f, g〉| ≤ π−n‖σ‖L1(R2n)‖f ‖L2(Rn)‖g‖L2(Rn) (11.4)

for all σ in S(R2n), and f and g in S(Rn). Hence, by (11.4),

‖Qσ‖∗ ≤ π−n‖σ‖L1(R2n), σ ∈ S(R2n). (11.5)

Let σ ∈ L1(R2n). Then we let {σk}∞k�1 be a sequence of functions in S(R2n) suchthat σk → σ in L1(R2n) as k → ∞. So, by (11.5),

‖Qσk − Qσl‖∗ ≤ π−n‖σk − σl‖L1(R2n) → 0

as k, l → ∞. Therefore, {Qσk}∞k�1 is a Cauchy sequence in B(L2(Rn)), and we candefine Qσ to be the limit of the sequence {Qσk}∞k�1 in B(L2(Rn)). This definitionis independent of the choice of the sequence {σk}∞k�1 in S(R2n). Indeed, let {τk}∞k�1be another sequence in S(R2n) such that τk → σ in L1(R2n) as k → ∞. Then, by(11.5),

‖Qσk − Qτk‖∗ ≤ π−n‖σk − τk‖L1(R2n) → 0

as k → ∞. Therefore, the limits in B(L2(Rn)) of the sequences {Qσk}∞k�1 and{Qτk}∞k�1 are equal. Let σ ∈ L1(R2n). Then we let {σk}∞k�1 be a sequence offunctions in S(R2n) such that σk → σ in L1(R2n) as k → ∞. Thus, by (11.5),

‖Qσ‖∗ � limk→∞

‖Qσk‖∗ ≤ limk→∞

π−n‖σk‖L1(R2n) � π−n‖σ‖L1(R2n). (11.6)

Furthermore, if f and g are in S(Rn), then by (4.10) and Theorem 4.3,

〈(Qσ )f, g〉 � limk→∞

〈(Qσk), g〉

� limk→∞

(2π )−n/2∫

Rn

∫Rn

σk(x, ξ )W (f, g)(x, ξ )dx dξ

� (2π )−n/2∫

Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ.

To prove the theorem for 1 ≤ r ≤ 2, we fix a function f in S(Rn). Let T be thelinear transformation with domain D consisting of all simple functions σ on R

2n

with the property that the Lebesgue measure of the set

{(x, ξ ) ∈ R2n : σ (x, ξ ) � 0}

is finite, and defined by

T σ � (Qσ )f, σ ∈ D. (11.7)

Then, by (4.9) and (11.7),

‖T σ‖L2(Rn) � ‖(Qσ )f ‖L2(Rn) ≤ (2π )−n/2‖f ‖L2(Rn)‖σ‖L2(R2n), (11.8)

11. Weyl Transforms with Symbols in Lr (R2n), 1 ≤ r ≤ 2 57

and by (11.6) and (11.7),

‖T σ‖L2(Rn) � ‖(Qσ )f ‖L2(Rn) ≤ π−n‖f ‖L2(Rn)‖σ‖L1(R2n) (11.9)

for all σ in D. So, by (11.8), (11.9), and the Riesz–Thorin theorem,

‖T σ‖L2(Rn) � ‖(Qσ )f ‖L2(Rn)

≤ {(2π )−n/2‖f ‖L2(Rn)}2− 2r {π−n‖f ‖L2(Rn)} 2

r−1‖σ‖Lr (R2n)

� 2−n

(2π

) nr

‖f ‖L2(Rn)‖σ‖Lr (R2n),

and hence

‖Qσ‖∗ ≤ 2−n

(2π

) nr

‖σ‖Lr (R2n), σ ∈ D. (11.10)

Now, let σ ∈ Lr (R2n). Then we pick a sequence {sk}∞k�1 of functions in D such thatsk → σ in Lr (R2n) as k → ∞. Then, by (11.10), {Qsk}∞k�1 is a Cauchy sequencein B(L2(Rn)), and we denote the limit in B(L2(Rn)) by Qσ . The proof that thelimit is independent of the choice of the sequence {sk}∞k�1 and the proof that Qσ

satisfies the conclusions of the theorem are the same as before. It is also obviousthat Q : Lr (R2n) → B(L2(Rn)) is the only bounded linear operator satisfying theconclusions of the theorem.

Remark 11.2. It is natural to denote Qσ by Wσ for any σ in Lr (R2n), 1 ≤ r ≤ 2.

Theorem 11.3. Let σ ∈ Lr (R2n), 1 ≤ r ≤ 2. Then Wσ : L2(Rn) → L2(Rn) is acompact operator.

To prove Theorem 11.3, we use the following lemma.

Lemma 11.4. Let Sh be the Hilbert–Schmidt operator on L2(Rn) correspondingto the kernel h in L2(R2n). Then Sh is compact.

Proof. By Proposition 6.2, we can get a sequence {tk}∞k�1 of finite linearcombinations of tensor products of functions in L2(Rn) such that

tk → h (11.11)

in L2(R2n) as k → ∞. By (5.1), (5.2), and (11.11),

‖Stk − Sh‖∗ ≤ ‖tk − h‖L2(R2n) → 0 (11.12)

as k → ∞. Hence, by (11.12), Sh is compact if we can prove that each Stk iscompact. To prove that each Stk is compact, it is enough to prove that Sa⊗b iscompact for all a and b in L2(Rn). But if a and b are in L2(Rn), then

(Sa⊗bf )(x) �∫

Rn

a(x)b(y)f (y)dy

� 〈f, b〉a(x), x ∈ Rn,

for all f in L2(Rn). So, Sa⊗b is of finite rank and hence compact. �

58 11. Weyl Transforms with Symbols in Lr (R2n), 1 ≤ r ≤ 2

Proof of Theorem 11.3. Let σ ∈ Lr (R2n), 1 ≤ r ≤ 2. Then we pick a sequence{σk}∞k�1 of functions in S(R2n) such that σk → σ in Lr (R2n) as k → ∞. ByTheorem 6.8, Wσk

: L2(Rn) → L2(Rn) is a Hilbert–Schmidt operator, and hence,by Lemma 11.4, compact for k � 1, 2, . . . . By (11.1) and Remark 11.2, Wσ is thelimit in B(L2(Rn)) of the sequence {Wσk

}∞k�1. Thus, Wσ : L2(Rn) → L2(Rn) iscompact. �

12Weyl Transforms with Symbols inL∞(R2n)

We begin this chapter by defining the notion of a Weyl transform with symbol inS ′(R2n). To this end, let σ be in S ′(R2n). Then, for all f in S(Rn), we define thelinear functional Wσf on S(Rn) by

(Wσf )(g) � (2π )−n/2σ (W (f, g)), g ∈ S(Rn), (12.1)

where W (f, g) is the Wigner transform of f and g defined by (3.12).

Theorem 12.1. For all σ in S ′(R2n) and f in S(Rn), Wσf is a tempereddistribution on R

n.

To prove Theorem 12.1, we need a lemma.

Lemma 12.2. Let f be in S(Rn) and {gk}∞k�1 a sequence of functions in S(Rn)such that gk → 0 in S(Rn) as k → ∞. Then, for all multi-indices α, β, γ , and δ,∫

Rn

|x||α||p||β|∣∣∣(∂γ f )

(x + p

2

)∣∣∣ |(∂δgk)(x − p

2)|dp → 0

uniformly with respect to x on Rn as k → ∞.

Proof. For k � 1, 2, . . . , we define the function Ik on Rn by

Ik(x) �∫

Rn

|x||α||p||β|∣∣∣(∂γ f )

(x + p

2

)∣∣∣ ∣∣∣(∂δgk)(x − p

2

)∣∣∣ dp, x ∈ Rn.

(12.2)

60 12. Weyl Transforms with Symbols in L∞(R2n)

Then, by (12.2),

Ik(x) ≤ 2|β|∫

Rn

(∣∣∣x + p

2

∣∣∣|α|+∣∣∣x − p

2

∣∣∣|α|)(∣∣∣x + p

2

∣∣∣|β|+∣∣∣x − p

2

∣∣∣|β|)

×∣∣∣(∂γ f )

(x + p

2

)∣∣∣ ∣∣∣(∂δgk)(x − p

2

)∣∣∣ dp (12.3)

for all x in Rn. Now,∫

Rn

∣∣∣x ± p

2

∣∣∣|α+β| ∣∣∣(∂γ f )(x + p

2

)∣∣∣ ∣∣∣(∂δgk)(x − p

2

)∣∣∣ dp → 0 (12.4)

and∫Rn

∣∣∣x ± p

2

∣∣∣|α| ∣∣∣x ∓ p

2

∣∣∣|β| ∣∣∣(∂γ f )(x + p

2

)∣∣∣ ∣∣∣(∂δgk)(x − p

2

)∣∣∣ dp → 0 (12.5)

uniformly with respect to x on Rn as k → ∞. Thus, by (12.3), (12.4), and (12.5),

Ik → 0 uniformly on Rn as k → ∞. �

Proof of Theorem 12.1. Let {gk}∞k�1 be a sequence of functions in S(Rn) suchthat gk → 0 in S(Rn) as k → ∞. Let α, β, γ , and δ be multi-indices. Then, forall x and ξ in R

n, we get, by (3.12),

xαξβ(∂γx ∂δ

ξ W (f, gk))(x, ξ )

� xαξβ∂γx ∂δ

ξ (2π )−n/2∫

Rn

e−ip·ξ f(x + p

2

)gk

(x − p

2

)dp

� xαξβ(2π )−n/2∫

Rn

(−ip)δe−ip·ξ ∑γ ′≤γ

(γ

γ ′

)(∂γ ′

f )(x + p

2

)(∂γ−γ ′

gk)

×(x − p

2

)dp

�∑γ ′≤γ

(γ

γ ′

)(2π )−n/2(−i)|δ|

∫Rn

1(−i)|β| (∂

βp e−ip·ξ )xαpδ(∂γ ′

f )(x + p

2

)

× (∂γ−γ ′gk)(x − p

2

)dp

�∑γ ′≤γ

(γ

γ ′

)(2π )−n/2(−i)|δ+β|

∫Rn

e−ip·ξ xα∂βp

{pδ(∂γ ′

f )(x + p

2

)(∂γ−γ ′

gk)

×(x − p

2

)}dp

�∑γ ′≤γ

∑β ′≤β

(γ

γ ′

)(β

β ′

)(2π )−n/2(−i)|δ+β|

(−1

2

)|β−β ′|

×∫

Rn

e−ip·ξ xα∂β ′p

{pδ(∂γ ′

f )(x + p

2

)}(∂β−β ′+γ−γ ′

gk)(x − p

2

)dp

12. Weyl Transforms with Symbols in L∞(R2n) 61

�∑γ ′≤γ

∑β ′≤β

∑β ′′≤β ′

(γ

γ ′

)(β

β ′

)(β ′

β ′′

)(2π )−n/2(−i)|δ+β|

(−1

2

)|β−β ′| (12

)|β ′−β ′′|

×∫

Rn

e−ip·ξ xα(∂β ′′p pδ)(∂β ′−β ′′+γ ′

f )(x + p

2

)(∂β−β ′+γ−γ ′

gk)(x − p

2

)dp

�∑γ ′≤γ

∑β ′≤β

∑β ′′≤β ′

(γ

γ ′

)(β

β ′

)(β ′

β ′′

)(2π )−n/2(−i)|δ+β|

(−1

2

)|β−β ′| (12

)|β ′−β ′′|

× β ′′!(

δ

β ′′

)

×∫

Rn

e−ip·ξ xαpδ−β ′′(∂β ′−β ′′+γ ′

f )(x + p

2

)(∂β−β ′+γ−γ ′

gk)(x − p

2

)dp,

and hence, by Lemma 12.2,

|xαξβ(∂γx ∂δ

ξ W (f, gk))(x, ξ )|

≤∑γ ′≤γ

∑β ′≤β

∑β ′′≤β ′

(γ

γ ′

)(β

β ′

)(β ′

β ′′

)β ′′!(

δ

β ′′

)

×∫

Rn

|x||α||p||δ−β ′′|∣∣∣(∂β ′−β ′′+γ ′

f )(x + p

2

)∣∣∣ ∣∣∣(∂β−β ′+γ−γ ′gk)(x − p

2

)∣∣∣ dp→ 0 (12.6)

uniformly with respect to (x, ξ ) on R2n as k → ∞. Since σ is in S ′(R2n), it follows

from (12.6) that σ (W (f, gk)) → 0 as k → ∞. �

Remark 12.3. Let σ ∈ Lr (R2n), 1 ≤ r ≤ 2. Then, for all f and g in S(Rn), weget, by Theorem 12.1,

(Wσf )(g) � (2π )−n/2σ (W (f, g))

� (2π )−n/2∫

Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ. (12.7)

Thus, for all σ in Lr (R2n), 1 ≤ r ≤ 2, we get, by (12.7),

(Wσf )(g) � 〈Wσf, g〉, f, g ∈ S(Rn), (12.8)

where Wσ on the right-hand side of (12.8) is the bounded linear operator Qσ onL2(Rn) provided by Theorem 11.1. By Theorem 11.3, Wσ : L2(Rn) → L2(Rn) iscompact if σ is in Lr (R2n), 1 ≤ r ≤ 2. Now, let σ be in Lr (R2n), 2 < r ≤ ∞, andlet f be in S(Rn). Is Wσf , which by Theorem 12.1 is a tempered distribution onR

n, always a function in L2(Rn)? If so, can Wσ be extended to a bounded or evencompact operator from L2(Rn) into L2(Rn)? We attempt to answer these questionsin this and the next chapter.

Theorem 12.4. There exists a function σ in L∞(R2n) for which Wσf is not afunction in L2(Rn) for all functions f in S(Rn) such that

∫Rn f (x)dx � 0.

62 12. Weyl Transforms with Symbols in L∞(R2n)

Proof. Let σ be the function on R2n defined by

σ (x, ξ ) � e2ix·ξ , x, ξ ∈ Rn. (12.9)

Then, for all f and g in S(Rn), we get, by (3.12), (12.1), (12.9), and the Fourierinversion formula,

(Wσf )(g) � (2π )−n/2σ (W (f, g))

� (2π )−n/2∫

Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ

� (2π )−n

∫Rn

{∫Rn

e2ix·ξ(∫

Rn

e−ip·ξ f(x + p

2

)g(x − p

2

)dp

)dξ

}dx

�∫

Rn

f (2x)g(0)dx � 2−n

(∫Rn

f (x)dx

)g(0),

and hence

Wσf � 2−n

(∫Rn

f (x)dx

)δ,

where δ : S(Rn) → C is the Dirac delta given by

δ(ϕ) � ϕ(0), ϕ ∈ S(Rn). �

13Weyl Transforms with Symbols inLr(R2n), 2 < r < ∞

The following result can be found in the paper [25] by Simon.

Theorem 13.1. For 2 < r < ∞, there exists a function σ in Lr (R2n) such thatthe Weyl transform Wσ , defined by (12.1), is not a bounded linear operator onL2(Rn).

To prove Theorem 13.1, we need some preparations.

Lemma 13.2. Suppose that for all σ in Lr (R2n), 2 < r < ∞, the Weyl transformWσ , defined by (12.1), is a bounded linear operator on L2(Rn). Then there existsa positive constant C such that

‖Wσ‖∗ ≤ C‖σ‖Lr (R2n), σ ∈ Lr (R2n). (13.1)

Proof. Suppose that for all σ in Lr (R2n), there exists a positive constant Cσ suchthat

‖Wσf ‖L2(Rn) ≤ Cσ‖f ‖L2(Rn), f ∈ L2(Rn). (13.2)

Let f and g be functions in S(Rn) such that ‖f ‖L2(Rn) � ‖g‖L2(Rn) � 1. Considerthe bounded linear functional Qf,g : Lr (R2n) → C defined by

Qf,gσ � 〈Wσf, g〉, σ ∈ Lr (R2n). (13.3)

Then, by (13.2) and (13.3),

sup |Qf,gσ | ≤ Cσ , σ ∈ Lr (R2n), (13.4)

where the supremum is taken over all functions f and g in S(Rn) such that‖f ‖L2(Rn) � ‖g‖L2(Rn) � 1. Thus, by (13.4) and the uniform boundedness

64 13. Weyl Transforms with Symbols in Lr (R2n), 2 < r < ∞

principle, there exists a positive constant C such that

‖Qf,g‖ ≤ C (13.5)

for all f and g in S(Rn) with ‖f ‖L2(Rn) � ‖g‖L2(Rn) � 1, where ‖Qf,g‖ is thenorm of the bounded linear functional Qf,g : Lr (R2n) → C. Therefore, by (13.3)and (13.5),

sup‖σ‖

Lr (R2n )�1|〈Wσf, g〉| ≤ C (13.6)

for all f and g in S(Rn) with ‖f ‖L2(Rn) � ‖g‖L2(Rn) � 1. So, by (13.6),

|〈Wσf, g〉| ≤ C‖σ‖Lr (R2n)‖f ‖L2(Rn)‖g‖L2(Rn) (13.7)

for all σ in Lr (R2n), and f and g in S(Rn). Thus, (13.1) follows from (13.7). �

Lemma 13.3. Let α ∈ (0, 1). Then

limρ→∞

∫ π2

0e−ρ cos θραdθ � 0.

Proof. Let θ � π2 − ϕ. Then

∫ π2

0e−ρ cos θραdθ �

∫ π2

0e−ρ sin ϕραdϕ. (13.8)

Since the sine function is concave down on [0, π2 ], it follows that

sin ϕ ≥ 2π

ϕ, ϕ ∈[0,

π

2

]. (13.9)

Thus, by (13.8) and (13.9),∫ π

2

0e−ρ cos θραdθ ≤

∫ π2

0e− 2ρ

πϕραdϕ

� ραe− 2ρ

πϕ

(− π

2ρ

) ∣∣∣∣π2

0

� π

2ρα−1(1 − e−ρ) → 0

as ρ → ∞. �

Lemma 13.4. Let α ∈ (0, 1). Then∫ ∞

0tα−1 cos tdt � cos

(πα

2

)�(α).

13. Weyl Transforms with Symbols in Lr (R2n), 2 < r < ∞ 65

Proof.

Fig. 2

Let �ρ,r be the simple closed curve in Fig. 2 traversed once in the positivedirection. Let f be the function on the cut plane C − (−∞, 0] defined by

f (z) � e−zzα−1 � e−ze(α−1)Log−πz, z ∈ C − (−∞, 0], (13.10)

where Log−πz � ln |z| + iArg−πz and −π < Arg−πz < π . Then, by Cauchy’stheorem,

0 �∫

�ρ,r

f (z)dz �∫

γ 1ρ,r

f (z)dz+∫

γρ

f (z)dz+∫

γ 2ρ,r

f (z)dz+∫

γr

f (z)dz (13.11)

for all real numbers ρ and r satisfying 0 < r < ρ. Of course,∫γ 1

ρ,r

f (z)dz �∫ ρ

r

e−xxα−1dx, 0 < r < ρ. (13.12)

Since ∫γρ

f (z)dz �∫

γρ

e−ze(α−1)Log−πzdz

�∫ π

2

0e−ρeiθ

e(α−1){ln ρ+iθ}iρeiθdθ

�∫ π

2

0e−ρeiθ

ραe(α−1)iθ ieiθ dθ,

it follows from Lemma 13.3 that∣∣∣∣∣∫

γρ

f (z)dz

∣∣∣∣∣ ≤∫ π

2

0e−ρ cos θραdθ → 0 (13.13)

66 13. Weyl Transforms with Symbols in Lr (R2n), 2 < r < ∞

as ρ → ∞. Next, ∫γ 2

ρ,r

f (z)dz �∫ r

ρ

e−it e(α−1){ln t+ iπ2 }idt

�∫ r

ρ

e−it tα−1eiπα

2 e− iπ2 idt

� eiπα

2

∫ r

ρ

e−it tα−1dt

� eiπα

2

∫ r

ρ

(cos t − i sin t)tα−1dt (13.14)

for all real numbers ρ and r satisfying 0 < r < ρ. Finally, for r > 0,∫γr

f (z)dz �∫

γr

e−ze(α−1)Log−πzdz

� −∫ π

2

0e−reiθ

e(α−1){ln r+iθ}ireiθdθ

� −∫ π

2

0e−reiθ

rαe(α−1)iθ ieiθ dθ. (13.15)

Thus, by (13.15), ∣∣∣∣∫

γr

f (z)dz

∣∣∣∣ ≤∫ π

2

0e−r cos θ rαdθ → 0 (13.16)

as r → 0. So, letting ρ → ∞ and r → 0, we get, by (13.10)–(13.14) and (13.16),∫ ∞

0e−xxα−1dx − e

iπα2

∫ ∞

0(cos t − i sin t)tα−1dt � 0. (13.17)

Therefore, by (13.17),

eiπα

2

∫ ∞

0(cos t − i sin t)tα−1dt �

∫ ∞

0e−xxα−1dx � �(α),

and the lemma is proved. �

Theorem 13.1 is a consequence of Lemma 13.2 and the following lemma.

Lemma 13.5. For 2 < r < ∞, there is no positive constant C such that (13.1) isvalid.

