A CONSTRUCTION OF RATIONAL WAVELETS AND FRAMES IN …€¦ · A CONSTRUCTION OF RATIONAL WAVELETS...

A CONSTRUCTION OF RATIONAL WAVELETS AND FRAMESIN HARDY–SOBOLEV SPACES WITH APPLICATIONS

TO SYSTEM MODELING∗

NICHOLAS F. DUDLEY WARD† AND JONATHAN R. PARTINGTON†

SIAM J. CONTROL OPTIM. c© 1998 Society for Industrial and Applied MathematicsVol. 36, No. 2, pp. 654–679, March 1998 009

Abstract. Using the Daubechies wavelet theory we establish rational wavelet decompositions ofthe Hardy–Sobolev classes on the half-plane. The decay of wavelet coefficients is analyzed and errorbounds for approximation are given. We give applications to the modeling of linear systems and tothe model reduction of infinite-dimensional systems.

Key words. wavelets, frames, atomic decompositions, matching pursuits, infinite-dimensionalsystems, Hardy–Sobolev spaces

AMS subject classifications. 41, 93

PII. S0363012996297339

1. Introduction.

1.1. Notation and conventions.C+ = s = x+ iy : x > 0 right half-plane,I = iy : y ∈ R imaginary axis.

For f belonging to L2(R) the Fourier transform f is defined using the followingconvention:

f(ξ) =

∫ ∞

−∞

f(t)e−iξtdt.

For g belonging to L2((0,∞)) we write G = (Lg)(s) for the Laplace transform of g:

G(s) = (Lg)(s) =

∫ ∞

0

g(t)e−stdt.

H2(C+) denotes the Hardy space of functions F (s) analytic in the right half-planeand such that

‖F‖2 = supx>0

∫ ∞

∞

|F (x+ iy)|2dy < ∞.

By the Paley–Wiener theorem every F (s) belonging to H2(C+) is the Laplace trans-form of some f(t) ∈ L2((0,∞)) such that ‖F‖2 = (2π)1/2‖f‖2. Upper-case letterswill be used to denote the Laplace transform of the corresponding lower-case letter,e.g., Ψ(s) = (Lψ)(s). (For further details of Hardy spaces defined on a half-plane see[28] or [36].)

Lettered results, e.g., Theorem A, will denote a known result in the literature.Numbered results will be proved.

∗Received by the editors January 17, 1996; accepted for publication (in revised form) January 17,1997. The research of the first author was supported by the EPSRC.

http://www.siam.org/journals/sicon/36-2/29733.html†School of Mathematics, University of Leeds, Leeds LS2 9JT, UK ([email protected],

[email protected]).

654

WAVELETS AND FRAMES IN HARDY–SOBOLEV SPACES 655

1.2. General background. The approximation and identification of transferfunctions of stable linear time-invariant infinite-dimensional systems by those of finite-dimensional systems is an important part of modern systems theory, and there are twonorms which are most commonly considered. The H∞ norm has found applicationsin robust control [20], whereas the H2 norm is the basis of linear quadratic Gaussiancontrol. However, it is widely regarded as desirable to be able to consider both H2

and H∞ criteria simultaneously: see, for example, the articles of Foias, Frazho, andTannenbaum [19] and Bernstein and Haddad [5]. One approach to this is by meansof Hardy–Sobolev classes.

Hardy–Sobolev classes for the disc were considered in [15] and [2]. For the upperhalf-plane the Hardy–Sobolev class H2,m, m ∈ R, is defined as follows: let H2,m,m ∈ R, be the class of functions F (s) analytic in the right half-plane such thatF (s) = (Lf)(s) and

‖F‖2,m =

(∫ ∞

0

|f(t)|2(1 + t)2mdt

)1/2

< ∞.(1.1)

If m = 0, H2,0 corresponds to the classical Hardy space H2 for the half-plane, and ifm is a positive integer, F (s) belonging to H2,m is equivalent to the first m derivativesof F (s) belonging to H2. An equivalent norm can then be defined on H2,m by theformula

‖F‖22,m = ‖F‖2

2 + ‖F ′‖22 + · · · + ‖Fm‖2

2.

In this investigation we shall be concerned mainly with the range |m| ≤ 1. The rangem > 1/2 is of particular interest since for F (s) belonging to H2,m we have ‖F‖∞ ≤Cm‖F‖2,m, where Cm is a constant, and so approximation in the Hardy–Sobolev normgives simultaneous approximations in both the uniform and the L2 norms. Indeed,we can say more: since we obtain, using the Cauchy–Schwarz inequality,

‖f‖L1 =

∫ ∞

0

|f(t)| dt

≤ ‖f(t)(1 + t)‖L2‖1/(1 + t)‖L2

= ‖F‖2,1,

we obtain simultaneous approximation in the L1 norm as well, itself important inbounded input, bounded output (BIBO) stable control theory (cf. [10]).

The approach to approximation that we shall take will involve the theory ofwavelets, and we now outline this.

1.3. Wavelet classes for H2. We give a definition of wavelet which is perhapssomewhat more geometric than is normal, and which includes the more usual analyticdefinition and is appropriate for both the disc and half-plane. By a lattice L we meana discrete set of points in the right half-plane which is both sufficiently dense andseparated with respect to the pseudohyperbolic metric: there exists 0 < δ1 ≤ δ2 < 1such that the union of balls B(S, δ2), S ∈ L, covers C+ (i.e., δ2 dense) and S, T ∈L, S 6= T =⇒ ρ(S, T ) > δ1 (i.e., δ1 separated), where

ρ(S, T ) =

∣

∣

∣

∣

S − T

S + T

∣

∣

∣

∣

.

(See [42] for details.)

656 NICHOLAS F. DUDLEY WARD AND JONATHAN R. PARTINGTON

In effect, to say that a set is separated in the above sense amounts to the factthat the ratio of the distance between neighboring points in the right half-plane tothe distance from the imaginary axis is approximately constant. In this paper we areinterested in lattices of the following form:

L =

Sj,k : Sj,k =1

2j+ i

k

2jb0; j, k ∈ Z

,

where b0 is some fixed positive constant. Clearly, L is separated and C1 dist(Sj,k, I) ≤dist(Sj,k, Sj′,k′) ≤ C2 dist(Sj,k, I). Associated with a given lattice L will be a waveletclass W = FW : W ∈ L. Notice that any point in L can be transformed ontoanother by a dilation followed by a translation. We construct W so that each FW isobtained from some fixed mother wavelet Ψ(y) ∈ H2 in a similar fashion. Thus Wactually consists of the system of functions Ψj,k(y) = 2j/2Ψ(2jy−kb0) where we havenormalized them in L2.

In this investigation we are interested in wavelet classes for H2 with motherwavelets of the form Ψ(y) = (1 + iy)p, where p is a fixed positive integer. Supposethat W = X + iY = 2−j + ikb02

−j ∈ L. Then the corresponding function Ψj,k canbe written

ΨW (y) =X−1/2

(1 + i(y − Y )/X)p

=Xp−1/2

((X − iY ) + iy)p

=Xp−1/2

(iy +W )p.

For p = 1 ΨW is just the normalized Cauchy kernel with pole at −W , and for p = 2ΨW is the Bergman kernel evaluated on the imaginary axis. From the practical pointof view the usefulness of the wavelet classes just given is based on the observation thatone has a system of rational functions whose poles cluster on the left of the imaginaryaxis. Much of the present investigation centers about the general problem of obtainingdecompositions by using wavelets as elementary building blocks in a sense which willbe made precise below.

Using a fundamental result of Daubechies we obtain decompositions of H2 wherethe mother wavelets consist of powers of the Cauchy kernel for the right half-plane.Consider for a moment the space H2,1 of the disc. It is easy to see that f(z) =∑∞

n=0 anzn ∈ H2,1 is equivalent to ‖f‖2

2,1 =∑∞

n=0(1 + n2)|an|2 < ∞. That is, onecan determine whether a function f(z) in H2 also has L2 bounded derivative byexamining decay of the Taylor coefficients an = 〈f, zn〉. We obtain similar conditionsfor H2,1(C+) where the building blocks are replaced by a nonorthogonal system ofrational wavelets. We also consider in section 6 the half-plane algebra A(C+) for theright half-plane and use a result of Hayman and Lyons to show that suitable sets ofCauchy kernels are fundamental for A(C+). In section 7 we consider error estimates,and in section 8 we deduce algorithms which may be applied to the model reductionof a system.

