AllS7

WIENER'S APPROXIMATION THEOREM

FOR LOCALLY COMPACT

ABELIAN GROUPS

THESIS

Presented to the Graduate Council of the

North Texas State University in Partial

Fulfillment of the Requirements

For the Degree of

MASTER OF SCIENCE

By

Ven-shion Shu, B. 5.

Denton, Texas

August, 1974

Shu, Ven-Shion, Wiener's Approximation Theorem for

Locally Compact Abelian Groups. Master of Science (Mathe-

matics), August, 1974, 46 pp., bibliography, 5 titles.

This study of classical and modern harmonic analysis

extends the classical Wiener's approximation theorem to

locally compact abelian groups.

The first chapter deals with harmonic analysis on the

n-dimensional Euclidean space. Included in this chapter are

some properties of functions in L1 (Rn) and 71(Rn), the

Wiener-L6vy theorem, and Wiener's approximation theorem.

The second chapter introduces the notion of standard

function algebra, cospectrum, and Wiener algebra. An abstract

form of Wiener's approximation theorem and its generalization

is obtained.

The third chapter introduces the dual group of a locally

compact abelian group, defines the Fourier transform of func-

tions in L (G), and establishes several properties of func-

tions in L (G) and 7(a). Wiener's approximation theorem

and its generalization for L1 (G) is established.

TABLE OF CONTENTS

Page

Chapter

I. CLASSICAL HARMONIC ANALYSIS ANDWIENER'S THEOREM IN LI(Rn) .0,0,0 *.. 0 1

II. FUNCTION ALGEBRAS AND GENERALIZATIONOF WIENER'S THEOREM . . . . . . ... ... 16

III. WIENER'S THEOREM INL (G). . . . . . . . . . 28

BIBLIOGRAPHY . - - . 0 * . . . . . . . . . . . . . 46

i ii

CHAPTER I

CLASSICAL HARMONIC ANALYSIS AND

WIENER'S THEOREM IN L (Rn)

1.1. Let f be a complex-valued Lebesgue integrable

function on Rn. We shall simply write ff(x)dx for the

Lebesgue integral over Rn. The Lebesgue integral has two

important properties:

(i) ff (x-a)dx = ff(x)dx, for all aE Rn,

(ii) S Pnf (px)dx = Sf(x)dx, for all p> 0.

The complex-valued Lebesgue integrable functions on Rn

form a commutative Banach algebra over the complex numbers C.

This algebra is denoted by Ll(Rn); the norm is

11f11l = SIf(x)Idx,

and the multiplication is defined by the convolution f*g,

f*g(x) = Jf(x-y)g(y)dy, for x ERn, f,g EL (Rn).

L1(Rn) contains the set COO(Rn) of all complex-valued con-

tinuous functions with compact support as a dense subalgebra.

In L1(Rn), we have two isometric operators, the trans-

lation operator La

Laf(x) f(x-a), for acRn

and the multiplication operator MP,

MPf(x) = nf(px), for p>0.

1

2

1.2. The Fourier transform of a function f cL (Rn) is

defined by

P(t) = Sf(x)K-x,t> dx, for t ERn

where <-x,t> = e-27i (xit+..+xntn) with x (x1,...xn),and

t = (t ,...t n)'

Fourier transforms have the following basic properties,

which we shall state without proof [1, 23.

(1) f is uniformly continuous, and jT(t)j 11f|| for

all t c Rn

(2) (c f + c2f2 )1= c 1 +c2?2 , for c1 ,c2 EC and

ff2E L2(R r).

(3) (f2*f2r 1 2f2forff2 EL1(Rn).

(4) (Laf5f (t) = K-a,t> -?(t), for aERn

(5) (Mf) (t) = f(P), for P >0.

(6) ( Yt f) = Lt 0, for t0 c Rn, where t(0X = KxtO>

(7) f = f, where f(x) = f(-x).

There are three fundamental theorems on Fourier trans-

forms of functions in L1(Rn).

(i) Riemann-Lebesgue lemma. If f EL (Rn), then

f (t) - 0 as 1={ +..+t

(ii) Inversion theorem. If both f and f are in Ll(Rn)5

then f(x) = SP(t)Kx,t>dt a.e.

(iii) Uniqueness theorem. If fE:L1(Rn) and f^(t) = 0

for all t c R, then f(x) = 0 a.e.

For the proofs, refer to Rudin and Stein [1, 21.

1.3. In this section, we shall construct a few special

Fourier transforms which are basic tools of our later devel-

opment in Chapter 1.

(i) For a> 0, let 0 be the characteristic function of

the interval [-a,aj. Then the Fourier transform of 0 is

' (t) = sin 2 acxt/rt.

(ii) Let A1 be the "triangle" function defined for

x c R by A,(x) = 1 - lxI for lxI 1, Al(x) = 0 for IxI 1.

Then A1 (t) (sinut)2 cL1 (R). By the inversion theorem, and

since A(x) A(-

As(x) ~ sinTt e2 ixtdt = (sin T t 2 -2ixtdt =STT(- .

Define a 1 (x) = (sinu x) , if x / 0, yo() = 1. Then

L 1 (Rn) and &1(t) = A1 (t).

(iii) Let T1 be the "trapezium" function defined for

x ER by T1 (x) = 1 for |xl 1, T1(x) = 2 -jxj for l J<xJ| 2,

Tl(x) = 0 for jxj !2. Then Tl(t) = sin 3t sint is in1 1 (i~t ) 2

L (R). As in (ii), applying the inversion theorem, let r

be defined by T,(X) _ sin 3Tx sin ux , if x / 0, T1(0) = 3.