Proof. Suppose that there exists a positive constant C such that (13.1) is valid.Then, by (12.1) and (13.1),

‖W (f, g)‖Lr′ (R2n) � sup‖σ‖

Lr (R2n )�1

∣∣∣∣∫

Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ

∣∣∣∣≤ sup

‖σ‖Lr (R2n )�1

(2π )n2 |〈Wσf, g〉|

13. Weyl Transforms with Symbols in Lr (R2n), 2 < r < ∞ 67

≤ sup‖σ‖

Lr (R2n )�1(2π )

n2 ‖Wσf ‖L2(Rn)‖g‖L2(Rn)

≤ (2π )n2 C‖f ‖L2(Rn)‖g‖L2(Rn), f, g ∈ S(Rn). (13.18)

Let f and g be in L2(Rn). Then, we let {fk}∞k�1 and {gk}∞k�1 be sequences offunctions in S(Rn) such that fk → f and gk → g in L2(Rn) as k → ∞. Thus,using the bilinearity of W ,

W (fk, gk) − W (fl, gl) � W (fk, gk) − W (fk, gl) + W (fk, gl) − W (fl, gl)� W (fk, gk − gl) − W (fk − fl, gl), k, l � 1, 2, . . . ,

and hence by (13.18),

‖W (fk, gk) − W (fl, gl)‖Lr′ (R2n)

≤ ‖W (fk, gk − gl)‖Lr′ (R2n) + ‖W (fk − fl, gl)‖Lr′ (R2n)

≤ (2π )n2 C{‖fk‖L2(Rn)‖gk − gl‖L2(Rn) + ‖fk − fl‖L2(Rn)‖gl‖L2(Rn)}

→ 0 as k, l → ∞. Therefore, {W (fk, gk)}∞k�1 is a Cauchy sequence in Lr ′ (R2n),and it is easy to see that its limit in Lr ′ (R2n) is equal to W (f, g). Thus, by (13.18),

‖W (f, g)‖Lr′ (R2n) ≤ (2π )n2 C‖f ‖L2(Rn)‖g‖L2(Rn) (13.19)

for all f and g in L2(Rn). Now, let f be in L2(Rn) such that

supp(f ) ⊆ {x ∈ Rn : |x| ≤ 1}, (13.20)

where supp(f ) is the support of the function f . Let x and ξ be in Rn. Then, by

(3.12), W (f )(x, ξ ) � 0 only if there is a point p in Rn such that |x ± p

2 | ≤ 1, andhence W (f )(x, ξ ) � 0 only if

|x| ≤ 12

∣∣∣(x + p

2

)+(x − p

2

)∣∣∣ ≤ 1.

Thus, |x| > 1 implies that W (f )(x, ξ ) � 0 for all ξ in Rn. Now, by (13.19),∫

Rn

∫Rn

|W (f )(x, ξ )|r ′dx dξ < ∞. (13.21)

Then, for any real-valued function θ on R2n, we get, by (13.20) and (13.21),{∫

Rn

∣∣∣∣∫

|x|≤1W (f )(x, ξ )eiθ (x,ξ )dx

∣∣∣∣r ′

dξ

} 1r′

≤∫

|x|≤1

{∫Rn

|W (f )(x, ξ )|r ′dξ

} 1r′

dx

≤{∫

|x|≤1dx

} 1r{∫

Rn

∫Rn

|W (f )(x, ξ )|r ′dx dξ

} 1r′

< ∞. (13.22)

Thus, by (13.22), ∫|x|≤1

W (f )(x, ξ )eiθ (x,ξ )dx ∈ Lr ′(Rn) (13.23)

68 13. Weyl Transforms with Symbols in Lr (R2n), 2 < r < ∞

as a function of ξ on Rn. Now, let

θ (x, ξ ) � 2x · ξ, x, ξ ∈ Rn. (13.24)

Then, by (3.12) and (13.24),∫|x|≤1

W (f )(x, ξ )eiθ (x,ξ )dx

� (2π )−n/2∫

Rn

{∫Rn

e−iξ ·pf(x + p

2

)f(x − p

2

)dp

}eiθ (x,ξ )dx

� (2π )−n/2∫

Rn

∫Rn

e2iξ ·(x− p

2 )f(x + p

2

)f(x − p

2

)dx dp (13.25)

for all ξ in Rn. Let u � x + p

2 and v � x − p

2 in (13.25). Then, using the Fourierinversion formula, ∫

Rn

W (f )(x, ξ )eiθ (x,ξ )dx

� (2π )−n/2∫

Rn

∫Rn

e2iξ ·vf (u)f (v)du dv

� (2π )−n/2∫

Rn

f (u){∫

Rn

e−2iξ ·vf (v)dv

}du

� (2π )n/2f (0)f (2ξ ), ξ ∈ Rn. (13.26)

Now, let Q � {x ∈ Rn : −a ≤ xj ≤ a, j � 1, 2, . . . , n} be a cube lying inside

{x ∈ Rn : |x| ≤ 1}. Let α ∈ (0, 1

2 ) and let f be the function on Rn defined by

f (x) �{∏n

j�1 |xj |−α, x ∈ Q, xj � 0, j � 1, 2, . . . , n,

0, otherwise.

Then

f (ξ ) � (2π )−n/2∫

Q

e−ix·ξn∏

j�1|xj |−αdx

� (2π )−n/2n∏

j�1

∫ a

−a

e−ixj ξj |xj |−αdxj , ξ ∈ Rn. (13.27)

Now, for ξj > 0,∫ a

−a

e−ixj ξj |xj |−αdxj �∫ a

−a

cos(xj ξj )|xj |−αdxj

� 2∫ a

0cos(xj ξj )x−α

j dxj

� 2(∫ aξj

0t−α cos t dt

)ξ−1+αj . (13.28)

13. Weyl Transforms with Symbols in Lr (R2n), 2 < r < ∞ 69

By Lemma 13.4, there is a positive constant A such that∣∣∣∣∫ aξj

0t−α cos t dt

∣∣∣∣ ≥ A (13.29)

for ξj large enough. Thus, by (13.27), (13.28), and (13.29),

|f (ξ )| ≥ (2π )−n/2

(n∏

j�1ξ−1+αj

)2nAn (13.30)

whenever ξ1, ξ2, . . . , ξn are all larger than some positive number R. Hence, by(13.30),∫

Rn

|f (ξ )|r ′dξ ≥ (2π )−

nr′2 2nr ′

Anr ′n∏

j�1

∫ ∞

R

ξ(−1+α)r ′j dξj � ∞ (13.31)

if (1 − α)r ′ < 1. Therefore,

‖Wf ‖Lr′ (R2n) ≤ (2π )n2 C‖f ‖2

L2(Rn) (13.32)

is impossible if

(1 − α)r ′ < 1, (13.33)

and (13.33) is obtained if we pick α to be some number in ( 1r, 1

2 ). �

14Compact Weyl Transforms

In view of the positive result, i.e., Theorem 11.1, and the negative results, i.e., The-orem 12.4 and Simon’s Theorem 13.1, it is of genuine interest to have a sufficientcondition on a function σ in Lr (R2n), 2 ≤ r ≤ ∞, such that the Weyl transformWσ , defined by (12.1), is a compact operator from L2(Rn) into L2(Rn). To do this,we introduce the space Lr

∗(R2n), 1 ≤ r ≤ ∞, defined by

Lr∗(R2n) � {σ ∈ Lr (R2n) : σ ∈ Lr ′

(R2n)}. (14.1)

To gain some insight into the space Lr∗(R2n), 1 ≤ r ≤ ∞, we use the following

theorem.

Theorem 14.1. (The Hausdorff–Young Inequality) The Fourier transform is abounded linear operator from Lr (Rn) into Lr ′ (Rn), 1 ≤ r ≤ 2. In fact,

‖f ‖Lr′ (Rn) ≤ (2π )−n2 (1− 2

r′ )‖f ‖Lr (Rn), f ∈ Lr (Rn). (14.2)

Proof. By the Plancherel theorem, we get

‖f ‖L2(Rn) � ‖f ‖L2(Rn), f ∈ L2(Rn), (14.3)

and using the definition of the Fourier transform, we get

‖f ‖L∞(Rn) ≤ (2π )−n/2‖f ‖L1(Rn), f ∈ L1(Rn). (14.4)

Let β1 � 12 , α1 � 1

2 , β2 � 0, and α2 � 1. Let M1 � 1 and M2 � (2π )−n/2. Letθ � 1 − 2

r ′ . Then, by (14.3), (14.4), and the Riesz–Thorin theorem,

‖f ‖Lr′ (Rn) ≤ (2π )−n2 (1− 2

r′ )‖f ‖Lr (Rn), f ∈ D, (14.5)

72 14. Compact Weyl Transforms

where D is the set of all simple functions f on Rn such that the Lebesgue measure

of the set

{x ∈ Rn : f (x) � 0}

is finite. Since D is dense in Lr (Rn), a limiting argument can be used to completethe proof that the Fourier transform is a bounded linear operator from Lr (Rn) intoLr ′ (Rn) and (14.2) is valid. �

As an immediate consequence of the Hausdorff–Young inequality, we give thefollowing corollary.

Corollary 14.2. Lr∗(R2n) � Lr (R2n), 1 ≤ r ≤ 2.

We can now give a result supplementing Theorems 11.1 and 11.3.

Theorem 14.3. Let σ ∈ Lr∗(R2n), 2 ≤ r ≤ ∞. Then Wσ : L2(Rn) → L2(Rn) is

a bounded linear operator. In fact,

‖Wσ‖∗ ≤ (2π )−n

r′ ‖σ‖Lr′ (R2n), σ ∈ Lr∗(R2n). (14.6)

Furthermore, Wσ : L2(Rn) → L2(Rn) is a compact operator.

To prove Theorem 14.3, we need some lemmas.

Lemma 14.4. Let f and g be in S(Rn). Then

W (f, g)(x, ξ ) � 2nV (f, g)(−2ξ, 2x), x, ξ ∈ Rn.

Proof. Let q � p

2 in (3.12). Then, by Proposition 2.3,

W (f, g)(x, ξ ) � (2π )−n/2∫

Rn

e−iξ ·pf(x + p

2

)g(x − p

2

)dp

� 2n(2π )−n/2∫

Rn

e−2iξ ·qf (x + q)g(x − q)dq

� 2n(2π )−n/2∫

Rn

e−2iξ ·qf (q + x)g(q − x)dq

� 2nV (f, g)(−2ξ, 2x), x, ξ ∈ Rn,

and the proof is complete. �

Lemma 14.5. Let σ ∈ Lr∗(R2n), 1 ≤ r ≤ ∞. Then the function τ on R

2n definedby

τ (x, ξ ) � σ (ξ, x), x, ξ ∈ Rn, (14.7)

is also in Lr∗(R2n), and

τ (q, p) � σ (p, q), q, p ∈ Rn. (14.8)

To prove Lemma 14.5, we need another lemma.

Lemma 14.6. Let f and g be in Lr (Rn), 1 ≤ r ≤ 2. Then∫Rn

f (x)g(x)dx �∫

Rn

f (x)g(x)dx.

14. Compact Weyl Transforms 73

Proof. Let f and g be in Lr (Rn), 1 ≤ r ≤ 2. Then we let {fk}∞k�1 and {gk}∞k�1be sequences of functions in S(Rn) such that fk → f and gk → g in Lr (Rn)as k → ∞. Then, by the Hausdorff–Young inequality, fk → f and gk → g inLr ′ (Rn) as k → ∞. Hence, by Proposition 1.10,∫

Rn

f (x)g(x)dx � limk→∞

∫Rn

fk(x)gk(x)dx

� limk→∞

∫Rn

fk(x)gk(x)dx �∫

Rn

f (x)g(x)dx. �

Proof of Lemma 14.5. For any ϕ in S(R2n), we get, by Lemma 14.6,

τ (ϕ) � τ (ϕ)

�∫

Rn

∫Rn

τ (x, ξ )ϕ(x, ξ )dx dξ

�∫

Rn

∫Rn

τ (x, ξ )ψ(ξ, x)dx dξ, (14.9)

where

ψ(q, p) � ϕ(p, q), q, p ∈ Rn. (14.10)

Thus, by (14.7), (14.9), (14.10), and Lemma 14.6,

τ (ϕ) �∫

Rn

∫Rn

σ (ξ, x)ψ(ξ, x)dξ dx

�∫

Rn

∫Rn

σ (p, q)ψ(p, q)dq dp

�∫

Rn

∫Rn

σ (p, q)ϕ(q, p)dq dp, ϕ ∈ S(R2n). (14.11)

Thus, by (14.11), τ ∈ Lr ′ (R2n), and (14.8) is valid. �

Proof of Theorem 14.3. Let f and g be in S(Rn). Then, by (12.1), Theorem3.1, (3.12), and Lemmas 14.4–14.6,

(Wσf )(g) � (2π )−n/2σ (W (f, g))

� (2π )−n/2∫

Rn

∫Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ

� (2π )−n/2∫

Rn

∫Rn

σ (x, ξ )V (f, g)(x, ξ )dx dξ

� (2π )−n/2∫

Rn

∫Rn

σ (q, p)V (f, g)(q, p)dq dp

� 2−n(2π )−n/2∫

Rn

∫Rn

σ (q, p)W (f, g)(p

2, −q

2

)dq dp

� 2n(2π )−n/2∫

Rn

∫Rn

σ (−2q, 2p)W (f, g)(p, q)dq dp

74 14. Compact Weyl Transforms

� 2n(2π )−n/2∫

Rn

∫Rn

σ (−2p, 2q)W (f, g)(q, p)dq dp

� 2n(2π )−n/2∫

Rn

∫Rn

τ (2q, −2p)W (f, g)(q, p)dq dp

� 2n(2π )−n/2∫

Rn

∫Rn

(δτ )(q, p)W (f, g)(q, p)dq dp, (14.12)

where

(δτ )(q, p) � τ (2q, −2p), q, p ∈ Rn. (14.13)

Since δτ ∈ Lr ′ (R2n), it follows from Theorem 11.1, (14.12), and (14.13) that

(Wσf )(g) � 2n〈Wδτf, g〉, f, g ∈ S(Rn),

and hence

|(Wσf )(g)| ≤(

2π

) n

r′‖δτ‖L2(R2n)‖f ‖L2(Rn)‖g‖L2(Rn) (14.14)

for all f and g in S(Rn). Therefore, by (14.14), Wσ : L2(Rn) → L2(Rn) is abounded linear operator and

‖Wσ‖∗ ≤(

2π

) n

r′‖δτ‖Lr′ (R2n). (14.15)

But, by (14.13),

‖δτ‖Lr′ (R2n) �(∫

Rn

∫Rn

|τ (2q, −2p)|r ′dq dp

) 1r′

�(∫

Rn

∫Rn

|σ (−2p, 2q)|r ′dq dp

) 1r′

� 2− 2n

r′ ‖σ‖Lr′ (R2n), (14.16)

and hence (14.6) follows from (14.15) and (14.16). To prove that Wσ : L2(Rn) →L2(Rn) is compact, let {τk}∞k�1 be a sequence of functions in S(R2n) such thatτk → σ in Lr ′ (R2n) as k → ∞. For k � 1, 2, . . . , let σk ∈ S(R2n) be such thatσk � τk . Then, by (14.6),

‖Wσ − Wσk‖∗ ≤ (2π )−

n

r′ ‖σ − σk‖Lr′ (R2n)

� (2π )−n

r′ ‖σ − τk‖Lr′ (R2n),

and hence Wσk→ Wσ in B(L2(Rn)) as k → ∞. By Theorem 6.8, Wσk

is aHilbert–Schmidt operator on L2(Rn), and hence, by Lemma 11.4, compact, fork � 1, 2, . . . . Thus, Wσ : L2(Rn) → L2(Rn) is compact. �

15Localization Operators

Our aim for Chapters 15–17 is to show that the localization operators introducedby Daubechies in the paper [3] as filters in signal analysis are examples of Weyltransforms that enjoy good mapping properties as compact operators from L2(Rn)into L2(Rn). In this chapter, we define the notion of a localization operator withsymbol in Lr (R2n), 1 ≤ r ≤ ∞, and prove that a localization operator with symbolin Lr (R2n), 1 ≤ r ≤ ∞, is a bounded linear operator from L2(Rn) into L2(Rn).

Let Z be the set of all integers. Then Cn × R/2πZ is a locally compact and

Hausdorff group in which the group law is given by

(q1, p1, t1) · (q2, p2, t2) � (q1 + q2, p1 + p2, t1 + t2 + q1 · p2)

for all (q1, p1, t1) and (q2, p2, t2) in Cn × R/2πZ, where q1 · p2 is the Euclidean

inner product of q1 and p2 in Rn; t1+t2 and t1+t2+q1 ·p2 are cosets in the quotient

group R/2πZ in which the group law is addition modulo 2π . On Cn × R/2πZ,

the left Haar measure coincides with the right Haar measure and can be identifiedwith the Lebesgue measure dq dp dt on the measurable space C

n × R/2πZ. Thelocally compact Hausdorff space C

n × R/2πZ endowed with the left (and right)Haar measure dq dp dt is hence unimodular. It is called the Weyl–Heisenberggroup, and we denote it by (WH )n.

Let π : (WH )n → B(L2(Rn)) be the mapping defined by

(π (q, p, t)f )(x) � ei(p·x+ 12 q·p+t)f (x − q), x ∈ R

n,

for all (q, p, t) in (WH )n and f in L2(Rn). That π : (WH )n → B(L2(Rn)) is anirreducible unitary representation is left as an exercise.

76 15. Localization Operators

Let ϕ be the function on Rn defined by

ϕ(x) � π− n4 e− |x|2

2 , x ∈ Rn. (15.1)

Then

‖ϕ‖L2(Rn) � 1, (15.2)

and it is also an easy exercise to prove that the number cϕ defined by

cϕ �∫ 2π

0

∫Rn

∫Rn

|〈ϕ, π(q, p, t)ϕ〉|2dq dp dt

is finite, and in fact

cϕ � (2π )n+1. (15.3)

The function ϕ is called an admissible wavelet for the irreducible unitary rep-resentation π : (WH )n → B(L2(Rn)), and the representation π : (WH )n →B(L2(Rn)) is called square integrable. It is left as an exercise that we have theresolution of the identity formula, i.e.,

〈f, g〉 � 1cϕ

∫ 2π

0

∫Cn

〈f, π(z, t)ϕ〉〈π (z, t)ϕ, g〉dz dt (15.4)

for all f and g in L2(Rn). The theory of the representation π : (WH )n →B(L2(Rn)) hitherto described is an important, albeit special, case of the theoryof square-integrable representations studied in the papers [11,12] by Grossmann,Morlet, and Paul, and the book [17] by Holschneider, among others.

For q and p in Rn, we define the function ϕq,p on R

n by

ϕq,p(x) � eip·xϕ(x − q), x ∈ Rn. (15.5)

Let F be a measurable function on Cn. Then we define σ : (WH )n → C by

σ (z, t) � F (z), z ∈ Cn, t ∈ [0, 2π ]. (15.6)

Let F be in L1(Cn) or L∞(Cn). Then, for any f in L2(Rn), we define the functionLF f on R

n by

〈LF f, g〉 � 1cϕ

∫ 2π

0

∫Cn

σ (z, t)〈f, π(z, t)ϕ〉〈π (z, t)ϕ, g〉dz dt

or, in view of (15.3) and (15.6),

〈LF f, g〉 � (2π )−n

∫Cn

F (z)〈f, ϕz〉〈ϕz, g〉dz, g ∈ L2(Rn). (15.7)

We have the following result.

Proposition 15.1. Let F ∈ L1(Cn). Then LF : L2(Rn) → L2(Rn) is a boundedlinear operator, and

‖LF ‖∗ ≤ (2π )−n‖F‖L1(Cn). (15.8)

15. Localization Operators 77

Proof. Since by (15.2) and (15.5),

‖ϕz‖L2(Rn) � 1, z ∈ Cn, (15.9)

it follows from (15.9) that for all z in Cn,

|〈f, ϕz〉〈ϕz, g〉| ≤ ‖f ‖L2(Rn)‖g‖L2(Rn), f, g ∈ L2(Rn). (15.10)

Since F ∈ L1(Cn), it follows from (15.7) and (15.10) that

|〈LF f, g〉| ≤ (2π )−n‖F‖L1(Cn)‖f ‖L2(Rn)‖g‖L2(Rn)

for all f and g in L2(Rn) and hence LF : L2(Rn) → L2(Rn) is a bounded linearoperator and (15.8) is valid. �

We also have the following.

Proposition 15.2. Let F ∈ L∞(Cn). Then LF : L2(Rn) → L2(Rn) is a boundedlinear operator, and

‖LF ‖∗ ≤ ‖F‖L∞(Cn). (15.11)

Proof. Let f and g be in L2(Rn). Then, by (15.7),

|〈LF f, g〉| ≤ (2π )−n‖F‖L∞(Cn)

(∫Cn

|〈f, ϕz〉|2dz

) 12(∫

Cn

|〈ϕz, g〉|2dz

) 12

.

(15.12)But using the resolution of the identity formula, i.e., (15.4),

‖f ‖2L2(Rn) � (2π )−n

∫Cn

|〈f, ϕz〉|2dz (15.13)

and

‖g‖2L2(Rn) � (2π )−n

∫Cn

|〈g, ϕz〉|2dz. (15.14)

So, by (15.12), (15.13), and (15.14),

|〈LF f, g〉| ≤ ‖F‖L∞(Cn)‖f ‖L2(Rn)‖g‖L2(Rn),

and the proof is complete. �

Remark 15.3. The bounded linear operators LF : L2(Rn) → L2(Rn) alreadyintroduced, and to be introduced, in this chapter are called localization operators.This terminology appears to be first used in the papers [3, 4] by Daubechies and[5] by Daubechies and Paul. To understand the terminology better, let us note thatwhen F is identically equal to 1 on C

n, the resolution of the identity formula, i.e.,(15.4), implies that the corresponding operator LF is equal to the identity. Thus, therole of the “symbol” F is to assign different weights to different parts of the phasespace C

n, i.e., localize on Cn, in order to produce a mathematically interesting

operator with applications in various disciplines in science and engineering. Towit, applications to signal analysis can be found in the above-mentioned papers,the paper [14] by He, and the paper [16] by He and Wong.