The following results will be established.THEOREM 1.1. Let Ψ(y) = (1 + iy)−3, let Ψj,k(y) = 2j/2Ψ(2jy − b0k) where

j, k ∈ Z, b0 > 0, and let 〈·, ·〉 be the usual L2-inner product.

If b0 > 0 is sufficiently small, then for each m with −1 ≤ m ≤ 1 there exist

positive constants Am and Bm such that for F (s) ∈ H2,m the wavelet coefficients


〈F,Ψj,k〉 satisfy the following pair of inequalities:

Am‖F‖22,m ≤

∑

j,k

| 〈F,Ψj,k〉 |2(1 + 2j)2m ≤ Bm‖F‖22,m.(1.2)

Our next result is an atomic decomposition of H2,m. Let ℓ2((1 + 22j)m) denotethe weighted ℓ2-space

ℓ2((1 + 2j)2m) =

λ = (λj,k) : ‖λ‖22,m =

∑

|λj,k|2(1 + 2j)2m < ∞

.

A sequence Ψj,k in H2,m will be called a set of atoms (with respect to ℓ2((1 +2j)2m)) for H2,m if the mapping S : ℓ2((1 + 2j)2m) → H2,m defined by

S : λ 7→∑

j,k

λj,kΨj,k

is a surjection. Let Ψj,k = 2j/2Ψ(2jy − b0k) where Ψ(t) = (1 + iy)−3.THEOREM 1.2. The system (Ψj,k), j, k ∈ Z, is a set of atoms for H2,m for

sufficiently small b0 in the sense defined above. Furthermore, for each F belonging to

H2,m we have the inequalities

1

B−m‖F‖2

2,m ≤ inf

‖(λj,k)‖22,m : F =

∑

j,k

λj,kΨj,k

≤ 1

A−m‖F‖2

2,m.(1.3)

THEOREM 1.3. Let T be the operator defined on H2,m, where m = 1 or m = −1by the formula

TF =∑

〈F,Ψj,k〉 Ψj,k, F ∈ H2,m.

Then, for sufficiently small b0, T is a bounded map H2,m → H2,m. Furthermore, Tis a surjection and invertible.

Finally, we have the following frame decomposition of H2,m for both m = 1 andm = −1.

COROLLARY 1.4. Every F ∈ H2,m, m = ±1, has the wavelet series representation

F = TT−1F =∑

j,k

⟨

T−1F,Ψj,k

⟩

Ψj,k.

We shall prove Theorem 1.1 for the cases m = ±1 in section 3, and for the generalcase at the end of section 5. Theorems 1.2 and 1.3 will be proved in sections 4 and5, respectively. We also obtain estimates for the various constants for a range of b0which appear in the given results by means of computer, and for these we refer to thetables.

2. Frames and wavelet decompositions of H2.

2.1. Brief exposition of the theory of frames. Frames were introduced byDuffin and Schaeffer in [17] in the context of nonharmonic Fourier series, and givea technique for expanding vectors in a Hilbert space by means of systems of non-orthogonal vectors.

We give below a brief exposition of the theory of frames. For further details andfuller proofs we refer to [17], [13], and [34]. Let H be a Hilbert space with inner


product 〈·, ·〉. A system (φj) of vectors in H will be called a frame for H if there existpositive constants A and B so that the following pair of inequalities obtain:

A‖f‖2 ≤∑

j

| 〈f, φj〉 |2 ≤ B‖f‖2, f ∈ H.(2.1)

We define the operator T on H by means of the formula

Tf =∑

j

〈f, φj〉φj , f ∈ H.

From (2.1) it follows that T is a positive operator on H such that 〈Tf, f〉 ≥ A‖f‖2.Next one can establish that T is a surjection and is invertible by means of the followingpiggyback closed range theorem.

LEMMA A. If U is a positive operator on H such that 〈Uf, f〉 ≥ α‖f‖2 for all

f ∈ H then U is invertible on H and its inverse U−1 satisfies ‖U−1‖ ≤ 1/α.

Proof. We first note that range(U) is a closed subspace of H. Let fn be a Cauchysequence in range(U). Then fn = U(gn), and by hypothesis,

‖gn − gm‖2 ≤ α−1 〈U(gn − gm), gn − gm〉 ≤ α−1‖U(gn − gm)‖ · ‖gn − gm‖.

Thus gn is also a Cauchy sequence in H with limit g. It follows from the continuityof U that U has closed range.

Next we show that range(U) is dense in H: if g ∈ H is such that 〈Uf, g〉 = 0 forall f ∈ H, it follows in particular that 〈Ug, g〉 = 0. We have α‖g‖2 ≤ 〈Ug, g〉, andhence we deduce that g = 0. Hence, by the first part, range(U) = H.

Finally, since any f ∈ H can be written as f = Ug we may define U−1f = g and

α‖U−1f‖2 ≤⟨

UU−1f, U−1f⟩

=⟨

f, U−1f⟩

≤ ‖f‖ · ‖U−1f‖,

whence ‖U−1f‖ ≤ α−1‖f‖. This completes the proof.In fact, one can show using (2.1) that I − 2(A + B)−1T is a contraction and so

compute T−1 by a Neumann series: suppose that f ∈ H, and define a residual vector

R(f) by the equation

R(f) = f − 2

A+B

∑

j

〈f, φj〉φj .

Then −B−AB+AI ≤ R ≤ B−A

B+AI, and so, since R is symmetric, ‖R‖ ≤ B−AB+A . If B is close

to A we obtain a good reconstruction of f . Otherwise we can write down an algorithm,viz., a Neumann series for the reconstruction of f with exponential convergence. (See[13] for details.)

Since T−1 is symmetric it follows that⟨

T−1f, φj

⟩

=⟨

f, T−1φj

⟩

and so f ∈ Hhas the expansions

f = TT−1f =∑

⟨

T−1f, φj

⟩

φj =∑

⟨

f, T−1φj

⟩

φj .

The system defined by φj = T−1φj is called the dual frame associated with φj . It can

also be shown that φj is itself a frame.


2.2. Fundamental theorem of Daubechies. Given a function Ψ ∈ L2(R),and a0 > 1 and b0 > 0, we define the system of functions

Ψj,k(t) = a0j/2Ψ(aj

0t− kb0), j, k ∈ Z.(2.2)

We ask for conditions on Ψ and b0 such that the system given by (2.2) is a frame forL2(R). That is, there exist positive constants A, B such that

A‖f‖22 ≤

∑

j,k

| 〈f,Ψj,k〉 |2 ≤ B‖f‖22, f ∈ L2(R).

Daubechies [12], [13], proves the following fundamental result.THEOREM B. Suppose that Ψ ∈ L2(R) and a0 > 1 are such that

inf1≤|ξ|≤a0

∞∑

−∞

|Ψ(aj0ξ)|2 > 0,

sup1≤|ξ|≤a0

∞∑

−∞

|Ψ(aj0ξ)|2 < ∞,

and if β(s) = sup1≤|ξ|≤a0

∑ |Ψ(aj0ξ)||Ψ(aj

0ξ+s)| decays at least as fast as (1+|s|)−(1+ǫ)

with ǫ > 0, then there exists a b0 > 0 such that the Ψj,k constitute a frame for all

b0 < b0. For such b0 the following equalities are frame bounds for the Ψj,k:

A=2π

b0

inf1≤|ξ|≤a0

∞∑

−∞

|Ψ(aj0ξ)|2 −

∞∑

k=−∞k 6=0

[

β

(

2π

b0k

)

β

(−2π

b0k

)]1/2

,

B=2π

b0

sup1≤|ξ|≤a0

∞∑

−∞

|Ψ(aj0ξ)|2 +

∞∑

k=−∞k 6=0

[

β

(

2π

b0k

)

β

(−2π

b0k

)]1/2

.

Since the Fourier transforms of H2 functions of the lower half-plane are functionsin L2(R) supported on the positive real axis, we may apply Theorem B to H2(P)where P = x + iy : y < 0. By rotating, we see that Theorem B is applicable toestablishing frames for H2(C+). In particular, it is routine to check that the estimatesgiven in the hypotheses of Theorem B are satisfied by mother wavelets of the formΨ(y) = (1 + iy)−p where p is a positive integer not less than 2 (since Ψ(y) is theLaplace transform of ψ(t) = tp−1e−t/(p− 1)!, t > 0).