78 15. Localization Operators

We can now associate a localization operator LF : L2(Rn) → L2(Rn) to everyF in Lr (Cn), 1 < r < ∞. The main result is the following theorem.

Theorem 15.4. Let F ∈ Lr (Cn), 1 < r < ∞. Then there exists a unique boundedlinear operator LF : L2(Rn) → L2(Rn) such that

‖LF ‖∗ ≤ (2π )−nr ‖F‖Lr (Cn), (15.15)

and 〈LF f, g〉, for all f and g in L2(Rn), is given by (15.7) for all simple functionsF on C

n for which the Lebesgue measure of the set {z ∈ Cn : F (z) � 0} is finite.

Proof. Let D be the set of all simple functions F on Cn such that the Lebesgue

measure of the set {z ∈ Cn : F (z) � 0} is finite. Let f ∈ L2(Rn) and let T be the

linear transformation from D into the set of all Lebesgue-measurable functions onR

n defined by

T F � LF f, F ∈ D. (15.16)

Then, by (15.8), (15.11), (15.16), and the Riesz–Thorin theorem,

‖LF f ‖L2(Rn) ≤ (2π )−nr ‖F‖Lr (Cn)‖f ‖L2(Rn) (15.17)

for all F in D. Hence (15.15) follows immediately from (15.17). Now, let F ∈Lr (Cn). Then there exists a sequence {Fk}∞k�1 of functions in D such that Fk → F

in Lr (Cn) as k → ∞. Thus, by (15.15),

‖LFk− LFj

‖∗ ≤ (2π )−nr ‖Fk − Fj‖Lr (Cn) → 0 (15.18)

as k, j → ∞. Therefore, by (15.18), {LFk}∞k�1 is a Cauchy sequence in B(L2(Rn)).

Thus, there is a bounded linear operator LF : L2(Rn) → L2(Rn) such that LFk→

LF in B(L2(Rn)) as k → ∞. That the limit is independent of the choice of thesequence {Fk}∞k�1 in D and that LF : L2(Rn) → L2(Rn) so defined is the uniquebounded linear operator satisfying the conclusions of the theorem should, by now,be obvious, or if not, it follows from a standard argument, which we leave as anexercise. �

16A Fourier Transform

In order to study localization operators in some detail, we first compute the Fouriertransform of a function on C

n.

Theorem 16.1. Let ϕ be the function on Rn defined by (15.1), and for any x in

Rn and f in S(Rn), let fx be the function on C

n defined by

fx(z) � 〈f, ϕz〉ϕz(x), z ∈ Cn. (16.1)

Then

fx(ζ ) � e− |ζ |24 (ρ(−ζ )f )(x), ζ ∈ C

n, (16.2)

where ρ(ζ ), for any ζ in Cn, is given by (2.1).

Proof. Let ε be any positive number. Then we define the function Iε on Cn by

Iε(ζ ) � (2π )−n

∫Rn

∫Rn

e−i(q·ξ+p·η)e− ε2 |p|22 〈f, ϕz〉ϕz(x)dz (16.3)

for all ζ � ξ + iη in Cn, where z � q + ip is a point in C

n. Then, by (15.1),(16.3), and the fact that the Fourier transform of the function ψ on R

n given by

ψ(x) � e− |x|22 , x ∈ R

n, (16.4)

is equal to itself, we get

(2π )nIε(ζ )

�∫

Rn

∫Rn

e−i(q·ξ+p·η)e− ε2 |p|22

(∫Rn

f (y)e−ip·y ϕ(y − q)dy

)eip·xϕ(x − q)dq dp

80 16. A Fourier Transform

�∫

Rn

∫Rn

e−iq·ξ f (y)ϕ(y − q)ϕ(x − q)(∫

Rn

e−ip·(η+y−x)e− ε2 |p|22 dp

)dq

� (2π )n/2∫

Rn

∫Rn

e−iq·ξ f (y)ϕ(y − q)ϕ(x − q)ε−ne− |η+y−x|2

2ε2 dq

� (2π )n/2∫

Rn

e−iq·ξϕ(x − q)(∫

Rn

ε−ne− |η+y−x|2

2ε2 f (y)ϕ(y − q)dy

)dq (16.5)

for all ζ � ξ + iη in Cn. Now, for each q in R

n, we define Fq : Rn → C by

Fq(y) � f (y)ϕ(y − q), y ∈ Rn. (16.6)

Then, by (16.4), (16.5), and (16.6),

Iε(ζ ) � (2π )−n/2∫

Rn

e−iq·ξϕ(x − q)(Fq ∗ ψε)(x − η)dq (16.7)

for all ζ � ξ + iη in Cn, where

ψε(x) � ε−nψ(x

ε

), x ∈ R

n, (16.8)

and ψ is the function on Rn given by (16.4). Now, for each fixed q in R

n, we get,by (16.4), (16.6), and (16.8),

Fq ∗ ψε →(∫

Rn

ψ(x)dx

)Fq � (2π )

n2 Fq (16.9)

uniformly on compact subsets of Rn as ε → 0. Furthermore, there exists a positive

constant C such that

|(Fq ∗ ψε)(w)| ≤ ‖Fq‖L∞(Rn)‖ψε‖L1(Rn)

≤ supy∈Rn

(|f (y)‖ϕ(y − q)|)‖ψ‖L1(Rn)

≤ C, w, q ∈ Rn. (16.10)

So, by (16.6), (16.7), (16.9), (16.10), and the Lebesgue dominated convergencetheorem,

limε→0

Iε(ζ ) �(∫

Rn

e−iq·ξϕ(x − q)ϕ(x − η − q)dq

)f (x − η) (16.11)

for all ζ � ξ + iη in Cn. But, using (16.1), (16.3), and the Lebesgue dominated

convergence theorem,

limε→0

Iε(ζ ) � (2π )−n

∫Rn

∫Rn

e−i(q·ξ+p·η)〈f, ϕz〉ϕz(x)dz � fx(ζ ) (16.12)

for all ζ � ξ + iη in Cn. Thus, by (16.11), (16.12), (2.1), and Proposition 2.3,

fx(ζ ) �(∫

Rn

e−iq·ξϕ(x − q)ϕ(x − η − q)dq

)f (x − η)

�(∫

Rn

e−i(x−q)·ξϕ(q)ϕ(q − η)dq

)f (x − η)

16. A Fourier Transform 81

� e−ix·ξ f (x − η)∫

Rn

ei(q+ η

2 )·ξϕ(q + η

2

)ϕ(q − η

2

)dq

� e−ix·ξ e12 iξ ·ηf (x − η)(2π )

n2 V (ϕ, ϕ)(ξ, η)

� (ρ(−ξ, −η)f )(x)(2π )n2 V (ϕ, ϕ)(ξ, η). (16.13)

But by Proposition 2.3, (15.1), and the fact that the Fourier transform of the functionψ on R

n defined by (16.4) is equal to itself, we get

V (ϕ, ϕ)(ξ, η) � (2π )−n/2∫

Rn

e−iy·ξϕ(y + η

2

)ϕ(y − η

2

)dy

� (2π )−n/2π− n2

∫Rn

eiy·ξ e− 12 |y+ η

2 |2e− 12 |y− η

2 |2dy

� (2π )−n/2π− n2

∫Rn

eiy·ξ e− 12 {2|y|2+ |η|2

2 }dy

� π− n2 e− |η|2

4 (2π )−n/2∫

Rn

eiy·ξ e−|y|2dy

� π− n2 e− |η|2

4 2− n2 (2π )−n/2

∫Rn

ei

y√2·ξe− |y|2

2 dy

� (2π )−n/2e− |η|24 e− |ξ |2

4 � (2π )−n/2e− |ζ |24 (16.14)

for all ζ � ξ + iη in Cn. Thus, (16.2) follows from (16.13) and (16.14). �

17Compact Localization Operators

The aim of this chapter is to use the Fourier transform computed in Chapter 16to prove that every localization operator is a Weyl transform and the fact that alocalization operator with symbol in Lr (R2n), 1 ≤ r < ∞, is compact.

Theorem 17.1. Let � be the function on Cn defined by

�(z) � π−ne−|z|2 , z ∈ Cn. (17.1)

Then, for all F in Lr (Cn), 1 ≤ r ≤ ∞, the Weyl transform WF∗�, initially definedon S(Rn), can be extended to a bounded linear operator from L2(Rn) into L2(Rn)that is equal to LF .

Proof. We begin with the case when F is in S(Cn). Then, for all f in S(Rn),we get, by (15.7) and (16.1),

(LF f )(x) � (2π )−n

∫Cn

F (z)fx(z)dz

� (2π )−n

∫Cn

F (ζ )fx(ζ ) dζ, x ∈ Rn, (17.2)

where F is the inverse Fourier transform of F . Thus, by (17.2),

(LF f )(x) � (2π )−n

∫Cn

F (ζ )e− 14 |ζ |2 (ρ(−ζ )f )(x)dζ

� (2π )−n

∫Cn

F (ζ )e− 14 |ζ |2 (ρ(ζ )f )(x)dζ, x ∈ R

n. (17.3)

84 17. Compact Localization Operators

But, by (9.4),

(Wσf )(x) � (2π )−n

∫Cn

σ (ζ )(ρ(ζ )f )(x)dζ, x ∈ Rn, (17.4)

for any σ in S(Cn). Then, by (17.1), (17.3), (17.4), and Proposition 1.7,

LF � WF∗�. (17.5)

Now, let F ∈ Lr (Cn), 1 ≤ r < ∞. Then there exists a sequence {Fk}∞k�1 offunctions in S(Cn) such that Fk → F in Lr (Cn) as k → ∞. Thus, by Proposition15.1, Theorem 15.4, and (17.5),

WFk∗� � LFk→ LF (17.6)

in B(L2(Rn)) as k → ∞. But for all f and g in S(Rn), we get, by (12.1) and(17.6),

(WFk∗�f )(g) � (2π )−n/2(Fk ∗ �)(W (f, g))→ (2π )−n/2(F ∗ �)(W (f, g)) � (WF∗�f )(g) (17.7)

as k → ∞. Thus, for all f in S(Rn), we get, by (17.7),

WFk∗�f → WF∗�f (17.8)

in S ′(Rn) as k → ∞. But of course, by (17.6), we get, for all f in S(Rn),

WFk∗�f → LF f (17.9)

in S ′(Rn) as k → ∞. Thus, for all f in S(Rn), we get, by (17.8) and (17.9),

WF∗�f � LF f (17.10)

in the sense of distributions. Thus, by (17.10), the Weyl transform WF∗�, initiallydefined on S(Rn), can be extended to a bounded linear operator from L2(Rn) intoL2(Rn) that is equal to LF . Now, let F ∈ L∞(Cn). Then we can find a sequence{Fk}∞k�1 of simple functions on C

n such that the Lebesgue measure of the set

{z ∈ Cn : Fk(z) � 0}

is finite for k � 1, 2, . . . , and Fk → F a.e. on Cn as k → ∞. Now, for all f and

g in S(Rn), we get, by (12.1), (17.5), and the Lebesgue dominated convergencetheorem,

(WF∗�f )(g) � (2π )−n/2∫

Cn

(F ∗ �)(z)W (f, g)(z)dz

� limk→∞

(2π )−n/2∫

Cn

(Fk ∗ �)(z)W (f, g)(z)dz

� limk→∞

(WFk∗�f )(g)

� limk→∞

(LFkf )(g) � (LF f )(g). (17.11)

So, by (17.11),

WF∗�f � LF f, f ∈ S(Rn). (17.12)

17. Compact Localization Operators 85

Therefore, by (17.12), the Weyl transform WF∗�, initially defined on S(Rn), canbe extended to a bounded linear operator from L2(Rn) into L2(Rn) that is equal toLF . �

Theorem 17.2. Let F ∈ Lr (Cn), 1 ≤ r < ∞. Then the localization operatorLF : L2(Rn) → L2(Rn) is compact.

Proof. Let {Fk}∞k�1 be a sequence of functions in S(Cn) such that Fk → F inLr (Cn) as k → ∞. Then, by Theorem 6.8 and Lemma 11.4, LFk

� WFk∗� isa Hilbert–Schmidt operator and hence compact because Fk ∗ � is a function inS(Cn) for k � 1, 2, . . . . But by Proposition 15.1 and Theorem 15.4, LFk

→ LF

in B(L2(Rn)) as k → ∞. Therefore, LF is compact. �

Remark 17.3. That Theorem 17.2 is false for r � ∞ can be seen easily by takingthe function F on C

n to be such that

F (z) � 1, z ∈ Cn.

For then, by the resolution of the identity formula, i.e., (15.4), LF is equal to theidentity operator on L2(Rn) and hence cannot be compact.

Remark 17.4. A more precise and more general result than Theorem 17.2 has beenproved. See Theorem 6.1 in the paper [15] by He and Wong in this connection.

18Hermite Polynomials

We are now interested in criteria for the boundedness and/or compactness of Weyltransforms on L2(R) using an orthonormal basis for L2(R2) consisting of Hermitefunctions on C (� R

2), i.e., Wigner transforms of Hermite functions on R. Theinverse Fourier transforms of functions in this orthonormal basis, i.e., the Fourier–Wigner transforms of Hermite functions on R, are shown in Chapter 22 to beeigenfunctions of a partial differential operator on C.

In this and the next two chapters we lay out the basic properties of Hermitepolynomials, Hermite functions, and Laguerre polynomials, which we shall use tostudy Weyl transforms. A good reference for these topics is Chapter 6 of the book[7] by Folland. The properties of Hermite functions on C to be used in this bookare given in Chapters 21, 22, and 23.

Let n � 0, 1, 2, . . . . Then we define the function Hn on R by

Hn(x) � (−1)nex2(

d

dx

)n

(e−x2), x ∈ R. (18.1)

We call Hn the Hermite polynomial of degree n. It is easy to see that H0(x) � 1,H1(x) � 2x, and so on.

Proposition 18.1. For n � 1, 2, . . . ,

Hn(x) � 2xHn−1(x) − H ′n−1(x), x ∈ R.

Proof. By (18.1),

e−x2Hn(x) � (−1)n

(d

dx

)n

(e−x2) � (−1)n

(d

dx

)(d

dx

)n−1

(e−x2)

88 18. Hermite Polynomials

� −(

d

dx

)((−1)n−1

(d

dx

)n−1

(e−x2)

)� − d

dx(e−x2

Hn−1(x))

� −(−2xe−x2Hn−1(x) + e−x2

H ′n−1(x))

� 2xe−x2Hn−1(x) − e−x2

H ′n−1(x), x ∈ R,

and Proposition 18.1 follows. �

Remark 18.2. Using Proposition 18.1 repeatedly, we see that the highest powerin Hn(x) is equal to (2x)n multiplied by the highest power in H0(x), which is thenequal to 2nxn.

Let w be the function on R defined by

w(x) � e−x2, x ∈ R. (18.2)

Then we define L2w(R) to be the set of all complex-valued functions on R such that∫ ∞

−∞|f (x)|2w(x)dx < ∞.

Then L2w(R) is a Hilbert space in which the inner product 〈 , 〉w and norm ‖ ‖w are,

respectively, given by

〈f, g〉w �∫ ∞

−∞f (x)g(x)w(x)dx (18.3)

and

‖f ‖w �(∫ ∞

−∞|f (x)|2w(x)dx

) 12

(18.4)

for all f and g in L2w(R).

Proposition 18.3. {Hn : n � 0, 1, 2, . . .} is an orthogonal set in L2w(R).

Moreover,

‖Hn‖2w � 2nn!

√π, n � 0, 1, 2, . . . .

Proof. Let m and n be nonnegative integers such that m ≤ n. Then, by (18.1),(18.2), and (18.3), we get

〈Hm, Hn〉w �∫ ∞

−∞Hm(x)Hn(x)e−x2

dx

�∫ ∞

−∞Hm(x)(−1)nex2

(d

dx

)n

(e−x2)e−x2

dx

� (−1)n∫ ∞

−∞Hm(x)

(d

dx

)n

(e−x2)dx

�∫ ∞

−∞H (n)

m (x)e−x2dx. (18.5)

18. Hermite Polynomials 89

Thus, by (18.5), 〈Hm, Hn〉w � 0 if m � n. Moreover, for n � 0, 1, 2, . . . , byRemark 18.2, (18.3), (18.4), and (18.5), we get

‖Hn‖2w �

∫ ∞

−∞H (n)

n (x)e−x2dx � 2nn!

∫ ∞

−∞e−x2

dx � 2nn!√

π

and hence complete the proof. �

We can strengthen Proposition 18.3 and get the following result.

Theorem 18.4. { 1‖Hn‖w

Hn : n � 0, 1, 2, . . .} is an orthonormal basis for L2w(R).

To prove Theorem 18.4, we use two lemmas.

Lemma 18.5. Let {pn : n � 0, 1, 2, . . .} be a sequence of nonzero polynomialssuch that the degree of pn is equal to n. Let P be any polynomial of degree k. Thenthere exist constants c0, c1, c2, . . . , ck such that

P �k∑

n�0cnpn.

Proof. Let P be a polynomial of degree zero. Then P (x) � α for all x in R,where α is a constant. Suppose that p0(x) � β for all x in R, where β is a nonzeroconstant. Then

P (x) � α

ββ � α

βp0(x), x ∈ R.

Suppose that Lemma 18.5 is true for all polynomials of degree at most k. Let P

be a polynomial of degree k + 1. Then we choose the constant ck+1 such that P

and ck+1pk+1 have the same highest power. Thus, P − ck+1pk+1 is a polynomialof degree at most k. Hence

P − ck+1pk+1 �k∑

n�0cnpn

for some constants c0, c1, c2, . . . , ck , and the proof is complete. �

Lemma 18.6. Let f be a measurable function on R such that∫ ∞

−∞|f (x)|e|xξ |e−x2

dx < ∞ (18.6)

for all ξ in R. If ∫ ∞

−∞f (x)P (x)e−x2

dx � 0 (18.7)

for all polynomials P , then f � 0 a.e. on R.

Proof. We begin by noting that∫ ∞

−∞eixξf (x)e−x2

dx �∫ ∞

−∞

∞∑n�0

(ixξ )n

n!f (x)e−x2

dx, ξ ∈ R. (18.8)

90 18. Hermite Polynomials

Now, ∣∣∣∣∣∞∑

n�0

(ixξ )n

n!f (x)e−x2

∣∣∣∣∣ ≤N∑

n�0

|xξ |nn!

|f (x)|e−x2

≤ e|xξ ||f (x)|e−x2, x, ξ ∈ R. (18.9)

Hence, using (18.6)–(18.9) and the Lebesgue dominated convergence theorem,∫ ∞

−∞eixξf (x)e−x2

dx �∞∑

n�0

(iξ )n

n!

∫ ∞

−∞xnf (x)e−x2

dx � 0. (18.10)

Thus, by (18.10), f (x)e−x2 � 0 for almost all x in R, and the proof iscomplete. �

Proof of Theorem 18.4. In view of Proposition 18.3, we need only prove that iff ∈ L2

w(R) is such that

〈f, Hn〉w � 0, n � 0, 1, 2, . . . , (18.11)

then f � 0 a.e. on R. But by the Schwarz inequality,∫ ∞

−∞|f (x)|e|tx|e−x2

dx ≤(∫ ∞

−∞|f (x)|2e−x2

dx

) 12(∫ ∞

−∞e2|tx|e−x2

dx

) 12

< ∞(18.12)

for all t in R. Let P be any polynomial of degree k. Then, by Lemma 18.5, we canfind constants c0, c1, c2, . . . , ck such that

P (x) �k∑

n�0cnHn(x), x ∈ R. (18.13)

Thus, by (18.11) and (18.13),∫ ∞

−∞f (x)P (x)e−x2

dx �k∑

n�0cn

∫ ∞

−∞f (x)Hn(x)e−x2

dx � 0. (18.14)

Hence, by (18.12) and (18.14), Lemma 18.6 can be used to conclude that f � 0a.e. on R. �

The following properties of Hermite polynomials will be useful to us.

Proposition 18.7. For all x in R,

(i) H ′0(x) � 0;

(ii) H ′n(x) � 2nHn−1(x), n � 1, 2, . . . ;

(iii) H ′′n (x) − 2xH ′

n(x) + 2nHn(x) � 0, n � 0, 1, 2, . . . .