In the main we will be interested in the system Ψj,k(y) = 2j/2Ψ(2jy− b0k) whereΨ is the mother wavelet Ψ(y) = (1+ iy)−3. However, in order to establish our results,we also consider some auxiliary systems of wavelets—in particular, ones generated byΦ(y) = (1 + iy)−2. With a0 = 2 and a given mother wavelet Ψ, one can then easilyestablish by means of a computer a range of b0 for which the corresponding systemΨj,k(y) is a frame for H2. See Tables 2.1 and 2.2 for the wavelets Ψ(y) = (1 + iy)−3

and Φ(y) = (1 + iy)−2.

3. Proof of Theorem 1.1 for m = ±1.


TABLE 2.1Estimates of frame bounds for the mother wavelet Ψ(y) = (1 + iy)−3.

b0 A B B/A0.25 13.579 13.615 1.0030.5 6.786 6.811 1.0041.0 3.114 3.685 1.183

2 0.460 2.939 6.3873 −0.591 2.857 −4.832

TABLE 2.2Estimates of frame bounds for the mother wavelet Ψ(y) = (1 + iy)−2.

b0 A B B/A0.25 9.064 9.066 1.0000.5 4.532 4.533 1.0001.0 2.208 2.325 1.053

2 0.738 1.528 2.0693 0.118 1.392 11.777

3.1. Proof of Theorem 1.1: m = 1. The proof of Theorem 1.1 for m = 1follows from the considerations of the previous section. Since Ψj,k(y) is a frame forH2 it follows that for F belonging to H2,

AΨ‖F‖22 ≤

∑

j,k

| 〈F,Ψj,k〉 |2 ≤ BΨ‖F‖22.(3.1)

If, in addition, F ′ also belongs to H2 (so that F ∈ H2,1) then since Φj,k(y) withΦ(y) = 2(1 + iy)−2 is also a frame for H2 we have

AΦ‖F ′‖22 ≤

∑

j,k

| 〈F ′,Φj,k〉 |2 ≤ BΦ‖F ′‖22.(3.2)

Up to a constant, Ψ is attained from Φ by differentiation with respect to y. Thereforewe may integrate by parts and obtain

〈F ′,Φj,k〉 =

∫ ∞

−∞

F ′(it)Φj,k(t)dt

= 2j

∫ ∞

−∞

F (it)Ψj,k(t)dt

= 2j 〈F,Ψj,k〉 .Therefore, (3.2) may be replaced by

Aφ‖F ′‖22 ≤

∑

j,k

| 〈F,Ψj,k〉 |222j ≤ Bφ‖F ′‖22.(3.3)

Adding (3.1) and (3.3) and noting that

1

2(1 + x)2 ≤ (1 + x2) ≤ (1 + x)2,

we obtain the conclusion of Theorem 1.1 with A1 = 1/4 minAΨ, Aφ and B1 =maxBΨ, Bφ.

Remark. One could prove Theorem 1.1 directly using the same technique asthe proof of Theorem 3.1 below. However, the proof given has the merit of beingelementary, although it will not give the best estimates for A1 and B1.


3.2. Proof of Theorem 1.1: m = −1. We wish to establish the existenceof positive constants A−1 and B−1 such that for G = Lg belonging to H2,−1, thefollowing inequalities obtain:

A−1‖G‖22,−1 ≤

∑

j,k

| 〈G,Ψj,k〉 |2(1 + 2j)−2m ≤ B−1‖G‖22,−1.

Since G = L(g) ∈ H2,−1 it follows that g(t) = g(t)(1 + t)−1 ∈ L2((0,∞)). If Φj,k =L(φj,k) is a frame for H2 there exist constants Aφ, Bφ such that

Aφ‖g‖22 ≤

∑

j,k

| 〈g, φj,k〉 |2 ≤ Bφ‖g‖22,

i.e.,

Aφ‖g‖22,−1 ≤

∑

j,k

| 〈g, φj,k〉 |2 ≤ Bφ‖g‖22,−1.

Now

〈g, φj,k〉 =⟨

g, (1 + t)−1φj,k

⟩

.

We would like to choose a frame Φj,k for H2 so that

φj,k = (1 + t)(1 + 2j)−1ψj,k,

where Ψ is the mother wavelet Ψ(y) = L(t2e−t/2) = (1 + iy)−3, Ψj,k = 2j/2Ψ(2jy −b0k), and Ψj,k = L(2−j/2(2−jt)2e−2−j(t−ib0k)).

Our next result is an analogue of Theorem B.THEOREM 3.1. Suppose that Ψ(y) ∈ H2(C+) and that Ψj,k(y) = 2j/2Ψ(2jy−kb0),

j, k ∈ Z. Define the system Φj,k by φj,k(t) = (1+ t)(1+2j)−1ψj,k(t). Suppose further

that

inf0<t<∞

(1 + t)2∞∑

−∞

(1 + 2j)−2|ψ(2−jt)|2 > 0,

sup0<t<∞

(1 + t)2∞∑

−∞

(1 + 2j)−2|ψ(2−jt)|2 < ∞,

and if γ(s) = sup0<t<∞(1 + t)2∑

(1 + 2j)−2|ψ(2−jt)||ψ(2−jt + s)| decays at least as

fast as (1+|s|)−(1+ǫ) with ǫ > 0, then there exists a b0 > 0 such that the Φj,k constitute

a frame for H2 for all b0 < b0. For such b0 the following equalities are frame bounds

for the Φj,k:

A−1 =2π

b0

inf0<t<∞

(1 + t)2∞∑

−∞

(1 + 2j)−2|ψ(2−jt)|2

−∞∑

k=−∞k 6=0

[

γ

(

2π

b0k

)

γ

(−2π

b0k

)]1/2

,

B−1 =2π

b0

sup0<t<∞

(1 + t)2∞∑

−∞

(1 + 2j)−2|ψ(2−jt)|2

+

∞∑

k=−∞k 6=0

[

γ

(

2π

b0k

)

γ

(−2π

b0k

)]1/2

.


Suppose that G = Lg ∈ H2. We proceed to analyze∑

j,k | 〈g, φj,k〉 |2 using themethod of Daubechies:

∑

j,k

| 〈g, φj,k〉 |2 =∑

j,k

∫ ∞

0

g(t)φj,k(t)dt

∫ ∞

0

g(t′)φj,k(t′)dt′

=∑

j,k

(1 + 2j)−2

∫ ∞

0

g(t)(1 + t)ψj,k(t)dt

×∫ ∞

0

g(t′)(1 + t′)ψj,k(t′)dt′

=∑

j,k

(1 + 2j)−22−j

∫ ∞

0

g(t)(1 + t)ψ(2−jt)e−i2−jb0ktdt

×∫ ∞

0

g(t′)(1 + t′)ψ(2−jt′)ei2−jb0kt′

dt′

=2π

b0

∑

j,k

(1 + 2j)−2

∫ ∞

0

∫ ∞

0

g(t)g(t′)(1 + t)ψ(2−jt)(1 + t′)ψ(2−jt′)

× δ(t′ − t− 2πk2jb−10 ) dt dt′

=2π

b0

∑

j,k

(1 + 2j)−2

∫ ∞

0

g(t)g(t− 2π2jb−10 k)(1 + t)ψ(2−jt)

× (1 + t− 2πk2jb−10 )ψ(2−jt− 2πkb−1

0 )dt

=2π

b0

∑

j

(1 + 2j)−2

∫ ∞

0

|g(t)|2(1 + t)2|ψ(2−jt)|2dt+ rest(g),

where in the fourth line we have used the Poisson summation formula∑

l∈Zexp(ilax) =

2πa−1∑

k∈Zδ(x− 2πka−1). This is justified by supposing first that g is smooth and

has compact support and then proceeding by a standard density argument. If

m = inf0<t<∞

(1 + t)2∞∑

−∞

(1 + 2j)−2|ψ(2−jt)|2,

M = sup0<t<∞

(1 + t)2∞∑

−∞

(1 + 2j)−2|ψ(2−jt)|2

we have

m‖g‖22 ≤

∑

j

(1 + 2j)−2

∫ ∞

0

|g(t)|2(1 + t)2|ψ(2−jt)|2dt ≤ M‖g‖22.(3.4)

3.3. Analysis of rest(g). We have

rest(g) =2π

b0

∑

j

∑

k 6=0

(1 + 2j)−2

∫ ∞

0

g(t)g(t− 2πk2jb−10 )(1 + t)ψ(2−jt)

×(1 + t− 2πk2jb−10 )ψ(2−jt− 2πkb−1

0 )dt.


TABLE 3.1Estimates of A−1 and B−1 in Theorem 1.1.

b0 A−1 B−1

0.25 13.699 65.5240.5 6.797 32.8141.0 0.775 19.0311.1 −0.324 18.330

By the Cauchy–Schwarz inequality for integrals we have

| rest(g)| ≤ 2π

b0

∑

j

∑

k 6=0

(1+2j)−2

∫ ∞

0

|g(t)|2(1 + t)2|ψ(2−jt)||ψ(2−jt− 2πkb−10 )|dt

1/2

×∫ ∞

0

|g(t− 2πk2jb−10 )|2(1 + t− 2πk2jb−1

0 )2|ψ(2−jt)||ψ(2−jt− 2πkb−10 )|dt

1/2

.

We make the substitution t− 2πk2jb−10 7→ t in the second integral and deduce that

| rest(g)| ≤ 2π

b0

∑

j

∑

k 6=0

(1+2j)−2

∫ ∞

0

|g(t)|2(1 + t)2|ψ(2−jt)||ψ(2−jt− 2πkb−10 )|dt

1/2

×∫ ∞

0

|g(t)|2(1 + t)2|ψ(2−jt)||ψ(2−jt+ 2πkb−10 )|dt

1/2

.

Next we apply the Cauchy–Schwarz inequality for sums to the sum indexed by j andobtain

| rest(g)| ≤ 2π

b0

∑

k 6=0

∑

j

(1 + 2j)−2

∫ ∞

0

|g(t)|2(1 + t)2|ψ(2−jt)||ψ(2−jt− 2πkb−10 )|dt

1/2

×

∑

j

(1 + 2j)−2

∫ ∞

0

|g(t)|2(1 + t)2|ψ(2−jt)||ψ(2−jt+ 2πkb−10 )|dt

1/2

.

Therefore, if we write

γ(s) = sup0<t<∞

(1 + t)2∑

j

(1 + 2j)−2|ψ(2−jt)||ψ(2−jt+ s)|,

we obtain

| rest(g)| ≤ 2π

b0

∞∑

k=−∞k 6=0

[

γ

(

2π

b0k

)

γ

(−2π

b0k

)]1/2

.(3.5)

Combining (3.4) with (3.5), we obtain the conclusion of Theorem 3.1.We apply Theorem 3.1 to the considerations above with Ψ(y) = (1+ iy)−3. Since

ψ(t) = 2−1t2e−t it is easy to see that the hypotheses of the theorem are satisfied. Werefer to Table 3.1 for estimates of frame bounds for different b0. This completes theproof of Theorem 1.1 in the case m = −1.


4. Proof of Theorem 1.2. Although H2,m is a Hilbert space and so, by theRiesz–Frechet theorem, is its own dual, we identify (H2,m)∗ with H2,−m by means ofthe following pairing:

〈F,G〉 =

∫

FG, F ∈ H2,m, G ∈ H2,−m.

In fact, the map M(1+t)2m : H2,m → H2,−m, F (s) 7→ (Lf(t)(1 + t)2m)(s) is anisometric isomorphism. (For further details, see [21].) Note also that | 〈F,G〉 | ≤‖F‖2,m‖G‖2,−m and that sup| 〈F,G〉 | : ‖G‖2,−m = 1 = ‖F‖2,m.

4.1. Atomic decompositions of H 2,m . A useful tool for establishing atomicdecompositions for a Banach space is Banach’s closed range theorem, which we statebelow for the convenience of the reader. We refer to the articles of Bonsall [6], [7] (seealso [14]) and the references contained therein for further information.

For a subset E of a normed space X let E⊥ = φ ∈ X∗ : φ(x) = 0 for x ∈ X;for a subset F of X∗ let F⊥ = x ∈ X : φ(x) = 0 for φ ∈ X∗.

THEOREM C (Banach’s closed range theorem). Let T be a bounded linear mapping

of a Banach space X into a Banach space Y .

(i) If T ∗Y ∗ is closed in X∗, then TX is closed in Y and TX = (kerT ∗)⊥.

(ii) If TX is closed in Y , then T ∗Y ∗ is closed in X∗ and T ∗Y ∗ = (kerT )⊥.

LEMMA 4.1. The adjoint S∗ from (H2,m)∗ = H2,−m to (ℓ2((1 + 2j)2m))∗ =ℓ2((1 + 2j)−2m) is given by the formula

S∗ : G 7→ 〈G,Ψj,k〉, G ∈ H2,−m.

Proof. Suppose that G ∈ H2,−m and let φG be the functional corresponding toG. Then, for any sequence λ = (λj,k) belonging to ℓ2((1 + 2j)2m), it follows from thedefinition of S∗ that

(S∗φG)(λ) = 〈G,Sλ〉=

∑

j,k

λj,k 〈G,Ψj,k〉 .

Thus S∗ maps G 7→ 〈G,Ψj,k〉.LEMMA 4.2. S∗ has zero kernel.

Proof. By definition, if G ∈ H2,−m,

‖S∗G‖22,−m = sup

λ∈ℓ2((1+2j)2m)

|(S∗G)(λ)|2‖λ‖2

2,m

= supλ∈ℓ2((1+2j)2m)

∣

∣

∣

∑

j,k λj,k 〈G,Ψj,k〉∣

∣

∣

2

‖λ‖22,m

.(4.1)

We now define the sequence (λj,k) by

λj,k = 〈G,Ψj,k〉(1 + 2j)−2m.

Then

‖(λj,k)‖22,m =

∑

j,k

| 〈G,Ψj,k〉 |2(1 + 2j)−4m(1 + 2j)2m

=∑

| 〈G,Ψj,k〉 |2(1 + 2j)−2m

≤ B−m‖G‖22,−m


by Theorem 1.1, and so is a candidate for the supremum in equation (4.1). Therefore

‖S∗G‖22,−m ≥

(

∑

j,k | 〈G,Ψj,k〉 |2(1 + 2j)−2m)2

∑

j,k |〈G,Ψj,k〉|2 (1 + 2j)−2m

=∑

j,k

| 〈G,Ψj,k〉 |2(1 + 2j)−2m

≥ A−m‖G‖22,−m,(4.2)

by Theorem 1.1. This completes the proof.LEMMA 4.3. S∗ has closed range.

Proof. Let ((λnj,k))n be a Cauchy sequence in ℓ2(1 + 2j)−2m in the range of S∗,

so that ((λnj,k))n → (λj,k). Then each sequence (λn

j,k) = S∗Gn, where Gn belongs to

H2,m. By the inequality derived in (4.2) it follows that (Gn) is a Cauchy sequencein H2,−m so that Gn → G. Therefore, it follows from the continuity of S∗ that(λj,k) = S∗(G) and we have the desired result.

The first part of Theorem 1.2 follows immediately from Lemmas 4.1, 4.2, 4.3, andBanach’s closed range theorem.

The lower bound in (1.3) follows immediately from an elementary duality argu-ment. To obtain the upper bound let N = kerS and X = ℓ2((1 + 2j)2m)/N anddefine U : X → H2,m by U [λ] = Sµ, (µ ∈ [λ] ∈ X). Plainly, U is a bounded linearbijection of X onto H2,m and so, by Banach’s isomorphism theorem, has a boundedinverse. Therefore U∗ is a bounded linear bijection from H2,−m onto X∗. From theproof of Lemma 4.2 we know that

A−m‖G‖22,−m ≤ sup‖(S∗G)(λ)‖2

2,−m : λ ∈ ℓ2((1 + 2j)2m)= sup‖ 〈F,U [λ]〉 : [λ] ∈ X, ‖[λ]‖ ≤ 1= ‖U∗G‖.

Hence ‖U−1‖ = ‖(U∗)−1‖ ≤ A−1−m. This completes the proof of Theorem 1.2.

Remark. The usefulness of Banach’s closed range theorem in establishing atomicdecompositions is clear from papers of D. H. Luecking (see, for example, [29] and [30]),who used it to prove decompositions of Bergman and Hardy spaces. In particular, aduality proof of the Coifman–Rochberg decomposition of Lp

a, p > 1, was given [9]. In[6] Bonsall gave an abstract formulation of this method giving various applications,and in [7] he gave an elementary proof of a general atomic decomposition theorem.

5. Proof of Theorem 1.3. We consider the operator

TF =∑

〈F,Ψj,k〉 Ψj,k, F ∈ H2,m,(5.1)

where 〈·, ·〉 is the L2-inner product. We proceed to show that T is bounded on bothH2,1 and H2,−1 and in fact is a bijection. Note that T is symmetric with respect to〈·, ·〉 and bounded on H2.