Proof. To prove part (iii), note that the equation is trivially true for n � 0. Now,suppose that

H ′′n−1(x) − 2xH ′

n−1(x) + 2(n − 1)Hn−1(x) � 0 (18.15)

18. Hermite Polynomials 91

for some positive integer n. By Proposition 18.1 and (18.15), we get

H ′n(x) � 2Hn−1(x) + 2xH ′

n−1(x) − H ′′n−1(x)

� 2Hn−1(x) + 2xH ′n−1(x) − 2xH ′

n−1(x) + 2(n − 1)Hn−1(x)� 2nHn−1(x), x ∈ R. (18.16)

So, by (18.16),

H ′′n (x) � 2nH ′

n−1(x), x ∈ R. (18.17)

Thus, by Proposition 18.1, (18.16) and (18.17),

H ′′n (x) − 2xH ′

n(x) + 2nHn(x)� 2nH ′

n−1(x) − 4nxHn−1(x) + 4nxHn−1(x) − 2nH ′n−1(x) � 0

for all x in R. Part (i) is obvious. To prove part (ii), note that by part (iii) with n

replaced by n − 1 and Proposition 18.1, we get

H ′n(x) � 2xH ′

n−1(x) + 2Hn−1(x) − 2xH ′n−1(x) + 2(n − 1)Hn−1(x)

� 2nHn−1(x), x ∈ R. �

19Hermite Functions

For n � 0, 1, 2, . . . , we define the function hn on R by

hn(x) � e− x22 Hn(x), x ∈ R. (19.1)

We call hn the Hermite function of order n.

Theorem 19.1. For all x in R,

(i) xhn(x) + h′n(x) � 2nhn−1(x), n � 1, 2, . . . ;

(ii) xhn(x) − h′n(x) � hn+1(x), n � 0, 1, 2, . . . ;

(iii) h′′n(x) − x2hn(x) + (2n + 1)hn(x) � 0, n � 0, 1, 2, . . . .

Remark 19.2. The differential operators x+ ddx

, x− ddx

, and − d2

dx2 +x2 in Theorem19.1 are, respectively, called the annihilation operator, the creation operator, andthe harmonic oscillator. The Hermite function hn of order n is an eigenfunction ofthe harmonic oscillator corresponding to the eigenvalue 2n + 1.

Proof of Theorem 19.1. For n � 1, 2, . . . , by (19.1) and part (ii) of Proposition18.7,

2nex22 hn−1(x) � 2nHn−1(x) � H ′

n(x)

� d

dx(e

x22 hn(x)) � xe

x22 hn(x) + e

x22 h′

n(x)

for all x in R, and part (i) is proved. For part (ii), using (19.1) and Proposition18.1, we get, for n � 1, 2, . . . ,

ex22 hn+1(x) � 2xe

x22 hn(x) − d

dx(e

x22 hn(x))

94 19. Hermite Functions

� 2xex22 hn(x) − xe

x22 hn(x) − e

x22 h′

n(x)

� xex22 hn(x) − e

x22 h′

n(x), x ∈ R,

and part (ii) follows. For part (iii), we use parts (i) and (ii) to get

hn(x) � xhn−1(x) − h′n−1(x)

� x

(12n

xhn(x) + 12n

h′n(x)

)−(

12n

hn(x) + 12n

xh′n(x) + 1

2nh′′

n(x))

� − 12n

h′′n(x) + 1

2nx2hn(x) − 1

2nhn(x), x ∈ R, (19.2)

for n � 1, 2, . . . . That part (iii) is also true for n � 0 is obvious. �

For n � 1, 2, . . . , we define the function en on R by

en(x) � 1(2nn!

√π ) 1

2hn(x), x ∈ R. (19.3)

Then we have the following important theorem.

Theorem 19.3. {en : n � 0, 1, 2, . . .} is an orthonormal basis for L2(R).

Proof. Let m and n be nonnegative integers. Then, by (19.1), (19.3), andProposition 18.3,∫ ∞

−∞em(x)en(x)dx � 1

(2m+nm!n!π )12

∫ ∞

−∞hm(x)hn(x)dx

� 1

(2m+nm!n!π )12

∫ ∞

−∞Hm(x)Hn(x)e−x2

dx

�{

0, m � n,1, m � n.

So, it remains to prove that if f ∈ L2(R) is such that

〈f, hn〉 � 0, n � 0, 1, 2, . . . , (19.4)

then f � 0 a.e. on R. To this end, let F be the function on R defined by

F (x) � f (x)ex22 , x ∈ R. (19.5)

Then F ∈ L2w(R), and for n � 0, 1, 2, . . . , we get, by (19.4),

〈F, Hn〉w � 〈f, hn〉 � 0. (19.6)

So, by Proposition 18.4 and (19.6), F � 0 a.e. on R, and hence, by (19.4), f � 0a.e. on R. �

20Laguerre Polynomials

Let α > −1. Then, for n � 0, 1, 2, . . . , we define the function Lαn on R by

Lαn(x) � x−αex

n!

(d

dx

)n

(e−xxα+n), x > 0.

We call Lαn the Laguerre polynomial of degree n and order α.

If we write out Lαn(x), x > 0, in detail, then we get

Lαn(x) � x−αex

n!

n∑k�0

(n

k

)(−1)ke−x

(d

dx

)n−k

(xα+n), x > 0. (20.1)

Thus,

Lαn(x)

� (−1)n

n!xn+

n−1∑k�0

(α + n)(α + n − 1) · · · (α + k + 1)(n − k)!k!

(−x)k, x > 0. (20.2)

For the highest power in Lαn(x), x > 0, we let k � n in (20.1) or use (20.2) to get

Lαn(x) � (−1)n

n!xn + · · · , x > 0. (20.3)

Let u be the function on (0, ∞) defined by

u(x) � xαe−x, x ∈ (0, ∞).

96 20. Laguerre Polynomials

Then we define L2u(0, ∞) to be the set of all complex-valued functions f on (0, ∞)

such that ∫ ∞

0|f (x)|2u(x)dx < ∞.

Then L2u(0, ∞) is a Hilbert space in which the inner product 〈 , 〉u and norm ‖ ‖u

are, respectively, given by

〈f, g〉u �∫ ∞

0f (x)g(x)u(x)dx

and

‖f ‖u �{∫ ∞

0|f (x)|2u(x)dx

} 12

for all f and g in L2u(0, ∞).

Proposition 20.1. {Lαn : n � 0, 1, 2, . . .} is an orthogonal set in L2

u(0, ∞).Moreover,

‖Lαn‖2

u � �(α + n + 1)n!

, n � 0, 1, 2, . . . .

Proof. Let m and n be nonnegative integers such that m < n. Then

〈Lαm, Lα

n〉u �∫ ∞

0Lα

m(x)Lαn(x)xαe−xdx

� 1n!

∫ ∞

0Lα

m(x)(

d

dx

)n

(e−xxα+n)dx

� (−1)n

n!

∫ ∞

0

(d

dx

)n

{Lαm(x)}e−xxα+ndx � 0.

Next, by (20.3) and the same computations as before,

‖Lαn‖2

u � 〈Lαn, Lα

n〉u � 1n!

∫ ∞

0e−xxα+ndx � �(α + n + 1)

n!for n � 1, 2, . . . , and that the same formula is valid for n � 0 follows from thedefinition of Lα

0 . �

Theorem 20.2. { 1‖Lα

n‖uLα

n : n � 0, 1, 2, . . .} is an orthonormal basis forL2u(0, ∞).

Proof. In view of Proposition 20.1, we only need to prove that if g ∈ L2u(0, ∞)

is such that

〈g, Lαn〉u � 0, n � 0, 1, 2, . . . , (20.4)

then g � 0 a.e. on (0, ∞). Now, for n � 0, 1, 2, . . . , we get, by Lemma 18.5,

xn �n∑

k�0ckL

αk (x), x > 0, (20.5)

20. Laguerre Polynomials 97

where c0, c1, c2, . . . , cn are constants. Thus, for n � 0, 1, 2, . . . , we get, by (20.4)and (20.5),∫ ∞

0g(x)xnxαe−xdx �

n∑k�0

ck

∫ ∞

0g(x)Lα

k (x)xαe−xdx � 0. (20.6)

Let x � y2. Then, by (20.6), we get, for n � 0, 1, 2, . . . ,

2∫ ∞

0g(y2)y2ny2α+1e−y2

dy � 0 ⇒ 2∫ ∞

0g(y2)y2n|y|2α+1e−y2

dy � 0

⇒∫ ∞

−∞g(y2)y2n|y|2α+1e−y2

dy � 0. (20.7)

Of course, ∫ ∞

−∞g(y2)y2n+1|y|2α+1e−y2

dy � 0, n � 0, 1, 2, . . . . (20.8)

So, by (20.7) and (20.8),∫ ∞

−∞g(y2)yn|y|2α+1e−y2

dy � 0, n � 0, 1, 2, . . . . (20.9)

Let P be any polynomial of degree k. Then

P (y) �k∑

n�0any

n, y ∈ R,

where a0, a1, a2, . . . , ak are constants. So, by (20.9),∫ ∞

−∞g(y2)P (y)|y|2α+1e−y2

dy � 0. (20.10)

Also, for all ξ in R,∫ ∞

−∞|g(y2)||y|2α+1e|yξ |e−y2

dy

≤{∫ ∞

−∞|g(y2)|2|y|2α+1e−y2

dy

} 12{∫ ∞

−∞|y|2α+1e2|yξ |e−y2

dy

} 12

�{∫ ∞

0|g(x)|2xαe−xdx

} 12{∫ ∞

−∞|y|2α+1e−2|yξ |e−y2

dy

} 12

< ∞. (20.11)

Thus, by Lemma 18.6, (20.10), and (20.11), g(y2)|y|2α+1 � 0 for almost all y inR. Therefore, g � 0 a.e. on (0, ∞). �

Theorem 20.3. For each fixed positive number x,∞∑

n�0Lα

n(x)zn � e− xz1−z

(1 − z)α+1 , |z| < 1,

where the series is uniformly and absolutely convergent on every compact subsetof {z ∈ C : |z| < 1}.

98 20. Laguerre Polynomials

Remark 20.4. We calle− xz

1−z

(1 − z)α+1 the generating function of the Laguerre

polynomials Lαn , n � 0, 1, 2, . . . .

Proof of Theorem 20.3.

�

x

� rγ

||||||||||||||

Fig. 3

Let γ be a circle with center at x and lying inside the right half plane (Fig. 3).Now,

∞∑n�0

Lαn(x)zn �

∞∑n�0

x−αex

n!

(d

dx

)n

(e−xxα+n)zn

� x−αex

2πi

∞∑n�0

zn

∫γ

e−ζ ζ α+n

(ζ − x)n+1 dζ, (20.12)

where the principal branch of ζ α+n is taken, i.e.,

ζ α+n � e(α+n)Log−πζ

and

Log−πζ � ln |ζ | + iArg−πζ, −π < Arg−πζ < π.

Next, for n � 1, 2, . . . ,∣∣∣∣ e−ζ ζ α+n

(ζ − x)n+1

∣∣∣∣ � e−Reζ e(α+n) ln |ζ |

rn+1

≤ e−(x−r)(x + r)α+n

rn+1

� e−(x−r) (x + r)α

r

(x + r

r

)n

.

20. Laguerre Polynomials 99

Then, for all z in C with |z| < rx+r

, the series∞∑

n�0

zne−ζ ζ α+n

(ζ − x)n+1 is uniformly and

absolutely convergent with respect to z on {z ∈ C : |z| < rx} and ζ on γ , whererx is any number in (0, r

x+r). Therefore, by (20.12),

∞∑n�0

Lαn(x)zn � x−αex

2πi

∫γ

e−ζ ζ α

ζ − x

∞∑n�0

(zζ

ζ − x

)n

dζ (20.13)

for |z| < rx . But ∣∣∣∣ zζ

ζ − x

∣∣∣∣ ≤ |z|(x + r)r

< 1,

and hence, by (20.13),∞∑

n�0Lα

n(x)zn � x−αex

2πi

∫γ

e−ζ ζ α

ζ − x

11 − zζ

ζ−x

dζ

� x−αex

2πi

∫γ

e−ζ ζ α

ζ − x − zζdζ

� x−αex

2πi

∫γ

e−ζ ζ α

(1 − z)ζ − xdζ

� x−αex

1 − z

12πi

∫γ

e−ζ ζ α

ζ − x1−z

dζ (20.14)

for |z| < rx . But for sufficiently small z, x1−z

is inside γ . So, by (20.14),

∞∑n�0

Lαn(x)zn � x−αex

1 − ze− x

1−z

(x

1 − z

)α

� e− xz

1−z

(1 − z)α+1

for sufficiently small z. Now, e− xz

1−z

(1 − z)α+1 is an analytic function on {z ∈ C : |z| <

1}. Thus, by the principle of analytic continuation,

∞∑n�0

Lαn(x)zn � e

−xz1−z

(1 − z)α+1 , |z| < 1,

and Theorem 20.3 is proved. �

21Hermite Functions on C

We can now give in this and the next two chapters a self-contained treatment ofHermite functions on C, i.e., the Fourier–Wigner transforms of Hermite functionson R, which we need in this book. A good reference for these topics is Chapter 1of the book [29] by Thangavelu.

For j, k � 0, 1, 2, . . . , we define the function ej,k on C by

ej,k(z) � V (ej , ek)(q, p) (21.1)

for all z � q + ip in C.

Proposition 21.1. {ej,k : j, k � 0, 1, 2, . . .} is an orthonormal set in L2(R2).

Proof. By (21.1), Theorem 19.3, the Moyal identity for the Fourier–Wignertransform, and the Plancherel theorem, we get, for all nonnegative integersj1, j2, k1, and k2,

〈ej1,k1 , ej2,k2〉 � 〈V (ej1 , ek1 ), V (ej2 , ek2 )〉� 〈ej1 , ej2〉〈ek1 , ek2〉 � 0

unless j1 � j2 and k1 � k2; and if j1 � j2 and k1 � k2, then

〈ej1,k1 , ej2,k2〉 � 1. �

Theorem 21.2. {ej,k : j, k � 0, 1, 2, . . .} is an orthonormal basis for L2(R2).

Proof. In view of Proposition 21.1, we only need to prove that if f ∈ L2(R2) issuch that

〈f, ej,k〉 � 0, j, k � 0, 1, 2, · · · , (21.2)

102 21. Hermite Functions on C

then f � 0 a.e. on R2. To this end, we let g ∈ L2(R2) be such that g � f . Then,

by Theorem 4.4, Lemma 14.6, and (21.1),

〈Wgej , ek〉 � (2π )−12

∫ ∞

−∞

∫ ∞

−∞g(x, ξ )W (ej , ek)(x, ξ )dx dξ

� (2π )−12

∫ ∞

−∞

∫ ∞

−∞f (q, p)V (ej , ek)(q, p)dq dp

� (2π )−12

∫ ∞

−∞

∫ ∞

−∞f (q, p)ej,k(q, p)dq dp (21.3)

for j, k � 0, 1, 2, . . . . By Theorem 19.3, (21.2), and (21.3),

Wgej � 0, j � 0, 1, 2, . . . . (21.4)

Now, let h ∈ L2(R) and ε be any positive number. Then, by Theorem 19.3, we canfind a finite linear combination

∑ajk

ejkof the ej ’s such that∥∥∥∑ ajk

ejk− h

∥∥∥L2(R)

< ε. (21.5)

So, by (21.4) and (21.5),

‖Wgh‖L2(R) ≤∥∥∥Wg

(h −

∑ajk

ejk

)∥∥∥L2(R)

+∥∥∥Wg

(∑ajk

ejk

)∥∥∥L2(R)

≤ ε‖Wg‖∗.

Since ε is arbitrary, it follows that

Wgh � 0, h ∈ L2(R).

Therefore, Wg � 0. But by part (v) of Theorem 7.5,

(2π )−12 ‖g‖L2(R2) � ‖Wg‖HS � 0.

Thus, g � 0 a.e. on R2. Consequently, f � 0 a.e. on R

2. �

22Vector Fields on C

In this chapter, we introduce the analogues of the annihilation operator, the cre-ation operator, and the harmonic oscillator for the Hermite functions ej,k, j, k �0, 1, 2, . . . ,onR

2. To do this, we define the vector fieldsZ andZ onC, respectively,by

Z � ∂

∂z+ 1

2z

and

Z � ∂

∂z− 1

2z,

where∂

∂z� ∂

∂q− i

∂

∂p

and∂

∂z� ∂

∂q+ i

∂

∂p.

We also define the vector field L on C by

L � −12

(ZZ + ZZ).

Theorem 22.1. For all z in C,

(i) (Zej,k)(z) � i(2k) 12 ej,k−1(z), j � 0, 1, 2, . . . , k � 1, 2, . . . ;

(ii) (Zej,k)(z) � i(2k + 2) 12 ej,k+1(z), j, k � 0, 1, 2, . . . .

104 22. Vector Fields on C

Proof. By (21.1), we get, for all q and p in R,

∂ej,k

∂q(q, p) � i(2π )−

12

∫ ∞

−∞yeiqyej

(y + p

2

)ek

(y − p

2

)dy. (22.1)

Now, using (22.1) and the formula

2y �(y + p

2

)+(y − p

2

),

we get, for all q and p in R,

∂ej,k

∂q(q, p) � J (+)(q, p) + J (−)(q, p), (22.2)

where

J (±)(q, p) � i

2(2π )−

12

∫ ∞

−∞eiqy

(y ± p

2

)ej

(y + p

2

)ek

(y − p

2

)dy. (22.3)

Next, for all q and p in R,

i∂ej,k

∂p(q, p) � K (1)(q, p) − K (2)(q, p), (22.4)

where

K (1)(q, p) � i

2(2π )−

12

∫ ∞

−∞eiqye′

j

(y + p

2

)ek

(y − p

2

)dy (22.5)

and

K (2)(q, p) � i

2(2π )−

12

∫ ∞

−∞eiqyej

(y + p

2

)e′k

(y − p

2

)dy. (22.6)

Now, by Theorem 19.1 and (19.3), we get, for k � 0, 1, 2, . . . ,((x − d

dx

)ek

)(x) � (2k + 2)

12 ek+1(x), x ∈ R, (22.7)

and, for k � 1, 2, . . . ,((x + d

dx

)ek

)(x) � (2k)

12 ek−1(x), x ∈ R. (22.8)

So, by (22.2)–(22.8), we get, for j � 0, 1, 2, . . . and k � 1, 2, . . . ,

∂ej,k

∂z(z) � (J (+)(q, p) − K (1)(q, p)) + (J (−)(q, p) + K (2)(q, p))

� i

2(2π )−

12

{∫ ∞

−∞eiqy(2j + 2)

12 ej+1

(y + p

2

)ek

(y − p

2

)dy

+∫ ∞

−∞eiqy(2k)

12 ej

(y + p

2

)ek−1

(y − p

2

)dy

}

� i

2{(2j + 2)

12 ej+1,k(z) + (2k)

12 ej,k−1(z)}, z ∈ C. (22.9)

22. Vector Fields on C 105

Also, by (22.2)–(22.8), we get, for j � 1, 2, . . . and k � 0, 1, 2, . . . ,

∂ej,k

∂z(z) � (J (+)(q, p) + K (1)(q, p)) + (J (−)(q, p) − K (2)(q, p))

� i

2(2π )−

12

{∫ ∞

−∞eiqy(2j )

12 ej−1

(y + p

2

)ek

(y − p

2

)dy

+∫ ∞

−∞eiqy(2k + 2)

12 ej

(y + p

2

)ek+1

(y − p

2

)dy

}

� i

2{(2j )

12 ej−1,k(z) + (2k + 2)

12 ej,k+1(z)}, z ∈ C. (22.10)

Now, by (21.1), (22.5), (22.6), and an integration by parts,

12qej,k(q, p) � − i

2(2π )

12

∫ ∞

−∞

{∂

∂yeiqy

}ej

(y + p

2

)ek

(y − p

2

)dy

� K (1)(q, p) + K (2)(q, p), q, p ∈ R. (22.11)

Using (21.1), (22.3), and the formula

p �(y + p

2

)−(y − p

2

),

we get

i

2pej,k(q, p) � J (+)(q, p) + J (−)(q, p), q, p ∈ R. (22.12)

So, by (22.1), (22.11), and (22.12), we get, for j � 1, 2, . . . and k � 0, 1, 2, . . . ,

12zej,k(z) � i

2(2π )−

12

{∫ ∞

−∞eiqy(2j )

12 ej−1

(y + p

2

)ek

(y − p

2

)dy

−∫ ∞

−∞eiqy(2k + 2)

12 ej

(y + p

2

)ek+1

(y − p

2

)dy

}

� i

2{(2j )

12 ej−1,k(z) − (2k + 2)

12 ej,k+1(z)} (22.13)

for all z in C. By (22.1), (22.11), and (22.12), we also get, for j � 0, 1, 2, . . . andk � 1, 2, . . . ,

12zej,k(z) � i

2(2π )−

12

{−∫ ∞

−∞eiqy(2j + 2)

12 ej+1

(y + p

2

)ek

(y − p

2

)dy

+∫ ∞

−∞eiqy(2k)

12 ej

(y + p

2

)ek−1

(y − p

2

)dy

}

� i

2{(2k)

12 ej,k−1(z) − (2j + 2)

12 ej+1,k(z)} (22.14)

for all z in C. Therefore, by (22.9) and (22.14),

Zej,k � i(2k)12 ej,k−1, j � 0, 1, 2, . . . , k � 1, 2, . . . ,

106 22. Vector Fields on C

and by (22.10) and (22.13),

Zej,k � i(2j + 2)12 ej,k+1, j � 1, 2, . . . , k � 0, 1, 2, . . . .

That the preceding formula is also true for e0,k , k � 0, 1, 2, . . . , should by nowbe obvious and is left as an exercise. �

From Theorem 22.1, we get the following theorem.

Theorem 22.2. Lej,k � (2k + 1)ej,k, j, k � 0, 1, 2, . . . .

Proof. By Theorem 22.1, we get, for j � 0, 1, 2, . . . and k � 1, 2, . . . ,

ZZej,k � i(2k + 2)12 Zej,k+1 � −(2k + 2)ej,k

and

ZZej,k � i(2k)12 Zej,k−1 � −2kej,k.