LEMMA 5.1. T is a bounded operator H2,m → H2,m for both m = 1 and m = −1.Proof. Let TN be the partial sum operator

TN =∑

|j|,|k|≤N

〈·,Ψj,k〉 Ψj,k.


Let F ∈ H2,m and G ∈ H2,−m. Then, by the Cauchy–Schwarz inequality,

| 〈TNF,G〉 |2 =∣

∣

∣

⟨

∑

〈F,Ψj,k〉 Ψj,k, G⟩∣

∣

∣

2

=∣

∣

∣

∑

〈F,Ψj,k〉〈G,Ψj,k〉∣

∣

∣

2

≤∑

| 〈F,Ψj,k〉 |2(1 + 2j)2m∑

| 〈G,Ψj,k〉 |2(1 + 2j)−2m

≤ Bm‖F‖22,m ×B−m‖G‖2

2,−m,

by Theorem 1.1. Now, by duality,

‖TNF‖22,m = sup

G∈H2,−m

| 〈TNF,G〉 |2‖G‖2

2,−m

≤ BmB−m‖F‖22,m,

so that TN is bounded (independently of N) on H2,m and so must T be. By thesymmetry of T (with respect to the duality pairing given by 〈·, ·〉) we conclude alsothat T is bounded on H2,−m.

Next we wish to establish that T is a bijection H2,m → H2,m, for m = 1 andm = −1. To do this we estimate ‖TF‖2

2,m from below. It follows from duality that

‖TF‖2,m = supG∈H2,−m

| 〈TF,G〉 |‖G‖2,−m

= supG∈H2,−m

|∑j,k 〈F,Ψj,k〉〈G,Ψj,k〉|‖G‖2,−m

.

We need to find a suitable candidate for G so that we may show that ‖TF‖22,m ≥

Km‖F‖22,m. We proceed to analyze

∑

j,k 〈F,Ψj,k〉〈G,Ψj,k〉 =∑

j,k 〈f, ψj,k〉〈g, ψj,k〉using the same technique as in the proof of Theorem 3.1. We obtain

∑

j,k

〈F,Ψj,k〉〈G,Ψj,k〉 =2π

b0

∑

j

∫ ∞

0

f(t)g(t)|ψ(2−jt)|2dt

+2π

b0

∑

j

∑

k 6=0

∫ ∞

0

f(t)g(t− 2πk2jb−10 )ψ(2−jt)ψ(2−jt− 2πkb−1

0 )dt.

(5.2)

It is clear in view of (5.2) that an appropriate candidate for G is given by g(t) =f(t)(1 + t)2m (so that ‖g‖2

2,−m = ‖f‖22,m). Next we analyze the second sum on the

right of (5.2) with this choice of g, which we denote by rest(f). We deduce

| rest(f)| ≤ 2π

b0

∑

j

∑

k 6=0

I1 × I2,

where

I1 =

∫ ∞

0

|f(t)|2(1 + t)2m(1 + t− 2πk2jb−10 )2m|ψ(2−jt)||ψ(2−jt− 2πkb−1

0 )|dt1/2

and

I2 =

∫ ∞

0

|f(t− 2πk2jb−10 )|2(1 + t− 2πk2jb−1

0 )2m(1 + t)−2m|ψ(2−jt)|

×|ψ(2−jt− 2πkb−10 )|dt

1/2

.


TABLE 5.1Estimates of the lower bound for ‖TF‖2,1.

b0 0.1 0.2 0.25 0.3 0.5K 33.947 16.974 13.578 11.297 −4.222

We define

µ = inf0<t<∞

∑

j

|ψ(2−jt)|2,

δm1 (s) = sup

0<t<∞

∑

j

(1 + t+ 2js)2m|ψ(2−jt)||ψ(2−jt+ s)|,

δm2 (s) = sup

0<t<∞

∑

j

(1 + t+ 2js)−2m|ψ(2−jt)||ψ(2−jt+ s)|.

If∑

k 6=0δm1 ( 2πk

b0)δm

2 ( 2πkb0

)1/2 =∑

k 6=0δ−m1 ( 2πk

b0)δ−m

2 ( 2πkb0

)1/2 converges and dimin-

ishes to 0 with b0, there exists b0 such that for b < b0, the quantity K = 2πb0

(µ −∑

k 6=0δm1 ( 2πk

b0)δm

2 ( 2πkb0

)1/2) > 0, and we obtain ‖TF‖2,m ≥ K‖F‖2,m. Plainly, we

can apply this argument to the mother wavelet Ψ(y) = (1 + iy)−3. We refer toTable 5.1 for estimates of the constant K for Ψ(y) and for different b0. Note thatfor b0 = 0.5 the value of K is negative, and so we cannot deduce that T is boundedbelow.

As in section 4 it is now plain that T : H2,m → H2,m has closed range. Next weobserve that TF : F ∈ H2,m is dense in H2,m. Suppose that G ∈ H2,−m and that

〈TF,G〉 = 〈F, TG〉 = 0, F ∈ H2,m.

If we take F = Lf defined by f(t) = g(t)(1 + t)−2m, we deduce that

0 =sup | 〈TF,G〉 |

‖G‖2,−m≥ K‖G‖2,−m,

and so G = 0. It follows from the Hahn–Banach theorem that the set TF : F ∈H2,m is dense in H2,m. Hence, since T has closed range in H2,m, we deduce thatrange(T ) = H2,m. This completes the proof of Theorem 1.3.

Remark. The proof given here should be compared with the proof of Proposition2.12 in [12], which unfortunately contains an error. However, we achieve similarlysymmetric lower and upper bounds for ‖TF‖2,m, and as Daubechies points out in

Remark 2, it follows by interpolation that T is bounded above and below on H2,m′

for −1 ≤ m′ ≤ 1. (See section IX.4, particularly Example 3 (rigged Hilbert spaces),in [41].)

5.1. Proof of Theorem 1.1: General case. To obtain Theorem 1.1 for −1 ≤m ≤ 1 it is simplest to proceed (as a referee has observed) by interpolation. Theupper bound

∑

j,k

| 〈F,Ψj,k〉 |2(1 + 2j)2m ≤ Bm‖F‖22,1(5.3)

is obtained by applying a theorem of Stein (Theorem 3.6 in Chapter 4 of [4]) to theoperator R defined by

RF = (〈F,Ψj,k〉),


which, as we have seen, can be regarded as a bounded operator between weightedHilbert spaces as follows:

R : L2(0,∞; (1 + t)(1−2θ)dt) → ℓ2((1 + 2j)(1−2θ))

for θ = 0 and 1. Hence, by Stein’s theorem, it is bounded for all intermediate valuesof θ, and we can take B(1−2θ) ≤ B1−θ

1 Bθ−1.

The lower bound is now most simply obtained by duality between H2,m andH2,−m. Namely, by Theorem 1.3 we can find a constant Cm > 0 such that ‖TF‖ ≥Cm‖F‖ for each F ∈ H2,m. Then there exists a normalized vector G ∈ H2,−m suchthat

| 〈TF,G〉 | ≥ Cm‖F‖2,m.

Hence, by the Cauchy–Schwarz inequality,

∑

| 〈F,Ψj,k〉 |2(1 + 2j)2m∑

| 〈G,Ψj,k〉 |2(1 + 2j)−2m ≥ C2m‖F‖2

2,m

and so

∑

| 〈F,Ψj,k〉 |2(1 + 2j)2m ≥ C2mB

−2−m‖F‖2

2,m,

as required.

6. Wavelets for the half-plane algebra. An interesting class of wavelets isobtained when p = 1 in section 1.3. That is, Ψ(y) = (1 + iy)−1 is the Cauchy kernel.This wavelet does not fall under the Daubechies theory since is does not have vanishingmean value, but the system Ψj,k does constitute a fundamental set for the half-planealgebra A(C+), and so one can use the Cauchy kernel to obtain approximations inthe uniform norm. In [16] it was shown using the Hayman–Lyons theory that certainsets of Cauchy kernels formed complete model sets in the disc algebra. By usinginequalities due to Borwein and Erdelyi [8] it was shown how these model sets couldbe used for worst-case identification.

In this section we consider the half-plane analogue of the main results in [16]. Sincethey are deduced from [16] by conformal mapping the proofs will be abbreviated.

Let D be the unit disc andA(D) = f(z) : f(z) is analytic in D and continuous in Dbe the disc algebra. Next let Qj,k be the Whitney cube partition of D defined forj = 1, 2, . . . and k = 1, 2, . . . , 2j by

Qj,k =

z : 1 − 1

2j≤ |z| ≤ 1 − 1

2j+1,2kπ

2j≤ arg z ≤ 2(k + 1)π

2j

.

If A ⊂ D we set Aj,k = A ∩Qj,k and zj,k =(

1 − 12j

)

exp 2πik2j . Finally, we define s(θ)

by

s(θ) = s(θ,A) =∑

Aj,k 6=∅

(

1 − |zj,k||zj,k − eiθ|

)2

.(6.1)

We say that A satisfies the Hayman–Lyons condition if, and only if, s(θ) = +∞ forall θ ∈ [0, 2π]. In [16] the following result was established.


THEOREM D. Suppose that A ⊂ D and that A satisfies the Hayman–Lyons con-

dition. Then if f ∈ A(D) and ǫ > 0, there exists λ1, . . . , λn ∈ C and a1, . . . , aN ∈ Asuch that

∥

∥

∥

∥

∥

f(z) −N

∑

k=1

λk1

1 − akz

∥

∥

∥

∥

∥

∞

< ǫ.

That is, the set W = Cw(z) : w ∈ A, where Cw(z) = (1 − wz)−1 is a fundamental

set for A(D).Our next result is a half-plane version of Theorem D. Let A(C+) be the half-plane

algebra for C+, that is, the set of F (s) analytic in C+, continuous up to the imaginaryaxis and such that limy→±∞ F (iy) exists. Plainly, the map Mz = (1 − z)(1 + z)−1

gives an isometric isomorphism between A(C+) and A(D).THEOREM 6.1. Let C ′

w(s) = (w+ s)−1 be the Cauchy kernel for C+ and L ⊂ C+

be the lattice

L = Zj,k : ℜ(Zj,k) = 2−j ,ℑ(Zj,k) = k2−j , j, k ∈ Z.

Then the set W = 1 ∪ C ′w(iy) : w ∈ L is a fundamental set for A(C). That is,

given F (s) ∈ A(C+) and ǫ > 0 there exists λ1, . . . , λN ,K ∈ C and w1, . . . , wN ∈ Lsuch that

∥

∥

∥

∥

∥

F (s) −N

∑

k=1

λk1

wk + s−K

∥

∥

∥

∥

∥

∞

< ǫ.

We sketch the proof. We need to reformulate the Hayman–Lyons condition. LetQj,k be the Whitney cube division of C+ defined by

Qj,k =

s = x+ iy :1

2j≤ x ≤ 1

2j−1,k

2j≤ y ≤ (k + 1)

2j

.

For a set A in C+ let Aj,k = A ∩ Qj,k and Zj,k = Xj,k + iYj,k = 2−j + ik2−j .It is easy to see by conformal transformation that the series in the Hayman–Lyonscondition (6.1) takes the form

S(iy) = S(iy, A) =∑

Aj,k 6=∅

(

Xj,k

|Zj,k − iy|

)2

.

If iy is infinite S(∞) is defined by

S(∞) = S(∞, A) =∑

Aj,k 6=∅

(

Xj,k

|Zj,k|

)2

.

It is then straightforward to show that if S(iy) = ∞ for all y in R where for each ywe sum over all those cubes Qj,k which meet a disc with center iy, that C ′

w(iy) :w ∈ A ∪ 1 is a fundamental set for A(C+). Suppose that G(s) ∈ A(C+). ThenG((1−z)(1+z)−1) ∈ A(D). Let A be the inverse image of A by (1−s)(1+s)−1. Then,since the hyperbolic metric is conformally invariant it follows that s(eiθ,A) = +∞


for θ ∈ [0, 2π], so that given ǫ > 0, there exists λ1, . . . , λN ∈ C and a1, . . . , aN ∈ Asuch that

∥

∥

∥

∥

∥

G

(

1 − z

1 + z

)

−N

∑

k=1

λk1

1 − akz

∥

∥

∥

∥

∥

∞

< ǫ.(6.2)

Suppose that ak = (1 − wk)(1 + wk)−1 and z = (1 − s)(1 + s)−1. Then

1

1 − akz=

1 + s

(1 − ak) + s(1 + ak)

=1 + wk

2

(

1 + (1 − wk)C ′wk

(s))

.

Hence (6.2) takes the form∥

∥

∥

∥

∥

G(s) −N

∑

k=1

λ′kC

′wk

(s) −K

∥

∥

∥

∥

∥

∞

< ǫ.

Plainly, the lattice L satisfies the Hayman–Lyons condition for the right half-plane,and so for A in the above argument we may take the lattice L and deduce that thecorresponding set of wavelets W is a fundamental set for A(C+)

Remark. The Hayman–Lyons condition can be formulated in a potential theoreticway: let A ⊂ C+ and define the sequence A′

j,k to be Aj,k if Aj,k 6= ∅ and ∅ otherwise.Let Zj,k be a point in A′

j,k and K(Zj,k, δ) be a hyperbolic ball with center Zj,k. ThenS(iy0, A) = ∞ if the union of hyperbolic balls K(Zj,k, δ) is not minimally thin at iy0(see [18] for more on this topic).

An interesting application of Theorem 6.1 is given to a problem in robust identifi-cation. Here one is given corrupted frequency response measurements ak = F (iyk) +ηk, k = 1, . . . , n, on the imaginary axis of an unknown function F (s) in the half-planealgebra A(C+), where ηk is the measurement error (sometimes called noise, althoughit could for example be due to deterministic effects such as nonlinearities), which isassumed to be small (say, |ηk| ≤ ǫ) for each k. Then the intention is to producean approximate model F ∈ A(C+) for F in such a way that the following robust

convergence condition is satisfied:

limn→∞ǫ→0

sup‖η‖≤ǫ

‖F − F‖∞ = 0 for all F ∈ A(C+).(6.3)

An algorithm for reconstructing F (s) is given by applying the following result [38].THEOREM E. Let X be a separable infinite-dimensional normed space over a field

F, and let φk∞k=1 be a uniformly bounded sequence of elements in the dual space X∗.

Then there exist maps Tn : F → X such that

limn→∞ǫ→0

sup‖η‖≤ǫ

‖xn − x‖ = 0 for all x ∈ X,

with xn = Tn((φk(x) + ηk)nk=1), if and only if there exists δ > 0 with

sup |φk(x)| ≥ δ‖x‖ for all x ∈ X.

In order to implement Theorem E we first note that for φk we can take evaluationfunctionals φk(F ) = F (iyk) and then we choose a complete model set Xpp≥1 forA(C+); that is, a sequence of finite-dimensional subspaces Xpp≥1 of A(C+) whichsatisfies the following two conditions:


(i) X1 ⊂ X2 ⊂ X3 ⊂ · · · ,(ii) ∪∞

p=1Xp is dense in A(C+).The vectors Tn(a1, . . . , an) are then constructed to be solutions Gp ∈ Xp of the

minimax problem:

s = minG∈Xp

max1≤k≤n

|φk(G) − ak|,

where ak = Fk(x) + ηk. Here p is chosen, depending on n, to be as large as possiblesuch that

max1≤k≤n

|φk(G)| ≥ (δ/2)‖G‖∞ for all G ∈ Xp.

Since this optimization problem can be solved by linear programming, the computa-tional burden is not excessive.

We have just shown in Theorem 6.1 that Cw(iy) : w ∈ L ∪ 1 is a completemodel set for A(C+), and we may for instance take Xp to be the finite-dimensionalspace spanned by the set

1 ∪ C ′w(iy) : s = 2−j + ik2−j ,−p ≤ k ≤ p, 0 ≤ j ≤ p.

On transforming to the disc, this subspace corresponds to a subspace Xp of the discalgebra spanned by Cauchy kernels Ca(z) = 1/(1 − az), where the points a lie on afinite lattice Lp, say. Now if w = u+ iv and a = (1 − w)/(1 + w), then the followingidentity holds:

1 −∣

∣

∣

∣

1 − w

1 + w

∣

∣

∣

∣

2

=4|u|

|1 + w|2 ,

and this implies that the points of Lp are not too close to the unit circle. Specifically,we easily obtain the uniform bound 1 − |a|2 ≥ (2/3)2−p, which implies that the polesof the functions in Xp lie in the region |z| ≥ R, where (R+ 1)/(R− 1) ≤ 6.2p.