Thus, for j � 0, 1, 2, . . . and k � 1, 2, . . . ,

Lej,k � −12

(ZZ + ZZ)ej,k � (2k + 1)ej,k.

That the preceding formula is also true for ej,0, j � 0, 1, 2, . . . , follows from thefact that

Zej,0 � 0, j � 0, 1, 2, . . . ,

and is left as an exercise. �

Remark 22.3. In view of Theorem 22.1, we call Z and Z the annihilation oper-ator and the creation operator, respectively, for the Hermite functions ej,k, j, k �0, 1, 2, . . . , on C. Theorem 22.2 says that for k � 0, 1, 2, . . . , 2k + 1 is aneigenvalue of the “harmonic oscillator” L and the Hermite functions ej,k, j, k �0, 1, 2, . . . , on C are eigenfunctions of L corresponding to the eigenvalue 2k + 1.The partial differential operator L is in fact a “Laplacian” with variable coefficientson C.

23Laguerre Formulas for HermiteFunctions on C

We are now interested in expressing some classes of Hermite functions on C interms of Laguerre polynomials. We begin with a formula.

Theorem 23.1. (Mehler’s Formula) For all x and y in R and all w in C with|w| < 1,

∞∑k�0

hk(x)hk(y)2kk!

wk � (1 − w2)−12 e

− 12

1+w21−w2 (x2+y2)+ 2w

1−w2 xy,

where the series is uniformly and absolutely convergent on {w ∈ C : |w| < 1}.Remark 23.2. To be specific, we use the principal branch of (1 − w2)− 1

2 , i.e.,

(1 − w2)−12 � e− 1

2 Log−π (1−w2),

where

Log−πζ � ln |ζ | + iArg−πζ, −π < Arg−πζ < π.

Thus, (1 − w2)− 12 is analytic on the cut plane C − {x ∈ R : x ≤ −1 or x ≥ 1}

and, for any w in R with |w| < 1, we get

(1 − w2)−12 � e− 1

2 ln(1−w2) > 0.

Proof of Theorem 23.1. We begin with the formula

e−x2 � 1√π

∫ ∞

−∞e−u2+2ixudu, x ∈ R. (23.1)

108 23. Laguerre Formulas for Hermite Functions on C

So, for k � 0, 1, 2, . . . ,

hk(x) � e− x22 (−1)kex2

(d

dx

)k 1√π

∫ ∞

−∞e−u2+2ixudu

� e− x22 (−1)kex2 1√

π

∫ ∞

−∞(2iu)ke−u2+2ixudu

� 1√π

(−2i)kex22

∫ ∞

−∞uke−u2+2ixudu, x ∈ R. (23.2)

Hence, for any x and y in R and any w in (−1, 1), we have, by (23.1) and (23.2),∞∑

k�0

hk(x)hk(y)2kk!

wk

� 1π

e12 (x2+y2)

∞∑k�0

(−2i)2k

2kk!

∫ ∞

−∞

∫ ∞

−∞e−u2−v2+2ixu+2iyvukvkwkdu dv

� 1π

e12 (x2+y2)

∞∑k�0

∫ ∞

−∞

∫ ∞

−∞

(−2uvw)k

k!e−u2−v2+2ixu+2iyvdu dv

� 1π

e12 (x2+y2)

∫ ∞

−∞

∫ ∞

−∞e−2uvwe−u2−v2+2ixu+2iyvdu dv

� 1π

e12 (x2+y2)

∫ ∞

−∞e−u2+2ixu

(∫ ∞

−∞e−v2+2iyv−2uvwdv

)du

� 1π

e12 (x2+y2)

∫ ∞

−∞e−u2+2ixu

(∫ ∞

−∞e−(v2+2uvw+u2w2)+u2w2+e2iyv

dv

)du

� 1π

e12 (x2+y2)

∫ ∞

−∞e−u2+2ixu+u2w2

(∫ ∞

−∞e−v2

e2iy(v−uw)dv

)du

� 1√π

e12 (x2+y2)e−y2

∫ ∞

−∞e−u2+2ixu+u2w2−2iyuwdu. (23.3)

Therefore, for any x and y in R and any w in (−1, 1), we have, by (23.1) and(23.3),

∞∑k�0

hk(x)hk(y)2kk!

wk � 1√π

e12 (x2−y2)

∫ ∞

−∞e−(1−w2)u2+2i(x−yw)udu. (23.4)

Let (1 − w2) 12 u � t . Then, for any x and y in R and any w in (−1, 1), we have,

by (23.1) and (23.4),∞∑

k�0

hk(x)hk(y)2kk!

wk � 1√π

e12 (x2−y2)

∫ ∞

−∞e−t2+2i

x−yw

(1−w2)1/2 tdt(1 − w2)−

12

� e12 (x2−y2)e

− (x−yw)2

1−w2 (1 − w2)−12

� (1 − w2)−12 e

− 12

1+w21−w2 (x2+y2)+ 2w

1−w2 xy. (23.5)

23. Laguerre Formulas for Hermite Functions on C 109

Now, the last term in (23.5) is analytic on the cut plane C−{x ∈ R : x ≤ −1 or x ≥1}, and the first term in (23.5) is analytic on the open disk {w ∈ C : |w| < 1}. Thetwo terms are equal on (−1, 1). Hence, by the principle of analytic continuation,the proof of the theorem is complete. �

Here is a formula expressing the Hermite functions ej,j , j � 0, 1, 2, . . . , on C

in terms of Laguerre polynomials.

Theorem 23.3. For j � 0, 1, 2, . . . and any z in C,

ej,j (z) � (2π )−12 L0

j

(12|z|2)

e− 14 |z|2 .

Proof. For j � 0, 1, 2, . . . , y, p in R, and all r on R with |r| < 1, we get, byMehler’s formula in Theorem 23.1,

∞∑k�0

ek

(y + p

2

)ek

(y − p

2

)rk �

∞∑k�0

hk(y + p

2 )hk

(y − p

2

)2kk!

√π

rk

� 1√π

(1 − r2)−12 e

− 12

1+r21−r2 (2y2+ p2

2 )+ 2r

1−r2 (y2− p24 )

� 1√π

(1 − r2)−12 e

− 1+r21−r2 y2− 1+r2

1−r2p24 + 2r

1−r2 y2− 2r

1−r2p24

� 1√π

(1 − r2)−12 e

− 1−2r+r21−r2 y2− 1+2r+r2

1−r2p24

� 1√π

(1 − r2)−12 e− 1−r

1+ry2− 1+r

1−r

p24 . (23.6)

Taking the inverse Fourier transform of the first and last terms in (23.6) with respectto y, we get, for all z � q + ip in C and all r in R with |r| < 1,

∞∑k�0

ek,k(z)rk � 1√π

(1 − r2)−12 e− 1+r

1−r

p24

1√2π

∫ ∞

−∞eiqye− 1−r

1+ry2

dy

� 1√2π

11 − r

e− 12 |z|2 r

1−r e− 14 |z|2 . (23.7)

So, by Theorem 20.3 and (23.7),∞∑

k�0ek,k(z)rk � 1√

2π

∞∑k�0

L0k

(12|z|2)

rke− 14 |z|2 ,

and hence

ek,k(z) � (2π )−1/2L0k

(12|z|2)

e− 14 |z|2 , z ∈ C,

for k � 0, 1, 2, . . . . �

Now, we can give another set of formulas expressing Hermite functions on C interms of Laguerre polynomials.

110 23. Laguerre Formulas for Hermite Functions on C

Theorem 23.4. For j, k � 0, 1, 2, . . . and any z in C,

(i) ej+k,j (z) � (2π )−1/2{ j !(j+k)! }

12 ( i√

2)k(z)kLk

j ( 12 |z|2)e− 1

4 |z|2 ,(ii) ej,j+k(z) � (2π )−1/2{ j !

(j+k)! }12 ( i√

2)kzkLk

j ( 12 |z|2)e− 1

4 |z|2 .

Remark 23.5. Let k � 0. Then Theorem 23.4 becomes Theorem 23.3.

Remark 23.6. We note that, for j, k � 0, 1, 2, . . . ,

ej,j+k(z) � V (ej , ej+k)(z) � W (ej , ej+k )(z) � (W (ej+k, ej ))(z)

� (W (ej+k, ej ))(z) � (W (ej+k, ej ))(−z)� (V (ej+k, ej ))(−z) � ej+k,j (−z), z ∈ C. (23.8)

Thus, if part (i) of Theorem 23.4 is true, then, by (23.8),

ej,j+k(z) � (2π )−1/2{

j !(j + k)!

} 12(

− i√2

)k

(−z)kLkj

(12|z|2)

e− 14 |z|2

� (2π )−1/2{

j !(j + k)!

} 12(

i√2

)k

zkLkj

(12|z|2)

e− 14 |z|2 , z ∈ C.

Thus, to prove Theorem 23.4, we only need to prove part (i).

The following lemma will be used in the proof of Theorem 23.4.

Lemma 23.7. For α > −1 and k � 1, 2, . . . ,

d

dx(Lα

k (x)) � −Lα+1k−1 (x), x > 0.

Proof. By (20.2),

d

dx(Lα

k (x)) � d

dx

k∑j�0

�(k + α + 1)�(k − j + 1)�(j + α + 1)

(−x)j

j !

� −k∑

j�1

�(k + α + 1)�(k − j + 1)�(j + α + 1)

(−x)j−1

(j − 1)!

� −k−1∑l�0

�(k − 1 + α + 1 + 1)�(k − 1 − l + 1)�(l + α + 1 + 1)

(−x)l

l!

� −Lα+1k−1 (x), x > 0. �

Proof of Theorem 23.4. In view of Remark 23.6, it is enough to prove part (i).By Theorem 23.3, the formula is true if k � 0. Suppose that the formula is truefor all nonnegative integers j and all nonnegative integers k with k ≤ l, say. Then,by part (i) of Theorem 22.1,

ej+k+1,j � −i(2j + 2)−12 Zej+k+1,j+1. (23.9)

23. Laguerre Formulas for Hermite Functions on C 111

Now, by the induction hypothesis, we have, for all z ∈ C,

ej+k+1,j+1(z) � (2π )−1/2{

(j + 1)!(j + k + 1)!

} 12(

i√2

)k

(z)kLkj+1

(12|z|2)

e− 14 |z|2 .

(23.10)Let fj be the function on C defined by

fj (z) � (z)kLkj+1

(12|z|2)

e− 14 |z|2 , z ∈ C. (23.11)

Then, for k ≥ 1,∂fj

∂q(z) � (z)k

{(∂Lk

j+1)(

12|z|2)

qe− 14 |z|2 + Lk

j+1

(12|z|2)(

−12q

)e− 1

4 |z|2}

+ k(z)k−1Lkj+1

(12|z|2)

e− 14 |z|2 , z ∈ C, (23.12)

and

i∂fj

∂p(z) � (z)k

{(∂Lk

j+1)(

12|z|2)

ipe− 14 |z|2 + Lk

j+1

(12|z|2)(

− i

2p

)e− 1

4 |z|2}

+ k(z)k−1Lkj+1

(12|z|2)

e− 14 |z|2 , z ∈ C. (23.13)

So, by (23.11), (23.12), and (23.13),

(Zfj )(z) � (z)k+1(∂Lkj+1)

(12|z|2)

e− 14 |z|2 , z ∈ C. (23.14)

It is easy to see that (23.14) is also true for k � 0. Thus, by (23.9)–(23.11) and(23.14),

ej+k+1,j (z)

� (2π )−1/2(−i)(2j + 2)−12

{(j + 1)!

(j + k + 1)!

} 12

×(

i√2

)k

(z)k+1(∂Lkj+1)

(12|z|2)

e− 14 |z|2

(23.15)

for all z in C. But, by (23.15) and Lemma 23.7,

ej+k+1,j (z)

� (2π )−1/2i(2j + 2)−12

{(j + 1)!

(j + k + 1)!

} 12(

i√2

)k

(z)k+1Lk+1j

(12|z|2)

e− 14 |z|2

� (2π )−1/2{

j !(j + k + 1)!

} 12(

i√2

)k+1

(z)k+1Lk+1j

(12|z|2)

e− 14 |z|2 , z ∈ C,

and the proof is complete. �

24Weyl Transforms on L2(R) with RadialSymbols

For Weyl transforms on L2(R) with radial symbols, we can give a sufficient andnecessary condition for boundedness. A sufficient and necessary condition forcompactness can also be given. In order to obtain these conditions, we need theWigner transforms of Hermite functions on R.

For j, k � 0, 1, 2, . . . , we define the function ψj,k on R2 by

ψj,k(x, ξ ) � W (ej , ek)(x, ξ ), x, ξ ∈ R.

Theorem 24.1. For j, k � 0, 1, 2, . . . , we get, for any ζ � x + iξ ,

(i) ψj+k,j (ζ ) � 2(−1)j (2π )− 12

{j !

(j+k)!

} 12 (

√2)k(ζ )kLk

j (2|ζ |2)e−|ζ |2 ,

(ii) ψj,j+k(ζ ) � 2(−1)j (2π )− 12

{j !

(j+k)!

} 12 (

√2)kζ kLk

j (2|ζ |2)e−|ζ |2 .

Proof. It is easy to see from Proposition 18.1 that ek is even or odd if k is,respectively, even or odd. Thus, for k � 0, 1, 2, . . . ,

ek(x) � ek(−x) � (−1)kek(x), x ∈ R. (24.1)

So, by (24.1), we get

V (ej , ek)(q, p) � (2π )−1/2∫ ∞

−∞eiqyej

(y + p

2

)ek

(y − p

2

)dy

� (−1)kV (ej , ek)(q, p), q, p ∈ R. (24.2)

Thus, by Lemma 14.4 and (24.2),

W (ej , ek)(x, ξ ) � (−1)k2V (ej , ek)(−2ξ, 2x)� (−1)k2ej,k(−2ξ, 2x), x, ξ ∈ R. (24.3)

114 24. Weyl Transforms on L2(R) with Radial Symbols

For all ζ � x + iξ in C, we get, by (24.3),

ψj,k(ζ ) � (−1)k2ej,k(2iζ ). (24.4)

Hence, by Theorem 23.4 and (24.4),

ψj+k,j (ζ ) � 2(−1)j ej+k,j (2iζ )

� 2(−1)j (2π )−1/2{

j !(j + k)!

} 12(

i√2

)k

2k(−i)k(ζ )kLkj (2|ζ |2)e−|ζ |2

� 2(−1)j (2π )−1/2{

j !(j + k)!

} 12

(√

2)k(ζ )kLkj (2|ζ |2)e−|ζ |2 , ζ ∈ C,

and

ψj,j+k(ζ ) � 2(−1)j+kej,j+k(2iζ )

� 2(−1)j+k(2π )−1/2{

j !(j + k)!

} 12(

i√2

)k

2kikζ kLkj (2|ζ |2)e−|ζ |2

� 2(−1)j (2π )−1/2{

j !(j + k)!

} 12

(√

2)kζ kLkj (2|ζ |2)e−|ζ |2 , ζ ∈ C. �

To see the role of the function ψj,k , j, k � 0, 1, 2, . . . , in the study of the Weyltransform, we use the following result in the paper [24] by Simon. We skip theproof.

Theorem 24.2. Let f be any function in S(R). Then

f �∞∑

k�0〈f, ek〉ek,

where the convergence is in S(R), i.e., for all nonnegative integers α and β,

supx∈R

∣∣∣∣∣xα(∂βf )(x) −N∑

k�0〈f, ek〉xα(∂βek)(x)

∣∣∣∣∣→ 0

as N → ∞.

Let σ be a tempered function on R2. Suppose that σ is radial, i.e.,

σ (x, ξ ) � σ (r), x, ξ ∈ R,

where r �√

x2 + ξ 2. Now, by Theorem 24.1, for j, k � 0, 1, 2, . . . and j ≥ k,

ψj,k(x, ξ ) � 2(−1)k(2π )−12

(k!j !

) 12

(√

2)j−k(ζ )j−kLj−k

k (2|ζ |2)e−|ζ |2 (24.5)

for all ζ � x + iξ in C. Now, for all f and g in S(R), we can use (12.1), Theorem24.2, the bilinearity of W , the proof of Theorem 12.1, and (24.5) to get

(Wσf )(g) � (2π )−1/2σ (W (f, g))

24. Weyl Transforms on L2(R) with Radial Symbols 115

� (2π )−1/2σ

(W

(f,

∞∑k�0

〈g, ek〉ek

))

� (2π )−1/2σ

( ∞∑k�0

〈g, ek〉W (f, ek)

)

� (2π )−1/2∞∑

k�0〈g, ek〉σ (W (f, ek))

� (2π )−1/2∞∑

k�0

∞∑j�0

〈g, ek〉〈f, ej 〉σ (ψj,k). (24.6)

Remark 24.3. We have shown only that (24.6) is valid in the sense that we sumwith respect to j first and then with respect to k.

Now, for j, k � 0, 1, 2, . . . and j ≥ k, we get, by (24.5),

σ (ψj,k) �∫ ∞

−∞

∫ ∞

−∞σ (x, ξ )ψj,k(x, ξ )dx dξ

�∫ 2π

0

∫ ∞

0σ (ρ)2(−1)k(2π )−

12

(k!j !

) 12

(√

2)j−kρj−ke−i(j−k)θLj−k

k (2ρ2)

× e−ρ2ρdρdθ

�∫ 2π

0e−i(j−k)θdθ

∫ ∞

0σ (ρ)

(k!j !

) 12

212 (j−k)+1(−1)k(2π )−

12 ρj−k+1

× Lj−k

k (2ρ2)e−ρ2dρ. (24.7)

Thus, by (24.7),

σ (ψj,k) � 0, j � k. (24.8)

So, by (24.6), (24.7), and (24.8), we get

(Wσf )(g) � (2π )−1/2∞∑

k�0〈f, ek〉〈g, ek〉σ (ψk,k), (24.9)

where

σ (ψk,k) � (2π )12 (−1)k2

∫ ∞

0σ (ρ)L0

k(2ρ2)e−ρ2ρdρ, k � 0, 1, 2, . . . , (24.10)

for all f and g in S(R).

Remark 24.4. The convergence in (24.9) is valid in the sense that the sequenceof partial sums of the series is convergent.

Theorem 24.5. Let σ be a tempered function on R2. Suppose that σ is radial,

i.e.,

σ (x, ξ ) � σ (ρ), x, ξ ∈ R,

116 24. Weyl Transforms on L2(R) with Radial Symbols

where ρ �√

x2 + ξ 2. For k � 0, 1, 2, . . . , let

ak �∫ ∞

0σ (ρ)L0

k(2ρ2)e−ρ2ρdρ.

Then

(i) Wσ is a bounded linear operator from L2(R) into L2(R) if and only if thesequence {ak}∞k�0 is bounded,

(ii) Wσ is a compact operator from L2(R) into L2(R) if and only if ak → 0 ask → ∞.

Proof. Suppose that there is a positive constant M such that

|ak| ≤ M, k � 0, 1, 2, . . . . (24.11)

Then, for all f and g in S(R), we get, by (24.9) and (24.10),

(Wσf )(g) � (2π )−1/2∞∑

k�0(2π )

12 (−1)k2ak〈f, ek〉〈g, ek〉, (24.12)

and hence, by the Schwarz inequality and (24.11),

|(Wσf )(g)| ≤ 2M

( ∞∑k�0

|〈f, ek〉|2) 1

2( ∞∑

k�0|〈g, ek〉|2

) 12

� 2M‖f ‖L2(R)‖g‖L2(R).

Thus,

‖Wσf ‖L2(R) ≤ 2M‖f ‖L2(R), f ∈ S(R).

Therefore, by a limiting argument, Wσ is a bounded linear operator from L2(R)into L2(R). Conversely, suppose that Wσ is a bounded linear operator from L2(R)into L2(R). Then, by (24.12),

(Wσej )(ej ) � (−1)j2aj , j � 0, 1, 2, . . . . (24.13)

Therefore, by (24.13) and the Schwarz inequality,

|aj | � |(Wσej )(ej )| � |〈Wσej , ej 〉|≤ ‖Wσej‖L2(R)‖ej‖L2(R) ≤ ‖Wσ‖∗, j � 0, 1, 2, . . . ,

and part (i) is proved. To prove part (ii), suppose that ak → 0 as k → ∞. ForN � 0, 1, 2, . . . , we define the bounded linear operator WN : L2(R) → L2(R) by

〈WNf, g〉 �N∑

k�0(−1)k2ak〈f, ek〉〈g, ek〉 (24.14)

for all f and g in L2(R). Then

WNf �N∑

k�0(−1)k2ak〈f, ek〉ek, f ∈ L2(R),

24. Weyl Transforms on L2(R) with Radial Symbols 117

for N � 0, 1, 2, . . . . Therefore, WN : L2(R) → L2(R) is a finite rank operator,and hence compact for N � 0, 1, 2, . . . . Now, for all f and g in S(R), by (24.12)and (24.14),

|〈(Wσ − WN )f, g〉| ≤ 2( supk≥N+1

|ak|)‖f ‖L2(R)‖g‖L2(R) (24.15)

for N � 0, 1, 2, . . . . Then, by (24.15),

‖Wσ − WN‖∗ ≤ 2( supk≥N+1

|ak|) → 0

as N → ∞. Therefore, Wσ is the limit in B(L2(R)) of a sequence of compactoperators from L2(R) into L2(R). So, Wσ is a compact operator from L2(R) intoL2(R). Conversely, suppose that Wσ is a compact operator from L2(R) into L2(R).Since ej → 0 weakly in L2(R) as j → ∞, it follows that Wσej → 0 in L2(R) asj → ∞. But by (24.13) and the Schwarz inequality,

2|aj | ≤ ‖Wσej‖L2(R)‖ej‖L2(R)

� ‖Wσej‖L2(R) → 0

as j → ∞. �

25Another Fourier Transform

We compute in this chapter the Fourier transform of a function related to the La-guerre polynomial of degree k and order 0, k � 0, 1, 2, . . . . This Fourier transformwill be used in the next chapter to obtain a criterion for the compactness of theWeyl transform on L2(R). We begin with some complex analysis.