We may now use the Borwein–Erdelyi extension of Bernstein’s inequality whichapplies to rational functions with poles in a region |z| ≥ R, namely,

‖g′‖∞ ≤ NR+ 1

R− 1‖g‖∞

where dim Xp = N [8], and perform a calculation similar to that of [16] to show that,provided that the interpolation points (iyk), k = 1, . . . , n(p), on the imaginary axisare chosen so that the maximum gap ∆n between their images (1 − iyk)/(1 + iyk) onthe unit circle is sufficiently small, one does indeed have

max |G(iyk)| ≥ (1/2)‖G‖∞ for all G ∈ Xp.

In the example we are considering this condition is satisfied if ∆n ≤ 1/(6N2p) andN = 2p2 + 3p+ 3.

This means that the Cauchy kernels can be used as a complete model set foridentification purposes along the lines of [38, 16].

7. Error estimates. There is extensive literature on the model reduction ofinfinite-dimensional linear systems, both in the H∞ norm [22, 23, 24, 26] and the H2

norm [25, 1]. Recent research has focused on two approaches which can be applied


to a wide class of systems. First, there is the use of orthogonal basis functions suchas Laguerre and Kautz models [26, 31, 37, 27, 43] where the poles normally lie in afixed finite set. Second, there is the approach based on rational wavelets [39, 40, 15]where the poles lie on some appropriate infinite lattice, and this is the approach wehave adopted in this paper.

In order to illustrate the techniques involved, we shall consider the class of re-tarded delay systems in the sense of Bellman and Cooke [3]. These have perhaps thesimplest irrational transfer functions that occur frequently and, unlike the transferfunctions obtained from partial differential equations, tend to be difficult to approxi-mate efficiently. Other examples, the discrete-time fractional filters, were consideredby analogous methods in [15], and delay systems were considered by transformingthem to the disc.

In this section we shall specialize to delay systems of the form R(s)e−sT , R(s)rational, because the results on approximation of such systems have a particularlysimple form. However, general retarded delay systems (of the form N(s)/D(s) whereN and D are sums of terms of the form p(s)e−Ts with p a polynomial and T > 0)can be analyzed by similar means (the techniques for decomposing such systems canbe found in [23, 25, 37]).

For a delay system of the form G(s) = R(s)e−sT , where R(s) is asymptotic toCs−k at infinity, the following approximation results are known [22, 23, 24, 25].

There exist constants A, B > 0 such that the minimal achievable error in H∞

and H2 rational approximation by degree-n systems satisfies

An−k ≤ E∞n (G) ≤ Bn−k

and

An−k+1/2 ≤ E2n(G) ≤ Bn−k+1/2.

This implies immediately that the H2,1 error must satisfy

E2,1n (G) ≥ An−k+1/2,

but it seems that a tight upper estimate is not known.Now if we consider the integral operator Ω on L2(0,∞) defined by

(Ωu)(t) =

∫ ∞

0

w(t)h(t+ τ)w(τ)u(τ) dτ,(7.1)

where w is a suitable weight function, then its rank is the same as the rank of theusual Hankel operator and its Hilbert–Schmidt norm squared is

∫ ∞

t=0

∫ ∞

τ=0

|w(t)|2|h(t+ τ)|2|w(τ)|2 dt dτ,

which equals∫ ∞

r=0

∫ r

t=0

|w(t)|2|h(r)|2|w(r − t)|2 dt dr.

Choosing w(t) =√

(2√t+ b/

√t) gives a Hilbert–Schmidt norm squared of

∫ ∞

0

(r2 + 4br + 2b2)|h(r)|2π/2 dr,


since one has∫ r

0

(2√t+ b/

√t)(2

√r − t+ b/

√r − t) dt = (r2 + 4br + 2b2)π/2.

We do not know a weight w such that

∫ r

0

|w(t)|2 |w(r − t)|2 dt = (r + 1)2,

or even r2 + 1, but by choosing b = 1/√

2 we have the bounds

(r + 1)2 ≤ r2 + 4br + 2b2 ≤ (1/2 + 1/√

2)(r + 1)2

for r ≥ 0, and so we have the fairly tight inequality

(π/2)‖h‖22,1 ≤ ‖Ω‖2

HS ≤ ((√

2 + 1)π/4)‖h‖22,1,

with corresponding bounds for rational approximation errors in theH2,1 norm. Namely,if the singular values of Ω are (ωr)r≥1, then we have the following error bound for‖h− hn‖2,1 when hn is the impulse response of any degree-n system and correspondsto an operator Ωn of rank n:

‖h− hn‖22,1 ≥ C‖Ω − Ωn‖2

HS

≥ C

∞∑

r=n+1

ω2r ,

where C = 4(√

2 − 1)/π. Similar weighted Hankel integral operators were consideredin [25] for the purposes of estimating error bounds in L2 approximation, and in [11]for the purposes of estimating the singular values of Hankel operators on Bergmanspaces. Here we have the additional complication that it does not seem possible tochoose w in closed form so that the Hilbert–Schmidt norm of Ω and the H2,1 normof h are equal, although we can obtain equivalence of norms.

The following result generalizes both Theorem 3.2 of [25] and Lemma 14 of [11].It applies directly to transfer functions of the form (e−sT − 1)/s with T > 0, and, bystandard decomposition techniques [37], to many other delay systems.

THEOREM 7.1. Let w(t) be a smooth positive function in L2(0, T ) and let h(t) =χ[0,T ](t). Then the singular values of the scaled Hankel operator Ω defined by (7.1)satisfy rωr → J(w), where

J(w) =1

π

∫ T

0

w(t)w(T − t) dt.

Proof. The proof here is a simple modification of the proof given in [25], so weshall omit many of the details. The operator Ω is self-adjoint and solving for itseigenvalues leads directly to the Sturm–Liouville differential equation

d

dt

(

v′(t)

w(T − t)2

)

+v(t)w(t)2

λ2= 0,

with boundary conditions v(T ) = 0 = v′(0). Analysis of this equation by theLiouville–Green method [33, 35] shows that its eigenvalues are asymptotic to those


of the equation z′′ + z/λ2 over the interval [0, X] with boundary conditions z(0) =z(X) = 0, where

X =

∫ T

0

w(t)w(T − t) dt,

and this gives the result.Unfortunately, for the weights w considered above, the integral J(w) cannot be

obtained in closed form, although it can be estimated numerically.We conclude this section with a further result which applies particularly to delay

systems, and can be regarded as an appendix to Theorem 1.1 above. There weconsidered the wavelet coefficients 〈F,Ψj,k〉 for F ∈ H2,1 regarded as functions of j.It would clearly be useful to be able to estimate them as functions of k as well (so thatan infinite series could be truncated to a finite sum by discarding the less significantterms) and this is what we now do.

LEMMA 7.2. Suppose that f(y), g(y) ∈ L2(R) satisfy |f(y)| ≤ A|y|−m and |g(y)| ≤B|y − C|−n, where m, n > 1/2 and C > 0. Then the following inequality holds:

〈f, g〉 ≤ B‖f‖2√2n− 1(C/2)n−1/2

+A‖g‖2√

2m− 1(C/2)m−1/2.

Proof. Using the Cauchy–Schwarz inequality we have, for any r with 0 < r < C,

〈f, g〉 ≤ ‖f‖2

(∫ r

−∞

|g(y)|2 dy)1/2

+ ‖g‖2

(∫ ∞

r

|f(y)|2 dy)1/2

,

and this easily gives the result when we take r = C/2.The conditions of the following theorem will apply to any strictly proper delay

system, and one can always take m ≥ 1. Clearly, the result is of greatest use whenj < 0, because then we obtain a good approximation with fewer translates Ψj,k.

THEOREM 7.3. Let Ψ(y) = (1 + iy)−3 and let Ψj,k(y) = 2j/2Ψ(2jy − kb0) where

j, k ∈ Z, and let 〈·, ·〉 be the usual L2-inner product.

Suppose that |F (iy)| ≤ A|y|−m, where m > 1/2. Then for k 6= 0 the wavelet

coefficients 〈F,Ψj,k〉 satisfy

〈F,Ψj,k〉 ≤ 25/2‖f‖2√5|kb0|5/2

+A‖Ψ‖2√

2m− 1(|kb0|2−j−1)m−1/2.

Proof. Clearly, we may assume without loss of generality that k > 0. We nowapply Lemma 7.2, with g = Ψj,k, noting that ‖Ψj,k‖2 = ‖Ψ‖2 and that

|Ψj,k(y)| ≤ 2−5j/2|y − kb02−j |−3.

8. Applications to system modeling. It is desired to approximate an infinite-dimensional model by a finite-dimensional model in the Hardy–Sobolev norm. Thatis, given a transfer function in H2,1, we desire to approximate F (s) by a degree nrational function. The framework developed in this paper gives a method by whichwe can approximate F (s) by a sum of degree 3 rational functions.

In order to implement the techniques of Theorems 1.2 or 1.3 in the model re-duction of a transfer function in the Hardy–Sobolev space H2,1, we present threealgorithms.


ALGORITHM 1. This is based on Theorem 1.2 and a matching pursuit (MP)algorithm of Mallat and Zhang [32]. With Ψ(y) as in Theorem 1.2, it follows that thelinear span of the collection D = Ψj,k(y) : j, k ∈ Z is dense in H2,1. Let 〈·, ·〉2,1 be

the H2,1-inner product. Take 0 < α ≤ 1 and F1 ∈ H2,1. Choose Ψj1,k1(y) such that

| 〈F1,Ψj1,k1〉2,1 | ≥ α sup

Ψj,k∈D| 〈F1,Ψj,k〉2,1 |.

There exists F2 in the orthogonal complement of the subspace spanned by Ψj1,k1with

F1 = 〈F1,Ψj1,k1〉2,1 Ψj1,k1

+ F2. Next we choose Ψj2,k2∈ D such that

| 〈F2,Ψj2,k2〉2,1 | ≥ α sup

Ψj,k∈D| 〈F2,Ψj,k〉2,1 |.

There exists F3 in the orthogonal complement of the subspace spanned by Ψj2,k2with

F2 = 〈F2,Ψj2,k2〉2,1 Ψj2,k2

+ F3. Proceeding in this manner we obtain a sequence

Ψjn,kn∞

n=1 and a sequence of residual vectors Fn∞n=1 such that

Fn = 〈Fn,Ψjn,kn〉2,1 Ψjn,kn

+ Fn+1

and

‖Fn‖22,1 = | 〈Fn,Ψjn,kn

〉2,1 |2 + ‖Fn+1‖22,1.

Therefore,

F1 =

m−1∑

n=1

(Fn − Fn+1) + Fm

=

m−1∑

n=1

〈Fn,Ψjn,kn〉2,1 Ψjn,kn

+ Fm,

and

‖F1‖22,1 =

m−1∑

n=1

(‖Fn‖22,1 − ‖Fn+1‖2

2,1) + ‖Fm‖22,1

=

m−1∑

n=1

| 〈Fn,Ψjn,kn〉2,1 |2 + ‖Fm‖2

2,1.

Mallat and Zhang (Theorem 1, [32]) prove that ‖Fm‖2,1 → 0, and so

F1 =

∞∑

n=1

〈Fm,Ψjn,kn〉2,1 Ψjn,kn

and ‖F1‖22,1 =

∑∞n=1 | 〈Fn,Ψjn,kn

〉2,1 |2.ALGORITHM 2. This is an adaptation of Algorithm 1. MP may converge some-

what slowly (see [15, section 4.2]), but it is possible to adapt it so that we obtaingeometric convergence in the H2,1 norm by projecting onto larger subspaces. Forexample, let Vn = spanΨj,k : −n ≤ j ≤ n,−2n ≤ k ≤ 2n and let PVn

be theprojection onto Vn in the L2 norm. Let 〈·, ·〉 denote the L2-inner product.


To proceed with the algorithm fix a positive ǫ and F1 ∈ H2,1. By Theorem 1.1we may choose n1 so large that

(A1 − ǫ)‖F1‖22,1 ≤

∑

|k|,|j|≤n1

| 〈F1,Ψj,k〉 |2(1 + 2j)2.(8.1)

The vector F2 = F1 − PVn1F1 is orthogonal to Vn1

in L2 and so 〈F2,Ψj,k〉 = 0.

Therefore 〈F1,Ψj,k〉 =⟨

PVn1F1,Ψj,k

⟩

, |j|, |k| ≤ n1. Therefore, by (8.1) and Theorem1.1, we have

(A1 − ǫ)‖F1‖22,1 ≤

∑

|k|,|j|≤n1

|⟨

PVn1F1,Ψj,k

⟩

|2(1 + 2j)2 ≤ B1‖PVn1F1‖2

2,1.

Thus B1‖PVn1F1‖2

2,1 ≥ (A1−ǫ)‖F1‖22,1. Increasing n1 if necessary, we obtain similarly

that B1‖PVn1F1‖2

2,1 ≥ (A1 − ǫ)‖PVn1F1 − F2‖2

2,1. By the parallelogram law we have

‖F2‖22,1 + ‖PVn1

F1‖22,1 =

1

2(‖F1‖2

2,1 + ‖PVn1F1 − F2‖2

2,1).

Therefore

‖F2‖22,1 ≤

(

1

2−

(

1 − B1

2(A1 − ǫ)

)

(A1 − ǫ)

B1

)

‖F1‖22,1

=

(

1 − (A1 − ǫ)

B1

)

‖F1‖22,1.(8.2)

With the same ǫ and F1 replaced by F2 we repeat the procedure. Continuing in thismanner we obtain a sequence Fkk≥1 such that Fk = PVnk

Fk + Fk+1, where the

PVnkFk and Fk+1 are orthogonal in the L2 norm and such that

‖Fk+1‖22,1 ≤

(

1 − A1 − ǫ

B1

)k

‖F1‖22,1.

In (8.2) we have assumed b0 is chosen so that B1 < 2A1. We may take, for example,b0 = 0.5 and estimate by computer A1 = 4.503 and B1 = 6.789.

Remark. It is perhaps surprising that the algorithm given is based on takingprojections in the L2 norm and obtaining an expansion which converges in the Hardy–Sobolev norm.

ALGORITHM 3. This is based on ideas in Theorem 1.3. Since the operator TFis not a positive operator on H2,1 it does not follow immediately that we can choosesome constant K such that ‖I −KT‖2,1 < 1. However, using a simple manipulationof Daubechies’ techniques one may establish the following elementary result.

PROPOSITION 8.1. Let Ψj,k be as in Theorem 1.3, and

RF = F −K∑

j,k

〈F,Ψj,k〉 Ψj,k, F ∈ H2,1.

Define

m = inf0<t<∞

∑

j

|ψ(2−jt)|2, M = sup0<t<∞

∑

j

|ψ(2−jt)|2,

β1(s) = sup0<t<∞

(1 + t)−2∑

j

|ψ(2−jt)||ψ(2−jt− s)|,

β2(s) = sup0<t<∞

(1 + t)2∑

j

|ψ(2−jt)||ψ(2−jt+ s)|.


If K = b0/π(m+M), then

‖RF‖2,1 ≤M −m+ 2

∑

k 6=0

β1

(

2πb0k)

β2

(

2πb0k)1/2

m+M‖F‖2,1.

Proof. The proof follows the lines of the proof of Theorem 1.3, and so the detailsare only sketched. We let G belong to H2,−1 and estimate the modulus of

〈RF,G〉 = 〈Rf, g〉= 〈f, g〉 −K

∑

j,k

〈f, ψj,k〉〈g, ψj,k〉

=

∫ ∞

0

f(t)g(t)dt− 2πK

b0

∑

j

∫ ∞

0

f(t)g(t)|ψ(2−jt)|2dt+ rest(f, g)

=

∫ ∞

0

f(t)g(t)

1 − 2πK

b0

∑

j

|ψ(2−jt)|2

dt+ rest(f, g).(8.3)

Analyzing the rest term we obtain with β1(s) and β2(s) defined above that

| rest(f, g)| ≤ 2π

b0

∑

k 6=0

β1

(

2π

b0k

)

β2

(

2π

b0k

)1/2

‖f‖2,1‖g‖2,−1.(8.4)

Therefore, substituting (8.4) into (8.3), we deduce

‖RF‖2,1 ≤

1 − 2πK

b0

m−∑

k 6=0

β1

(

2π

b0k

)

β2

(

2π

b0k

)1/2

‖F‖2,1.

With K = b0/2π(m+M) we obtain the conclusion of the proposition.In a forthcoming paper it is intended to present some numerical results on the

model reduction of several systems using the methods which appear in this study.

Acknowledgments. In conclusion, we would like to express our gratitude tothe following friends and colleagues who have helped us in conversations and corre-spondence to understand frames and wavelets: Professors Ingrid Daubechies, MauriceDodson, Walter Hayman F.R.S., and E. Christopher Lance. It will be plain to theinformed reader the debt we owe to the work of Daubechies. We are also very grate-ful to one of the referees for reading a manuscript very carefully and making somepertinent comments. Finally, we would like to acknowledge a special debt to IngridDudley Ward for writing some programs in C to estimate the frame constants givenin this paper.

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