Let f be a continuous function on [0, ∞) such that we can find a positiveconstant A and a constant c for which

|f (t)| ≤ Aect , t ≥ 0. (25.1)

Then we define the function F on the region {z ∈ C : Rez > c} by

F (z) �∫ ∞

0e−ztf (t)dt, Rez > c. (25.2)

The function F is in fact the Laplace transform of f .

Theorem 25.1. F is analytic on the region {z ∈ C : Rez > c}.To prove Theorem 25.1, we need some preparations.

Lemma 25.2. Let ϕ be an entire function. Let g be a continuous function on aclosed and bounded interval [a, b] and let G be the function on C defined by

G(z) �∫ b

a

g(t)ϕ(zt)dt, z ∈ C. (25.3)

Then G is continuous on C.

Proof. Let z0 ∈ C. Let {zk}∞k�1 be a sequence of complex numbers such thatzk → z0 as k → ∞. Let D be a fixed disk centered at z0 such that z0t ∈ D for all t

120 25. Another Fourier Transform

in [a, b]. Then ϕ is uniformly continuous on D. So, for any given positive numberε, there exists a positive number δ such that

|ϕ(z) − ϕ(w)| < ε

for all z and w in D with |z − w| < δ. Thus, there exists a positive integer K suchthat

k ≥ K ⇒ |ϕ(zkt) − ϕ(z0t)| < ε

for all t in [a, b]. So,

g(t)ϕ(zkt) → g(t)ϕ(z0t)

uniformly with respect to t on [a, b] as k → ∞. Therefore, G(zk) → G(z0) ask → ∞. �

Lemma 25.3. The function G on C defined by (25.3) is entire.

Proof. Let C be a simple closed curve in C. Then∫C

G(z)dz �∫

C

(∫ b

a

g(t)ϕ(zt)dt

)dz

�∫ b

a

(∫C

g(t)ϕ(zt)dz

)dt �

∫ b

a

g(t)(∫

C

ϕ(zt)dz

)dt

� 0.

So, by Lemma 25.2 and Morera’s theorem, G is entire. �

Proof of Theorem 25.1. We see that by (25.1) and (25.2), F (z) exists for all z

in the region {z ∈ C : Rez > c}. Indeed,∫ ∞

0|e−ztf (t)|dt �

∫ ∞

0e−(Rez)t |f (t)|dt

≤ A

∫ ∞

0e−(Rez−c)t dt � A

Rez − c< ∞.

Next, for j � 1, 2, . . . , we define the function Fj on C by

Fj (z) �∫ j

0e−ztf (t)dt, z ∈ C. (25.4)

By Lemma 25.3, Fj is entire for j � 1, 2, . . . . Now, let c1 > c. Then, for Rez > c1,we get, by (25.1) and (25.4),

|Fj (z) − Fl(z)| ≤∫ j

l

e−(Rez−c)t dt ≤ A

∫ j

l

e−(c1−c)t dt

� A

(e−(c1−c)l − e−(c1−c)j

c1 − c

)→ 0

as j, l → ∞. Thus, Fj → F uniformly on the region {z ∈ C : Rez > c1} asj → ∞, and hence F is analytic on {z ∈ C : Rez > c1}. But c1 is an arbitrarynumber larger than c. So, F is analytic on the region {z ∈ C : Rez > c}. �

25. Another Fourier Transform 121

Corollary 25.4. For k � 0, 1, 2, . . . , the function Fk defined on the region {z ∈C : Rez > 0} by

Fk(z) �∫ ∞

0e−zt t kdt, Rez > 0,

is analytic.

Proof. Let ε be any positive number. Then we can find a positive constant Aε

such that

|tk| ≤ Aεeεt , t ≥ 0.

So, by Theorem 25.1, Fk is analytic on the region {z ∈ C : Rez > ε}. Since ε is anarbitrary positive number, it follows that Fk is analytic on {z ∈ C : Rez > 0}. �

For k � 0, 1, 2, . . . , we define the function lk on R by

lk(x) �{

e− x2 L0

k(x), x > 0,

0, x ≤ 0.(25.5)

The aim of this chapter is to compute the Fourier transform of the function lk ,k � 0, 1, 2, . . . .

Theorem 25.5. For k � 0, 1, 2, . . . ,

lk(ξ ) �√

2π

(2iξ − 12iξ + 1

)k 12iξ + 1

, ξ ∈ R.

Proof. By (25.5) and the definition of Lk(x), x > 0,

lk(ξ ) � (2π )−12

∫ ∞

0e−ixξ lk(x)dx

� (2π )−12

∫ ∞

0e−ixξ e− x

2ex

k!

(d

dx

)k

(e−xxk)dx

� (−1)k

k!(2π )−

12

∫ ∞

0

(d

dx

)k

(ex2 (1−2iξ ))e−xxkdx

� (−1)k

k!(2π )−

12

∫ ∞

0e

x2 (1−2iξ )

(1 − 2iξ

2

)k

e−xxkdx

� (−1)k

k!

(1 − 2iξ

2

)k

(2π )−12

∫ ∞

0e− x

2 (1+2iξ )xkdx (25.6)

for all ξ in R. Now, for z > 0,∫ ∞

0e−zt t kdt �

∫ ∞

0e−s

(s

z

)kds

z� 1

zk+1 �(k + 1) � k!zk+1 . (25.7)

By Corollary 25.4,∫∞

0 e−zt t kdt is analytic on {z ∈ C : Rez > 0}. So, by (25.7)and the principle of analytic continuation,∫ ∞

0e−zt t kdt � k!

zk+1 , Rez > 0. (25.8)

122 25. Another Fourier Transform

So, by (25.8), we get∫ ∞

0e−x( 1+2iξ

2 )xkdx � k!( 1+2iξ

2 )k+1, ξ ∈ R. (25.9)

Therefore, by (25.6) and (25.9),

lk(ξ ) � (2π )−12 (−1)k

(1 − 2iξ

2

)k 2k+1

(1 + 2iξ )k+1

�√

2π

(2iξ − 12iξ + 1

)k 12iξ + 1

, ξ ∈ R. �

26A Class of Compact Weyl Transformson L2(R)

The space Lr∗(R), 1 ≤ r < ∞, defined by (14.1) will be used again in this chapter

to give a criterion for the compactness of the Weyl transform on L2(R).Let g ∈ Lr

∗(R), 1 ≤ r < ∞, and let σ be the function on R2 defined by

σ (x, ξ ) � g(ρ2), x, ξ ∈ R,

where ρ �√

x2 + ξ 2. Then σ ∈ Lr ′ (R2). Indeed,∫ ∞

−∞

∫ ∞

−∞|σ (x, ξ )|r ′

dx dξ �∫ 2π

0

∫ ∞

0|g(ρ2)|r ′

ρdρdθ

� π

∫ ∞

0|g(ρ)|r ′

dρ ≤ π‖g‖r ′Lr′ (R2) < ∞.

Theorem 26.1. Let g ∈ Lr∗(R), 1 ≤ r < ∞, and let σ be the function on R

2

defined by

σ (x, ξ ) � g(ρ2), x, ξ ∈ R,

where ρ �√

x2 + ξ 2. Then Wσ : L2(R) → L2(R) is a compact operator.

Proof. By part (ii) of Theorem 24.5, we only need to prove that the sequence{ak}∞k�1 of complex numbers defined by

ak �∫ ∞

0σ (ρ)L0

k(2ρ2)e−ρ2ρdρ, k � 0, 1, 2, . . . ,

where σ (ρ) � g(ρ2), is such that limk→∞ ak � 0. To do this, let lk and Ik befunctions on R defined, respectively, by (25.5) and

Ik(x) � lk(2x), x ∈ R, (26.1)

124 26. A Class of Compact Weyl Transforms on L2(R)

for k � 0, 1, 2, . . . . Then, by Theorem 25.5 and (26.1),

Ik(ξ ) � 12lk

(ξ

2

)� 1√

2π

(iξ − 1iξ + 1

)k 1iξ + 1

, ξ ∈ R, (26.2)

for k � 0, 1, 2, . . . . Now, for 1 < s < ∞, we get, by (26.2),

‖Ik‖Ls (R) � 1√2π

(∫ ∞

−∞

1|iξ + 1|s dξ

) 1s

(26.3)

and

‖Ik‖L∞(R) ≤ 1√2π

(26.4)

for k � 0, 1, 2, . . . . Thus, for 1 < s ≤ ∞, by (26.3) and (26.4) we can get apositive constant Cs such that

‖Ik‖Ls (R) ≤ Cs, k � 0, 1, 2, . . . . (26.5)

Note that for k � 0, 1, 2, . . . ,

ak �∫ ∞

0σ (ρ)L0

k(2ρ2)e−ρ2ρdρ �

∫ ∞

0g(ρ2)L0

k(2ρ2)e−ρ2ρdρ

� 12

∫ ∞

0g(t)L0

k(2t)e−t dt � 12

∫ ∞

−∞g(t)Ik(t)dt. (26.6)

Now, if ∫ ∞

−∞g(t)Ik(t)dt �

∫ ∞

−∞g(t)Ik(t)dt, k � 0, 1, 2, . . . , (26.7)

then, by (26.2), (26.6), and (26.7),

ak � 12√

2π

∫ ∞

−∞g(ξ )

(iξ − 1iξ + 1

)k 1iξ + 1

dξ, k � 0, 1, 2, . . . . (26.8)

Let ξ � tan θ , −π2 < θ < π

2 . Then, by (26.8), we get, for k � 0, 1, 2, . . . ,

ak � 12√

2π

∫ π2

− π2

g(tan θ )(

i tan θ − 1i tan θ + 1

)k 1i tan θ + 1

sec2 θdθ. (26.9)

But for k � 0, 1, 2, . . . and −π2 < θ < π

2 ,(i tan θ − 1i tan θ + 1

)k

� (−1)k(cos θ − i sin θ )k

(cos θ + i sin θ )k� (−1)ke−2ikθ . (26.10)

Thus, for k � 0, 1, 2, . . . , we have, by (26.9) and (26.10),

ak � (−1)k

2√

2π

∫ π

−π

f (θ )e−2ikθ dθ, (26.11)

26. A Class of Compact Weyl Transforms on L2(R) 125

where

f (θ ) �⎧⎨⎩

0, −π < θ < −π2 ,

g(tan θ ) sec2 θ

i tan θ+1 , −π2 < θ < π

2 ,0, π

2 < θ < π .

Note that

f ∈ L1[−π, π] (26.12)

because by Holder’s inequality there exists a positive constant Cr such that∫ π2

− π2

∣∣∣∣g(tan θ ) sec2 θ

i tan θ + 1

∣∣∣∣ dθ �∫ ∞

−∞

∣∣∣∣ g(ξ )iξ + 1

∣∣∣∣ dξ

≤ ‖g‖Lr (R)

(∫ ∞

−∞

1|iξ + 1|r ′ dξ

) 1r′

< ∞.

Thus, by (26.11), (26.12), and the Riemann–Lebesgue lemma, ak → 0 as k → ∞.Thus, the proof is complete, provided that we can prove (26.7). To do this, we firstnote that for 1 ≤ r ≤ 2, (26.7) follows from Lemma 14.6. For 2 < r < ∞, by(26.5) we can find a sequence {ψj }∞j�1 of functions in S(R) such that ψj → Ik inLr ′ (R) as j → ∞. Now, let {ϕl}∞l�1 be a sequence of functions in S(R) such thatϕl → g in Lr (R) as l → ∞. Then ϕl → g in S ′(R) as l → ∞, so that ϕl → g inS ′(R) as l → ∞. Therefore,∫ ∞

−∞g(t)Ik(t)dt � lim

l→∞

∫ ∞

−∞ϕl(t)Ik(t)dt

� liml→∞

limj→∞

∫ ∞

−∞ϕl(t)ψj (t)dt

� liml→∞

limj→∞

∫ ∞

−∞ϕl(t)ψj (t)dt. (26.13)

Now, if we can prove that

liml→∞

limj→∞

∫ ∞

−∞ϕl(t)ψj (t)dt � lim

j→∞liml→∞

∫ ∞

−∞ϕl(t)ψj (t)dt, (26.14)

then, by (26.13) and (26.14),∫ ∞

−∞g(t)Ik(t)dt � lim

j→∞

∫ ∞

−∞g(t)ψj (t)dt. (26.15)

But by the Fourier inversion formula and the Hausdorff–Young inequality, ψj →Ik in Lr (R) as j → ∞. So, by (26.15),∫ ∞

−∞g(t)Ik(t)dt �

∫ ∞

−∞g(t)Ik(t)dt.

So, it remains to prove (26.14), or equivalently,

liml→∞

limj→∞

∫ ∞

−∞ϕl(t)ψj (t)dt � lim

j→∞liml→∞

∫ ∞

−∞ϕl(t)ψj (t)dt. (26.16)

126 26. A Class of Compact Weyl Transforms on L2(R)

To prove (26.16), let ε be any positive number. Then we pick N1 to be any positiveinteger such that

j ≥ N1 ⇒ ‖ψj − Ik‖Lr′ (R) <ε

2M1, (26.17)

where

M1 � sup{‖ϕl‖Lr (R) : l � 0, 1, 2, . . .}. (26.18)

Let N2 be another positive integer such that

l ≥ N2 ⇒ ‖ϕl − g‖Lr (R) <ε

2‖Ik‖Lr′ (R). (26.19)

Thus, for j, l ≥ max(N1, N2), we get, by (26.17), (26.18), and (26.19),∣∣∣∣∫ ∞

−∞ϕl(t)ψj (t)dt −

∫ ∞

−∞g(t)Ik(t)dt

∣∣∣∣≤∣∣∣∣∫ ∞

−∞ϕl(t)(ψj (t) − Ik(t))dt

∣∣∣∣+∣∣∣∣∫ ∞

−∞(ϕl(t) − g(t))Ik(t)dt

∣∣∣∣≤ ‖ϕl‖Lr (R)‖ψj − Ik‖Lr′ (R) + ‖ϕl − g‖Lr (R)‖Ik‖Lr′ (R)

< M1ε

2M1+ ε

2‖Ik‖Lr′ (R)‖Ik‖Lr′ (R) <

ε

2+ ε

2� ε.

Therefore, ∫ ∞

−∞ϕl(t)ψj (t)dt →

∫ ∞

−∞g(t)Ik(t)dt

as l, j → ∞, and so the left-hand side and the right-hand side of (26.16) ex-ist. So, we can conclude that (26.16) holds, and this completes the proof of thetheorem. �

27A Class of Bounded Weyl Transformson L2(R)

In order to give another sufficient condition for the Weyl transform to be a boundedlinear operator from L2(R) into L2(R), we begin with two basic facts in Fourierseries.

Lemma 27.1. Let f be a C1 function on [−π, π] such that f (−π ) � f (π ). Fork � 0, ±1, ±2, . . . , let ck and c′

k be numbers defined by

ck � 12π

∫ π

−π

f (θ )e−ikθ dθ

and

c′k � 1

2π

∫ π

−π

f ′(θ )e−ikθ dθ.

Then

c′k � ikck, 0, ±1, ±2, . . . .

Proof. For k � 0, ±1, ±2, . . . ,

c′k � 1

2π

∫ π

−π

f ′(θ )e−ikθ dθ

� 12π

f (θ )e−ikθ |π−π + ik

2π

∫ π

−π

f (θ )e−ikθ dθ � ikck. �

Lemma 27.2. Let f and ck be as in Lemma 27.1. Then∞∑

k�−∞|ck| < ∞.

128 27. A Class of Bounded Weyl Transforms on L2(R)

Proof. By Lemma 27.1, the Schwarz inequality, and the Parseval identity in theL2 theory of Fourier series,

∞∑k�−∞

|ck| � |c0| +∑k �0

|ck| � |c0| +∑k �0

|c′k|

1k

� |c0| +(∑

k �0|c′

k|2) 1

2(∑

k �0

1k2

) 12

� |c0| + ‖f ′‖L2[−π,π ]

(∑k �0

1k2

) 12

< ∞. �

We can now give a class of bounded Weyl transforms on L2(R).

Theorem 27.3. Let σ be a function on R2 defined by

σ (x, ξ ) � R(r)�(θ ), x, ξ ∈ R2 − {(0, 0)},

where � is a C1 function on [−π, π] such that �(−π ) � �(π ), and R is atempered function on (0, ∞), i.e.,∫ ∞

0

|R(ρ)|(1 + ρ)N

dρ < ∞

for some nonnegative integer N . For j, k � 0, 1, 2, . . . , let ajk be the numberdefined by

ajk �

⎧⎪⎪⎨⎪⎪⎩

2(−1)k(2π )− 12 ( k!

j ! )12 (

√2)j−k

∫∞0 R(ρ)ρj−k+1L

j−k

k (2ρ2)e−ρ2dρ,

j ≥ k, (27.1)2(−1)j (2π )− 1

2 ( j !k! )

12 (

√2)k−j

∫∞0 R(ρ)ρk−j+1L

k−j

j (2ρ2)e−ρ2dρ,

j < k. (27.2)

Suppose that there exists a positive constant C such that

|ajk| ≤ C, j, k � 0, 1, 2, . . . .

Then Wσ : L2(R) → L2(R) is a bounded linear operator.

Proof. By Theorem 24.1,

ψj,k(ζ ) �

⎧⎪⎪⎨⎪⎪⎩

2(−1)k(2π )− 12 ( k!

j ! )12 (

√2)j−k(ζ )j−kL

j−k

k (2|ζ |2)e−|ζ |2 ,j ≥ k, (27.3)

2(−1)j (2π )− 12 ( j !

k! )12 (

√2)k−j ζ k−jL

k−j

j (2|ζ |2)e−|ζ |2 ,k > j. (27.4)

So, by (27.1)–(27.4),

σ (ψj,k) � ajk

∫ π

−π

�(θ )e−i(j−k)θdθ

� ajk(2π )�(j − k), j, k � 0, ±1, ±2, . . . , (27.5)

27. A Class of Bounded Weyl Transforms on L2(R) 129

where

�(l) � 12π

∫ π

−π

�(θ )e−ilθ dθ, l � 0, ±1, ±2, . . . . (27.6)

Let f and g be in S(R). Then, by (24.6) and (27.5), there exists a positive constantC ′ such that

|(Wσf )(g)| � (2π )−1/2

∣∣∣∣∣∞∑

k�0

∞∑j�0

〈g, ek〉〈f, ej 〉σ (ψj,k)

∣∣∣∣∣≤ C ′

∞∑k�0

∞∑j�0

|�(j − k)‖〈f, ej 〉‖〈g, ek〉|

� C ′∞∑

k�0|〈g, ek〉|

∞∑j�0

|�(k − j )‖〈f, ej 〉|

≤ C ′( ∞∑

k�0|〈g, ek〉|2

) 12⎛⎝ ∞∑

k�0

{ ∞∑j�0

|�(k − j )‖〈f, ej 〉|}2⎞⎠

12

� C ′‖g‖L2(R)

⎛⎝ ∞∑

k�0

{ ∞∑j�0

|�(k − j )‖〈f, ej 〉|}2⎞⎠

12

, (27.7)

where

�(θ ) � �(θ ), −π ≤ θ ≤ π. (27.8)

By Lemma 27.2, (27.6), and (27.8),∞∑

k�−∞|�(k)| < ∞. (27.9)

So, by (27.7), (27.8), (27.9), and Young’s inequality,

|(Wσf )(g)| ≤ C ′‖g‖L2(R)

( ∞∑k�−∞

|�(k)|)( ∞∑

k�0|〈f, ek〉|2

) 12

� C ′( ∞∑

k�−∞|�(k)|

)‖f ‖L2(R)‖g‖L2(R) (27.10)

for all f and g in S(R). Therefore, by (27.10),

‖Wσf ‖L2(R) ≤ C ′( ∞∑

k�−∞|�(k)|

)‖f ‖L2(R), f ∈ S(R),

and the proof is complete. �

28A Weyl Transform with Symbol inS ′(R2)

We can compute in this chapter the eigenvalues and eigenfunctions of a specificWeyl transform with symbol in S ′(R2) as a compact and self-adjoint operator onL2(R).

Let ρ be a positive real number. Then we define the linear functional δρ :S(R2) → C by

δρ(ϕ) �∫ π

−π

ϕ(ρeiθ )ρdθ, ϕ ∈ S(R2).

Then it is easy to check that δρ is a tempered distribution onR2. For k � 0, 1, 2, . . . ,

we define the number λk by

λk � 2(−1)kρL0k(2ρ2)e−ρ2

. (28.1)

Theorem 28.1. Wδρ: L2(R) → L2(R) is a compact and self-adjoint operator.

Furthermore, the nonzero eigenvalues of Wδρ: L2(R) → L2(R) are precisely

equal to the numbers defined by (28.1), and∞∑

k�0|λk|r < ∞, r > 4.

Proof. Let f and g be in S(R). Then, by (24.6),

(Wδρf )(g) � (2π )−1/2

∞∑k�0

∞∑j�0

〈g, ek〉〈f, ej 〉δρ(ψj,k). (28.2)

132 28. A Weyl Transform with Symbol in S ′(R2)

For j, k � 0, 1, 2, . . . , by (27.3) and (27.4), we get

δρ(ψj,k)

�

⎧⎪⎪⎪⎨⎪⎪⎪⎩

2(−1)k(2π )− 12(

k!j !

) 12 (

√2)j−k

∫ π

−πρj−k+1e−i(j−k)θL

j−k

k (2ρ2)e−ρ2dθ,

j ≥ k,

2(−1)j (2π )− 12(

j !k!

) 12 (

√2)k−j

∫ π

−πρk−j+1e−i(j−k)θL

k−j

j (2ρ2)e−ρ2dθ,

k ≥ j. (28.3)Thus, by (28.3),

δρ(ψj,k) �{

0, j � k, (28.4)2(−1)k(2π ) 1

2 ρL0k(2ρ2)e−ρ2

, j � k. (28.5)

So, for all f and g in S(R), by (28.1), (28.2), (28.4), and (28.5),

(Wδρf )(g) �

∞∑k�0

λk〈f, ek〉〈g, ek〉. (28.6)

We now use an asymptotic formula for the Laguerre polynomials in the book [28]by Szego, which states that for arbitrary but fixed positive numbers ε and ω we get

L0k(x) � π− 1

2 ex2 x− 1

4 k− 14 cos

{2(kx)

12 − π

4

}+ O(k− 3

4 ), x > 0, (28.7)

as k → ∞, where the O-term is uniform with respect to x on [ε, ω]. Thus, by(28.1) and (28.7), there exists a positive constant C such that

|λk| ≤ Ck− 14 (28.8)

for k large enough, say, for k ≥ k0. Therefore, for r > 4, by (28.8), we get∞∑

k�k0

|λk|r ≤ Cr∞∑

k�k0

k− r4 < ∞. (28.9)

Now, for every positive integer N , the linear operator WN : L2(R) → L2(R)defined by

WNf �N∑

k�0λk〈f, ek〉ek, f ∈ L2(R), (28.10)

is of finite rank and hence compact. Moreover, for all f and g in S(R), by (28.6)and (28.10),

|(Wδρf )(g) − (WNf )(g)| ≤

∣∣∣∣∣∞∑

k�N+1λk〈f, ek〉〈g, ek〉

∣∣∣∣∣≤ ( sup

k≥N+1|λk|)‖f ‖L2(R)‖g‖L2(R). (28.11)

Thus, by (28.9) and (28.11),

‖Wδρ− WN‖∗ ≤ sup

k≥N+1|λk| → 0

28. A Weyl Transform with Symbol in S ′(R2) 133

as N → ∞. Therefore, Wδρ: L2(R) → L2(R) is compact. To see that Wδρ

:L2(R) → L2(R) is self-adjoint, we note that for all f and g in S(R), by (28.6),

〈Wδρf, g〉 � (Wδρ

f )(g) �∞∑

k�0λk〈f, ek〉〈g, ek〉

and

〈f, Wδρg〉 � 〈Wδρ

g, f 〉 � (Wδρg)(f ) �

∞∑k�0

λk〈f, ek〉〈g, ek〉.

For k � 0, 1, 2, . . . , λk is an eigenvalue of Wδρ: L2(R) → L2(R) because for any

g in S(R), by (28.6), we get

(Wδρej )(g) �

∞∑k�0

λk〈ej , ek〉〈g, ek〉 � (λjej )(g), j � 0, 1, 2, . . . .

Therefore, λj is an eigenvalue of Wδρ: L2(R) → L2(R) for j � 0, 1, 2, . . . .

Finally, let λ be an eigenvalue of Wδρ: L2(R) → L2(R) and let f ∈ L2(R) be a

corresponding eigenfunction. Then

Wδρ

( ∞∑j�0

〈f, ej 〉ej

)� λ

∞∑j�0

〈f, ej 〉ej

⇒∞∑

j�0〈f, ej 〉Wδρ

ej �∞∑

j�0〈f, ej 〉λej

⇒∞∑

j�0〈f, ej 〉λjej �

∞∑j�0

〈f, ej 〉λej

⇒ 〈f, ej 〉λj � 〈f, ej 〉λ, j � 0, 1, 2, . . . . (28.12)

Since {ej : j � 0, 1, 2, . . .} is an orthonormal basis for L2(R) and f � 0, it followsthat 〈f, ek〉 � 0 for some k � 0, 1, 2, . . . . Thus, by (28.12), λ � λk . So, everyeigenvalue of Wδρ

: L2(R) → L2(R) is equal to λk for some k � 0, 1, 2, . . . . �

To put Theorem 28.1 in a proper perspective of quantum mechanics, it is im-perative to note that due to quantization described at the end of Chapter 4, thenumerical values of measurements of an observable in classical mechanics withphase space R

2n are replaced by the spectrum of the self-adjoint Weyl transformrepresenting the observable. This explains why the computation of the spectrumof a self-adjoint Weyl transform on L2(Rn) is an important chapter in appliedmathematics.

29The Symplectic Group

It is shown in this chapter that the Weyl transform is invariant with respect to thesymplectic group, i.e., the group of all symplectic linear transformations from C

n

into Cn. A linear transformation a : C

n → Cn satisfying

[a(z), a(z′)] � [z, z′], z, z′ ∈ Cn, (29.1)

where [ , ] is the symplectic form on Cn defined by (8.1), is said to be symplectic. We

provide a sufficiently self-contained treatment of symplectic linear transformationsfrom C

n into Cn and the symplectic group in this chapter. Related matters can be

found in the books [9], [13], and [34] by Greub, Halmos, and Weyl, respectively.

Remark 29.1. Let a : Cn → C

n be a linear transformation. We call the matrix ofa with respect to the standard basis for C

n (� R2n) the standard matrix of a. We

shall identify a with its standard matrix. A point (q, p) in Cn will also be denoted

by(q

p

).

Proposition 29.2. Let a : Cn → C

n be a linear transformation such that the

standard matrix of a is equal to(

A B

C D

), where A, B, C, and D are n × n

matrices with real entries. Then a is symplectic if and only if

AtD − CtB � I, (29.2)AtC � CtA, (29.3)

and

BtD � DtB, (29.4)

where ( )t is the transpose of the matrix ( ) and I is the n × n identity matrix.

136 29. The Symplectic Group

Proof. For j � 1, 2, . . . , n, we let εj be the point in Rn such that all coordinates

except the j th coordinate are equal to zero, and the j th coordinate is equal to 1.Then, for j, k � 1, 2, . . . , n,

a(εj , 0) �(

A B

C D

)(εj

0

)�(

Aεj

Cεj

), (29.5)

a(0, εk) �(

A B

C D

)(0εk

)�(

Bεk

Dεk

), (29.6)

and hence, by (8.1), (29.5), and (29.6),

[a(εj , 0), a(0, εk)] � 2(εtkB

tCεj − εtkD

tAεj ). (29.7)

Since by (8.1),

[(εj , 0), (0, εk)] � −2εtkεj , j, k � 1, 2, . . . , n, (29.8)

it follows from (29.1), (29.7), and (29.8) that DtA−BtC � I , and (29.2) is proved.Next, for j, k � 1, 2, . . . , n, we get, by (8.1) and (29.5),

[a(εj , 0), a(εk, 0)] � 2(εtkA

tCεj − εtkC

tAεj ) (29.9)

and

[(εj , 0), (εk, 0)] � 0. (29.10)

So, by (29.1), (29.9), and (29.10), (29.3) is proved. Finally, for j, k � 1, 2, . . . , n,we get, by (8.1) and (29.6),

[a(0, εj ), a(0, εk)] � 2(εtkB

tDεj − εtkD

tBεj ) (29.11)

and

[(0, εj ), (0, εk)] � 0. (29.12)

So, by (29.1), (29.11), and (29.12), (29.4) is proved. Conversely, for z � x + iξ

and z′ � x ′ + iξ ′ in Cn, we get

a(z) �(

A B

C D

)(x

ξ

)�(

Ax + Bξ

Cx + Dξ

)(29.13)

and

a(z′) �(

A B

C D

)(x ′

ξ ′

)�(

Ax ′ + Bξ ′

Cx ′ + Dξ ′

). (29.14)

Hence, by (8.1), (29.13), and (29.14),

[a(z), a(z′)] � 2(x ′)t (AtC − CtA)x + 2(ξ ′)t (BtD − DtB)ξ+ 2(x ′)t (AtD − CtB)ξ − 2xt (AtD − CtB)ξ ′. (29.15)

So, by (8.1), (29.2), (29.3), (29.4), and (29.15),

[a(z), a(z′)] � 2(x ′ · ξ − x · ξ ′) � [z, z′],

and hence a is symplectic. �

29. The Symplectic Group 137

Proposition 29.3. Let Sp(n, R) be the set of all symplectic linear transformationsfrom C

n into Cn. Then Sp(n, R) is a group with respect to the usual composition

of mappings.

Proof. Let(

A1 B1C1 D1

)and

(A2 B2C2 D2

)be in Sp(n, R). Then

(A1 B1C1 D1

)(A2 B2C2 D2

)�(

A B

C D

),

where

A � A1A2 + B1C2,

B � A1B2 + B1D2,

C � C1A2 + D1C2,

and

D � C1B2 + D1D2.

So, some easy computations and Proposition 29.2 give

AtD − CtB � I,

AtC � CtA,

and

BtD � DtB,

and hence, by Proposition 29.2 again,(A1 B1C1 D1

)(A2 B2C2 D2

)∈ Sp(n, R).

The associative law follows from the usual association law for the compositions

of mappings. The matrix(

I 00 I

)in Sp(n, R) is obviously the identity element.

Finally, let a �(

A B

C D

)∈ Sp(n, R). Let

b �(

Dt −Bt

−Ct At

).

Then, by Proposition 29.2,(Dt −Bt

−Ct At

)(A B

C D

)�(

I 00 I

).

So,(

A B

C D

)is invertible, and

(A B

C D

)−1

�(

Dt −Bt

−Ct At

).

138 29. The Symplectic Group

Therefore, it remains to prove that b ∈ Sp(n, R). To do this, we note that for all z

and z′ in Cn, we get, by (29.1),

[z, z′] � [(ab)(z), (ab)(z′)] � [a(b(z)), a(b(z′))]� [b(z), b(z′)],

i.e., b is symplectic. �

Corollary 29.4. Let(

A B

C D

)∈ Sp(n, R). Then ABt � BAt and CDt � DCt .

Proof. From the proof of Proposition 29.3,(A B

C D

)−1

�(

Dt −Bt

−Ct At

)∈ Sp(n, R).

Hence, by Proposition 29.2, the proof is complete. �

Proposition 29.5. Let Rλ be the irreducible and unitary representation of theHeisenberg group Hn on L2(Rn) defined by (8.5). Let a ∈ Sp(n, R). Then themapping Rλ

a : Hn → G defined by

Rλa (z, t) � Rλ(a(z), t), (z, t) ∈ Hn, (29.16)

is also an irreducible and unitary representation of Hn on L2(Rn).

Proof. Let (z, t) and (z′, t ′) be in Hn. Then, by (8.2) and (29.1),

Rλa ((z, t) · (z′, t ′)) � Rλ

a (z + z′, t + t ′ + [z, z′])� Rλ(a(z + z′), t + t ′ + [z, z′])� Rλ(a(z) + a(z′), t + t ′ + [a(z), a(z′)])� Rλ((a(z), t) · (a(z′), t ′))� Rλ(a(z), t)Rλ(a(z′), t ′)� Rλ

a (z, t)Rλa (z′, t ′).

Next, it is easy to see that for all f in L2(Rn), we get, by (29.16),

Rλa (z, t)f � Rλ(a(z), t)f → f

in L2(Rn) as (z, t) → (0, 0). Thus, Rλa : Hn → G is a unitary representation of Hn

on L2(Rn). Finally, let M be a closed subspace of L2(Rn) such that M is invariantunder all the operators Rλ

a (z, t), (z, t) ∈ Hn. Let (z, t) ∈ Hn and let w ∈ Cn be

such that z � a(w). Then M is invariant under the unitary operator Rλa (w, t). But

by (29.16),

Rλa (w, t) � Rλ(a(w), t) � Rλ(z, t).

So, M is invariant under all the operators Rλ(z, t), (z, t) ∈ Hn, and using theirreducibility of the unitary representation Rλ of Hn on L2(Rn), we conclude thatM � L2(Rn) or M � {0}. Therefore, Rλ

a is irreducible. �

Proposition 29.6. The irreducible and unitary representations Rλa and Rλ of Hn

on L2(Rn) are equivalent for all a in Sp(n, R).

29. The Symplectic Group 139

Proof. We first note that by (8.5),

Rλa (0, t) � Rλ(a(0), t) � Rλ(0, t) � e

14 iλt , t ∈ R.

So, by Proposition 8.6 and Remark 8.7, the proof is complete. �

Remark 29.7. Let λ � 1. Then, for all a in Sp(n, R), we get, by Proposition29.6, a unitary operator Va on L2(Rn) such that

R1(a(z), t) � R1a(z, t) � VaR

1(z, t)V −1a , (z, t) ∈ Hn,

and hence, by Remark 8.8,

ρ(a(z)) � R1(a(z), 0) � R1a(z, 0) � Vaρ(z)V −1

a , z ∈ Cn.

Proposition 29.8. Let a ∈ Sp(n, R). Then there exists a unitary operator Ua onL2(Rn) such that

ρ(a(z)) � Uaρ(z)U−1a , z ∈ C

n,

where a � (at )−1.

Proof. In view of Remark 29.7, we only need to prove that a ∈ Sp(n, R). Inview of Proposition 29.3, we only need to prove that the transpose of any a in

Sp(n, R) is in Sp(n, R). To this end, let a �(

A B

C D

)∈ Sp(n, R) and let

J �(

0 I

−I 0

). (29.17)

Then, by Proposition 29.2, it is easy to see that J ∈ Sp(n, R). It has been shownin the proof of Proposition 29.3 that

J−1 �(

0 −I

I 0

)(29.18)

and

a−1 �(

Dt −Bt

−Ct At

). (29.19)

So, by (29.17), (29.18), and (29.19),

at �(

At Ct

Bt Dt

)�(

0 −I

I 0

)(Dt −Bt

−Ct At

)(0 I

−I 0

)� J−1a−1J,

and hence at ∈ Sp(n, R). �

Remark 29.9. For each a in Sp(n, R), the unitary operator Ua on L2(Rn), gener-ated by Proposition 29.8, is uniquely determined up to a constant multiple in thesense that if Ua and Va are unitary operators on L2(Rn) such that

ρ(a(z)) � Uaρ(z)U−1a , z ∈ C

n, (29.20)

and

ρ(a(z)) � Vaρ(z)V −1a , z ∈ C

n, (29.21)

140 29. The Symplectic Group

then Va � caUa for some constant ca . Indeed, by (29.20) and (29.21), we get

Uaρ(z)U−1a � Vaρ(z)V −1

a , z ∈ Cn,

and hence

ρ(z)U−1a Va � U−1

a Vaρ(z), z ∈ Cn. (29.22)

So, by (8.5) and (29.22),

R1(z, t)U−1a Va � U−1

a VaR1(z, t), (z, t) ∈ Hn. (29.23)

Since R1 is an irreducible and unitary representation of Hn on L2(Rn), it followsthat the only bounded linear operators on L2(Rn) that commute with all operatorsR1(z, t), (z, t) ∈ Hn, are constant multiples of the identity I on L2(Rn). Hence,by (29.23), we get a constant ca such that

U−1a Va � caI,

or

Va � caUa.

Proposition 29.10.

(i) Let a �(

0 I

−I 0

). Then Ua � F .

(ii) Let a �(

I 0C I

), where C is a symmetric matrix. Then

(Uaf )(x) � e12 i(Cx)·xf (x), x ∈ R

n,

for all f in L2(Rn).

(iii) Let a �(

A 00 (At )−1

), where A is an invertible matrix. Then

(Uaf )(x) � f (A−1x)| det A|− 12 , x ∈ R

n,

for all f in L2(Rn).

Proof. For all z in Cn and all f in S(Rn), we get, by (2.1),

(ρ(z)f )(x) � eiq·x+ 12 iq·pf (x + p)

� eiq·x+ 12 iq·p(2π )−n/2

∫Rn

ei(x+p)·ξ f (ξ )dξ

� eiq·x+ 12 iq·p(2π )−n/2

∫Rn

ei(x+p)·(η−q)f (η − q)dη

� eiq·x+ 12 iq·p(2π )−n/2

∫Rn

eix·η−ix·q+ip·η−iq·pf (η − q)dη

� (2π )−n/2∫

Rn

eix·ηeip·η− 12 iq·pf (η − q)dη

� (ρ(p, −q)f )(x), x ∈ Rn. (29.24)

29. The Symplectic Group 141

Also,

a(q, p) �(

0 I

−I 0

)(q

p

)�(

p

−q

), q, p ∈ R

n. (29.25)

So, by (29.24) and (29.25),

ρ(a(z)) � Fρ(z)F−1, z ∈ Rn,

and hence Ua � F . To prove part (ii), we first note that

a � (at )−1 �(

I C

0 I

)−1

�(

I −C

0 I

),

and hence, for all z � q + ip in Cn,

a(z) �(

I −C

0 I

)(q

p

)�(

q − Cp

p

). (29.26)

Now, we only need to prove that the Ua given by the equation in part (ii) satisfies

(Uaρ(z)U−1a f )(x) � (ρ(a(z))f )(x), x ∈ R

n, (29.27)

for all z in Cn and all f in L2(Rn). But, by (29.26),

(Uaρ(z)U−1a f )(x) � e

12 i(Cx)·x(ρ(z)U−1

a f )(x)

� e12 i(Cx)·xeiq·xe

12 iq·pe− 1

2 i(C(x+p))·(x+p)f (x + p)

� e12 i(Cx)·x+iq·x+ 1

2 iq·p− 12 i(Cx)·x−i(Cx)·p− 1

2 i(Cp)·pf (x + p)

� ei(q−Cp)·x+ 12 i(q−Cp)·pf (x + p)

� (ρ(q − Cp, p)f )(x)� (ρ(a(z))f )(x), x ∈ R

n,

for all z in Cn and all f in L2(Rn). Thus, (29.27) follows. To prove part (iii), we

note that

a �(

At 00 A−1

)−1

�(

(At )−1 00 A

),

and hence, for all z � q + ip in Cn,

a(z) �(

(At )−1 00 A

)(q

p

)�(

(At )−1q

Ap

). (29.28)

We only need to prove that the Ua given by the equation in part (iii) satisfies

(Uaρ(z)U−1a f )(x) � (ρ(a(z))f )(x), x ∈ R

n, (29.29)

for all z in Cn and all f in L2(Rn). But

(U−1a f )(x) � f (Ax)| det A| 1

2 , x ∈ Rn, (29.30)

142 29. The Symplectic Group

and hence, by (29.28) and (29.30),

(Uaρ(z)U−1a f )(x) � (ρ(z)U−1

a f )(A−1x)| det A|− 12

� eiq·(A−1x)e12 iq·p(U−1

a f )(A−1x + p)| det A|− 12

� ei((At )−1q)·x+ 12 iq·pf (x + Ap)

� ei((At )−1q)·x+ 12 i((At )−1q)·(Ap)f (x + Ap)

� (ρ((At )−1q, Ap)f )(x)� (ρ(a(z))f )(x), x ∈ R

n,

for all z in Cn and all f in L2(Rn). Thus, (29.29) is proved. �

Remark 29.11. It can be proved that the group Sp(n, R) is finitely generated bymatrices of the three types given in Proposition 29.10. So, we are able to computeUa , up to a constant multiple, for all a in Sp(n, R). It can also be proved that theconstants can be chosen such that

UaUb � ±Uab, a, b ∈ Sp(n, R).

Thus, the mapping from Sp(n, R) into G given by a �→ Ua is a “metaplecticrepresentation” of Sp(n, R) on L2(Rn). See Chapter 4 of the book [6] by Follandfor a discussion of the metaplectic representation and related matters.

Remark 29.12. The unitary operators Ua provided by Proposition 29.10 areobviously homeomorphisms from S(Rn) onto S(Rn). It is also easy to see that theunitary operators Ua of the types given by parts (i) and (ii) of Proposition 29.10 arehomeomorphisms from S ′(Rn) onto S ′(Rn). Let Ua be a unitary operator of thetype given by part (iii) of Proposition 29.10. Then, for all f in S ′(Rn), we defineUaf : S(Rn) → C by

(Uaf )(ϕ) � f (U−1a ϕ), ϕ ∈ S(Rn).

It is then easy to check that Ua is a homeomorphism from S ′(Rn) onto S ′(Rn).

Let σ ∈ S ′(R2n) and a ∈ Sp(n, R). Then we define the mapping σ ◦ a :S(R2n) → C by

(σ ◦ a)(ϕ) � σ (ϕ ◦ a−1), ϕ ∈ S(R2n). (29.31)

We leave it as an exercise to prove that σ ◦ a ∈ S ′(R2n).We can now give the main result in this chapter.

Theorem 29.13. Let a ∈ Sp(n, R). Then, for all σ in S ′(R2n),

Wσ◦a � U−1a WσUa,

where Ua is any unitary operator guaranteed by Proposition 29.8.

Proof. Let σ ∈ S(R2n). Then, for all ϕ and ψ in S(Rn), we get, by (2.3), (3.12),Remark 29.12, Theorem 3.1, Theorem 4.3, and the adjoint formula for the Fourier

29. The Symplectic Group 143

transform,

〈U−1a WσUaϕ, ψ〉 � 〈WσUaϕ, Uaψ〉

� (2π )−n/2∫

Cn

σ (x, ξ )W (Uaϕ, Uaψ)(x, ξ )dx dξ

� (2π )−n/2∫

Cn

σ (z)V (Uaϕ, Uaψ)(z)dz

� (2π )−n

∫Cn

σ (z)〈ρ(z)Uaϕ, Uaψ〉dz

� (2π )−n

∫Cn

σ (z)〈U−1a ρ(z)Uaϕ, ψ〉dz. (29.32)

So, by (29.32), Proposition 29.8, and Remark 29.11,

〈U−1a WσUaϕ, ψ〉 � (2π )−n

∫Cn

σ (z)〈ρ(at (z))ϕ, ψ〉dz

� (2π )−n

∫Cn

σ (a(z))〈ρ(z)ϕ, ψ〉dz

� (2π )−n

∫Cn

(σ ◦ a)(z)〈ρ(z)ϕ, ψ〉dz

� 〈Wσ◦aϕ, ψ〉for all ϕ and ψ in S(Rn). So,

Wσ◦a � U−1a WσUa, σ ∈ S(R2n). (29.33)

Let σ ∈ S ′(R2n). Then, by Proposition 1.18, we can find a sequence {σj }∞j�1 offunctions in S(R2n) such that σj → σ in S ′(R2n) as j → ∞. Then, by (29.31),(29.33), and Remark 29.12, we get, for all ϕ in S(Rn),

Wσj ◦aϕ → Wσ◦aϕ

and

U−1a Wσj

Uaϕ → U−1a WσUaϕ

in S ′(Rn) as j → ∞, and hence

Wσ◦a � U−1a WσUa. �

30Symplectic Invariance of WeylTransforms

We make precise in this chapter the fact that the symplectic invariance propertyin Theorem 29.13 characterizes the quantization σ �→ σ (x, D), described at theend of Chapter 4, as the Weyl transform Wσ . The main result, i.e., Theorem 30.1,obtained by Shale in [23], is another testimonial to the vision of Hermann Weylthat Wσ is the correct quantization of the classical observable σ .

We begin with a proposition.

Proposition 30.1. Let σ be the function on R2n defined by

σ (x, ξ ) � ei∑n

j�1 xj , x, ξ ∈ Rn. (30.1)

Then, for all q � (q1, q2, . . . , qn) and p � (p1, p2, . . . , pn) in Rn with qj �

0, j � 1, 2, . . . , n, there exists an element a in Sp(n, R) such that

(σ ◦ a)(x, ξ ) � σ (a(x, ξ )) � ei(q·x+p·ξ ), x, ξ ∈ Rn. (30.2)

Proof. By (30.1),

σ (x, ξ ) � ei(u,0)·(x,ξ ), x, ξ ∈ Rn, (30.3)

where u is the point in Rn in which all coordinates are equal to 1. Let

a �(

Q P

0 Q−1

), (30.4)

where

Q �

⎛⎜⎝

q1 0 · · · 00 q2 · · · 0

· · ·0 0 · · · qn

⎞⎟⎠

146 30. Symplectic Invariance of Weyl Transforms

and

P �

⎛⎜⎝

p1 0 · · · 00 p2 · · · 0

· · ·0 0 · · · pn

⎞⎟⎠ .

By Proposition 29.2, a ∈ Sp(n, R). Now, by (30.4),

a(x, ξ ) �(

Q P

0 Q−1

)(x

ξ

)�(

Qx + Pξ

Q−1ξ

), x, ξ ∈ R

n. (30.5)

Thus, by (30.3) and (30.5),

(σ ◦ a)(x, ξ ) � ei(u,0)·(Qx+Pξ ) � ei(q·x+p·ξ ), x, ξ ∈ Rn,

and the proposition is proved. �

Let L(S(Rn), S ′(Rn)) be the set of all continuous linear mappings from S(Rn)into S ′(Rn). Let {Ak}∞k�1 be a sequence of elements in L(S(Rn), S ′(Rn)) and let A

be in L(S(Rn), S ′(Rn)). We say that Ak → A in L(S(Rn), S ′(Rn)) as k → ∞ if

Akϕ → Aϕ

in S ′(Rn) as k → ∞ for all ϕ in S(Rn).We can now state and prove the main result in the last chapter of the book.

Theorem 30.2. Let A : S ′(R2n) → L(S(Rn), S ′(Rn)) be a linear mapping. Wesuppose that it is continuous in the sense that

σk → σ in S ′(R2n) ⇒ Aσk → Aσ

in L(S(Rn), S ′(Rn)) as k → ∞. Moreover, we suppose that

((Aσ )ϕ)(x) � σ (x)ϕ(x), x ∈ Rn, (30.6)

for all ϕ in S(Rn) and all σ in L∞(Rn), and

A(σ ◦ a) � U−1a (Aσ )Ua (30.7)

for all σ in S ′(R2n) and all a in Sp(n, R), where Ua is a unitary operator on L2(Rn)given by Proposition 29.8. Then

Aσ � Wσ, σ ∈ S ′(R2n).

To prove Theorem 30.2, we need a lemma.

Lemma 30.3. For any q and p in Rn, let eq,p be the function on R

2n such that

eq,p(x, ξ ) � ei(q·x+p·ξ ), x, ξ ∈ Rn.

Then

Weq,p� ρ(q, p), q, p ∈ R

n,

where ρ(q, p) is given by (2.1).

30. Symplectic Invariance of Weyl Transforms 147

Proof. Let q and p be in Rn. Then, by (2.3), (3.12), Theorem 3.1, and (12.1),

we get

(Weq,pϕ)(ψ) � (2π )−n/2

∫Rn

∫Rn

ei(q·x+p·ξ )W (ϕ, ψ)(x, ξ )dx dξ

� (2π )n/2W (ϕ, ψ )(q, p)� (2π )n/2V (ϕ, ψ)(q, p)� 〈ρ(q, p)ϕ, ψ〉, ϕ, ψ ∈ S(Rn),

and hence the proof is complete. �

Proof of Theorem 30.2. We begin with the observation that by Theorem 29.13,(30.1), (30.2), (30.6), (30.7), and Lemma 30.3,

Aeq,p � Weq,p� ρ(q, p) (30.8)

for all q and p in Rn such that every coordinate in q is nonzero. Let {Qk}∞k�1 be

a sequence of concentric cubes in Cn with centers at the origin and edges parallel

to the coordinate axes, and limk→∞ |Qk| � ∞, where |Qk| is the volume of Qk ,k � 1, 2, . . . .Then, using (9.4) and the Lebesgue dominated convergence theorem,we get, for all σ in S(R2n) and all ϕ in S(Rn),

(Wσϕ)(x) � limk→∞

(2π )−n

∫Qk

(ρ(q, p)ϕ)(x)σ (q, p)dq dp, x ∈ Rn. (30.9)

Let σ ∈ S(R2n). Then, by (30.9), Fubini’s theorem, and the Lebesgue dominatedconvergence theorem,

(2π )−n

∫Qk

(ρ(q, p)ϕ)(·)σ (q, p)dq dp → Wσϕ (30.10)

in S ′(Rn) as k → ∞ for all ϕ in S(Rn). Now, for k � 1, 2, . . . , we let σk be thefunction on R

2n defined by

σk(x, ξ ) � (2π )−n

∫Qk

eq,p(x, ξ )σ (q, p)dq dp, x, ξ ∈ Rn.

Then σk → σ in S ′(R2n). Indeed, using Fubini’s theorem and the Fourier inversionformula, we get

σk(ψ) � (2π )−n

∫Qk

σ (q, p)(∫

Cn

eq,p(x, ξ )ψ(x, ξ )dx dξ

)dq dp

�∫

Qk

σ (q, p)ψ(q, p)dq dp, ψ ∈ S(R2n), (30.11)

and hence, by the Lebesgue dominated convergence theorem and the adjointformula for the Fourier transform,

σk(ψ) → σ (ψ)

148 30. Symplectic Invariance of Weyl Transforms

as k → ∞ for all ψ in S(R2n). So, by continuity,

(Aσk)(ϕ) → (Aσ )(ϕ) (30.12)

in S ′(Rn) as k → ∞ for all ϕ in S(Rn). Now, by (30.8),

(2π )−n

∫Qk

(ρ(q, p)ϕ)(x)σ (q, p)dq dp � liml→∞

Sl(x), x ∈ Rn, (30.13)

where Sl(x) is a Riemann sum of the form

Sl(x) � (2π )−n∑

(ρ(qj , pj )ϕ)(x)σ (qj , pj )δj � ((Asl)ϕ)(x), x ∈ Rn,

(30.14)where (qj , pj ) is a point in Qk chosen in such a way that each coordinate in qj isnonzero;

sl(x, ξ ) � (2π )−n∑

eqj ,pj(x, ξ )σ (qj , pj )δj , x, ξ ∈ R

n, (30.15)

and

sl(x, ξ ) → (2π )−n

∫Qk

eq,p(x, ξ )σ (q, p)dq dp, x, ξ ∈ Rn,

as l → ∞. But by (30.11) and (30.15), we get

sl(ψ) � (2π )−n∑

σ (qj , pj )(∫

Cn

eqj ,pj(x, ξ )ψ(x, ξ )dx dξ

)δj

�∑

σ (qj , pj )ψ(qj , pj )δj

→∫

Qk

σ (q, p)ψ(q, p)dq dp � σk(ψ) (30.16)

as l → ∞ for all ψ in S(R2n). Thus, by (30.16), sl → σk in S ′(R2n) as l → ∞,and hence, by continuity,

(Asl)(ϕ) → (Aσk)(ϕ) (30.17)

in S ′(Rn) as l → ∞ for all ϕ in S(Rn). Next, for all ψ in S(Rn), we get, by (2.3),(30.14), and Fubini’s theorem,

Sl(ψ) � (2π )−n∑

σ (qj , pj )(∫

Rn

(ρ(qj , pj )ϕ)(x)ψ(x)dx

)δj

→ (2π )−n

∫Qk

σ (q, p)(∫

Rn

(ρ(q, p)ϕ)(x)ψ(x)dx

)dq dp

� ((2π )−n

∫Qk

(ρ(q, p)ϕ)(·)σ (q, q)dq dp)(ψ)

as l → ∞, and hence

Sl → (2π )−n

∫Qk

(ρ(q, p)ϕ)(·)σ (q, q)dq dp (30.18)

30. Symplectic Invariance of Weyl Transforms 149

in S ′(Rn) as l → ∞. So, by (30.14), (30.17), and (30.18),

(Aσk)(ϕ) � (2π )−n

∫Qk

(ρ(q, p)ϕ)(·)σ (q, q)dq dp. (30.19)

Thus, by (30.10), (30.12), and (30.19),

(Aσ )(ϕ) � Wσϕ, ϕ ∈ S(Rn),

and hence

Aσ � Wσ, σ ∈ S(R2n). (30.20)

Finally, let σ ∈ S ′(R2n). Then, by Proposition 1.18, there is a sequence {σj }∞j�1 offunctions in S(R2n) such that σj → σ in S ′(R2n) as j → ∞. So, for all ϕ and ψ

in S(Rn), we get, by (12.1) and (30.20),

(Wσϕ)(ψ) � (2π )−n/2σ (W (ϕ, ψ))� lim

j→∞(2π )−n/2σj (W (ϕ, ψ))

� limj→∞

(Wσjϕ)(ψ)

� limj→∞

((Aσj )ϕ)(ψ),

and hence

(Aσj )(ϕ) → Wσϕ (30.21)

in S ′(Rn) as j → ∞. But by continuity,

(Aσj )(ϕ) → (Aσ )(ϕ) (30.22)

in S ′(Rn) as j → ∞. So, by (30.21) and (30.22), Aσ � Wσ , and the proof iscomplete. �

References

[1] W. Ambrose, “Structure theorems for a special class of Banach algebras,”Trans. Amer. Math. Soc. 57 (1945), 364–386.

[2] F.A. Berezin and M.A. Shubin, The Schrodinger Equation, KluwerPublishers, 1991.

[3] I. Daubechies, “Time–frequency localization operators: a geometric phasespace approach,” IEEE Trans. Inform. Theory 34 (1988), 605–612.

[4] I. Daubechies, “The wavelet transform, time–frequency localization andsignal analysis,” IEEE Trans. Inform. Theory 36 (1990), 961–1005.

[5] I. Daubechies and T. Paul, “Time–frequency localisation operators—a geo-metric phase space approach: II. The use of dilations,” Inverse Prob. 4 (1988),661–680.

[6] G.B. Folland, Harmonic Analysis in Phase Space, Princeton University Press,1989.

[7] G.B. Folland, Fourier Analysis and Its Applications, Wadsworth andBrooks/Cole, 1992.

[8] R.R. Goldberg, Fourier Transforms, Cambridge University Press, 1961.[9] W. Greub, Linear Algebra, Fourth Edition, Springer-Verlag, 1995.

[10] A. Grossmann, G. Loupias, and E.M. Stein, “An algebra of pseudodifferen-tial operators and quantum mechanics in phase space,” Ann. Inst. Fourier(Grenoble) 18 (1968), 343–368.

[11] A. Grossmann, J. Morlet, and T. Paul, “Transforms associated to squareintegrable group representations I: General results,” J. Math. Phys. 26 (1985),2473–2479.

152 References

[12] A. Grossmann, J. Morlet, and T. Paul, “Transforms associated to squareintegrable group representations II: Examples,” Ann. Inst. Henri Poincare 45(1986), 293–309.

[13] P.R. Halmos, Finite-Dimensional Vector Spaces, Second Edition, Springer-Verlag, 1996.

[14] Z. He, “A unique continuation property for wavelet transforms,” Panamer.Math. J. 8(1) (1998), 65–77.

[15] Z. He and M.W. Wong, “Localization operators associated to squareintegrable group representations,” Panamer. Math. J. 6(1) (1996), 93–104.

[16] Z. He and M. W. Wong, “Wavelet multipliers and signals,” J. AustralianMath. Soc. Series B-Appl. Math., to appear.

[17] M. Holschneider, Wavelets: An Analysis Tool, Oxford University Press, 1995.[18] G.M. Iancu and M.W. Wong, “Semilinear wave equations and quantization,”

Comm. Appl. Nonlinear Anal, to appear.[19] J.E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Camb. Phil.

Soc. 45 (1949), 99–124.[20] J.C.T. Pool, “Mathematical aspects of the Weyl correspondence,” J. Math.

Phys. 7 (1966), 66–76.[21] E. Prugovecki, Quantum Mechanics in Hilbert Space, Second Edition,

Academic Press, 1981.[22] M. Schechter, Operator Methods in Quantum Mechanics, North Holland,

1981.[23] D. Shale, “Linear symmetries of free Boson fields,” Trans. Amer. Math. Soc.

103 (1962), 149–167.[24] B. Simon, “Distributions and Their Hermite Expansions,” J. Math. Phys. 12

(1971), 140–148.[25] B. Simon, “The Weyl transform and Lp functions on phase space,” Proc.

Amer. Math. Soc. 116 (1992), 1045–1047.[26] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and

Oscillatory Integrals, Princeton University Press, 1993.[27] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean

Spaces, Princeton University Press, 1971.[28] G. Szego, Orthogonal Polynomials, Third Edition, American Mathematical

Society, 1967.[29] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Princeton

University Press, 1993.[30] E.C. Titchmarsh, “A contribution to the theory of Fourier transforms,” Proc.

London Math. Soc. (2), 23 (1923), 279–289.[31] E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Third

Edition, Chelsea, 1986.[32] J. von Neumann, Mathematical Foundations of Quantum Mechanics,

Princeton University Press, 1996.[33] H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, 1950.[34] H. Weyl, The Classical Groups: Their Invariants and Representations,

Princeton University Press, 1997.

References 153

[35] E. Wigner, “On the quantum correction for thermodynamic equilibrium,”Phys. Rev. 40 (1932), 749–759.

[36] M.W. Wong, An Introduction to Pseudo-Differential Operators, WorldScientific, 1991.

[37] M.W. Wong, “Minimal and maximal operator theory with applications,”Canad. J. Math. 43 (1991), 617–627.

[38] M.W. Wong, “Spectral theory of pseudo-differential operators,” Adv. Appl.Math. 15 (1994), 437–451.

Notation Index

B(L2(Rn)), 22

C∞0 , 2

ej,k , 101en, 94

f , 4f , 6Ff , 4F−1f , 6f ∗ g, 3f ∗λ g, 43f ⊗ g, 29

G, 39

Hn, 10Hn, 87hn, 93

lk , 121LF , 76Lα

n , 95Lr

∗, 71

L(S(Rn), S ′(Rn)), 146

Rλ, 39Rλ

a , 138

S, 2Sh, 25Sp(n, R), 137

Tσ , 8

V (f, g), 10

Wσ , 19W (f ), 13W (f, g), 15(WH )n, 75

∂α , 2ρ(q, p), 9σ (x, D), 24ψj,k , 113[z, w], 37‖ ‖∗, 22‖ ‖HS , 35

Index

Adjoint formula, 5, 44, 142, 147admissible wavelet, 76annihilation operator, 93, 103, 106

Bracket operation, 38

Canonical commutation relations, 38classical mechanics, 23, 133commutator, 38convolution, 2, 3creation operator, 93, 103, 106

Differential operator, 1, 2dilation operator, 5Dirac delta, vi, 62

Eigenfunction, vi, 93, 106, 131, 133eigenvalue, vi, 93, 106, 131, 133

Fourier inversion formula, 5, 8, 62, 68,125, 147

Fourier series, 127, 128Fourier transform, vi, 4–6, 13, 30, 44,

71, 72, 79, 81, 83, 119, 121Fourier–Wigner transform, 9, 10, 13,

16, 101functional calculus, 24, 25, 33

H∗-algebra, 33–35harmonic oscillator, 93, 103, 106Hausdorff–Young inequality, 71–73,

125Heisenberg group, vi, 37, 38, 138Hermite function, vi, 87, 93, 101, 103,

106, 107, 109Hermite polynomial, 87, 90Hilbert–Schmidt operator, 25, 26, 32,

34, 57, 58, 74, 85

Inverse Fourier transform, 6, 30, 83,109

irreducible representation, 40, 42, 75,76, 138, 140

Joint probability distribution, 13

Kernel, 25, 26, 32, 57

Laguerre polynomial, 87, 95, 98, 107,109, 132

Laplace transform, 119Laplacian, 106left-invariant vector field, 38Lie algebra, 38localization operator, vi, 77–79, 83, 85

158 Index

Mehler’s formula, 107, 109metaplectic representation, 142modulation operator, 5momentum, 1, 13, 23Moyal identity, 15, 16, 101multiplication operator, 23

Nondeterministic statistical dynamics,15

Observable, 1, 24, 35, 133, 145

Partial differential operator, 2, 8, 20, 87,106

phase space, 23, 77, 133Plancherel’s theorem, 6, 15, 46, 71, 101Planck’s constant, 1, 23position, 13, 23projective representation, 10pseudo-differential operator, v, vi, 7, 8

Quantization, vi, 19, 23, 35, 133, 145quantum mechanics, v, 15, 24, 37, 38,

133

Radial symbol, 113–115regularization, 3representation, vi, 19, 76, 142resolution of the identity, 76, 77, 85Riemann–Lebesgue lemma, 4, 125Riesz–Thorin theorem, 47, 49, 57, 71,

78

Schrodinger representation, 10, 39signal analysis, 75, 77

spectrum, 133square-integrable representation, vi, 76state, 13Stone–von Neumann theorem, 42structure equations, 38symbol, vi, 2, 8, 19–21, 29, 33, 37, 43,

55, 59, 75, 77, 83, 113–115symplectic form, 37, 38, 43, 135symplectic group, vi, 135, 137symplectic invariance, 145

Tempered distribution, vi, 6, 7, 59, 61,131

tempered function, 7, 114, 115, 128tensor product, 29, 57three lines theorem, 47translation operator, 5twisted convolution, vi, 37, 43, 44twisting operator, 30

Unitarily equivalent representations, 42unitary representation, 39, 40, 42, 44,

75, 76, 138, 140

Vector field, 103

Weyl calculus, 33Weyl–Heisenberg group, 75Weyl transform, v, vi, 7, 9, 19, 21, 29,

32, 35, 37, 43, 44, 55, 59, 63, 71,83–85, 87, 113, 114, 123, 127,128, 131, 145

Wigner transform, vi, 9, 13, 15, 21, 59

Young’s inequality, 3, 129

Universitext (continued)

Rotman: Galois TheoryRubel/CoIIiander: Entire and Meromorphic FunctionsSagan: Space-Filling CurvesSamelson: Notes on Lie AlgebrasSchiff: Normal FamiliesShapiro: Composition Operators and Classical Function TheorySimonnet: Measures and ProbabilitySmith: Power Series From a Computational Point of ViewSmorynski: Self-Reference and Modal LogicStillwell: Geometry of SurfacesStroock: An Introduction to the Theory of Large DeviationsSunder: An Invitation to von Neumann AlgebrasTondeur: Foliations on Riemannian ManifoldsWong: Weyl TransformsZong: Strange Phenomena in Convex and Discrete Geometry