Then T1 cL1 (R) and 91(t) = T1(t).

(iv) Let AN (t) = T ) = (), then (M= A,

and (MT = T. The functions A , T, and their translates

are of much significance, as we can see later.

4

(v) For x (xi,...xn) E Rn, put f(x) = f1 (x)f 2 2(x9)f ' (xn1 -i1nwhere f1 ,f2 .'"fn EL (R). Then f cL (Rn) and

f(t) = fl(t L) ...fn(t). By this fact, we can use functions

in L (R) with certain properties to obtain functions in

L (Rn) with analogous properties concerning Fourier trans-

forms.

1.4. By 1 .2 (1), (2), (3), the Fourier transforms of

the functions in L (Rn) form an algebra of complex-valued

continuous functions with ordinary pointwise algebraic opera-

tions. We denote it by 71(Rn). From the uniqueness theorem,

1.2 (iii), L1(Rn) and 71(Rn) are isomorphic by the map f - P.

We carry the L1 -norm over to ?'(Rn) by putting jjf|| = ||f|| 1 ,f ci(Rn). Then 71(Rn) becomes a commutative Banach algebra

which is homeomorphic to L (Rn). There is an isometric opera-

tor on i1(Rn), the translation operator; in fact, by 1.2 (6),

liL t fll = 'lf Il.L0

1.5. Theorem. Let f E L (Rn). The map x - Lx f is a

uniformly continuous map of Rn into Ll(Rn).

Proof. Let e >0 be given. Find a gcC 00 (Rn) with com-

pact support K and |1f-g| 1|< '. Let V be a neighborhood of

0 cRn such that K+V is contained in another compact set K'.The uniform continuity of g implies that there is a neighbor-

hood U of 0 such that Uc V and lg-L g|il< - 1m(K')]~fo

all xc U, where m(K') is the Lebesgue measure of K'. Hence,

for all xcU, lg-L gll< , and

5

f -L L f fl <1f-g 1 1|g-L g|1 1Lxg-L fIl<l .

Finally, Lxf-Lyffl1 = |iLx(f-L Yf)l|I = |f-Ly xfi<C Gif

y-x EU, and the proof is complete.

Remark. The above result is also true for LP(Rn),

l< p <co. The proof is similar to the one for Ll(Rn).

1.6. Theorem. If f,hcEL1 (Rn), then

1 (MP h)*f - [Jh(x)dxIfI1 1 -- 0, as p co

Proof. ((M h)*f- [Sh(x)dxlf)(y)P

= fM h(x)f(y-x)dx - [Jh(x)dxlf(y)

= SM h(x)[f(y-x) -f(y)]dx.

Hence, l|(MPh)*f - [Sh(x)dx]f 1

S1SMph(x)[f(y-x) -f(y)]dx I dy

5 |M h(x)l If(y-x) -f(y)j dx dy

SIMPh(x)ISILxf(y)-f(y)Idy dx

SIMPh(x)l ILxf-f|f1dx.

Let O(x) = ||Lxf-fl 11. Then 0 is uniformly continuous, by

1.5, and $(0) = 0. Note that O< o(x) :521|ffI|1 . Hence,

1(M Ph)*f-[Sh(x)dxIf|Il <SIM Ph (x)|1 0(x) dx

=Jpnh( px)| 0(x) dx

= fIh(x)I (L) dx.

We want to prove Slh(x)1 s(L) dx- 0 as p - o. We prove

first that this is true for hCC0 0 (Rn). Let K be the support

of h. Since 0 is uniformly continuous and 0(0) = 0,

(X <b.nG for all xK and p p0 . It follows that

6

5Ih(x)I@( )dy = |h(x)|K( )dxP KP

< 7 Kh(x)dx = e for all p tpo'1K

Now, if hcEL1(Rn), there exists gcE% 00(Rn) such that

h-g 11| < 1.<Therefore,

h(x) I (' )dx < f h(x) -g(x)|I$( )dx+jlg(x) I ( )dx

Jh2||f|| -|h(x)-g(x)|dx +I|g(x)| (L)dx

< C+I Ig(x)|I (?)dx2 p< E, if p is large enough.

1.7. Theorem. Let fEL1 (Rn) and c > 0 be given. Then

there is a aELl(Rn) such that "acCoo(Rn) and 11|a*f-f1 1 <cE.

Proof. Let heLl1(Rn) be such that ^hEC000 (Rn), and

fi(O) = 1 (cf. 1.3). Using 1.6, let p be chosen large enough

so that

(M h)*f-[Sh(x)dx]f| 1, = f|(M h)*f - f(O)ff||< C.

Let a = M h. Thenp

G*f -f l =|(M h) *f - hi(O)f||<E.

Remark. This theorem says that Ll(Rn) (or 9l(Rn)) pos_

sesses approximate units. There is no unit in Ll(Rn) (or

,71(Rn)). This fact comes directly from the Riemann-Lebesgue

lemma.

1.8. Theorem. Let f cL1(Rn). Given E >0, there is a

function T 9L(Rn) such that

(i) 9 has the value 1 near the origin.

(ii) |f *,- [ f (x) dx ] r||1 < E.

7

Proof. Let T be any function satisfying (i) (cf. 1.3).

By 1.7, |IM f)*Ti- [f (x)dxlTIl< c for large p. Applying

the operator M1 to the function in KI, we getp

(M M T 1)- [f f (x)dx]M1T < C.

p pp

Iff*(M1 T1) - [ff(x)dx]M 11T111 .<

P p

Put T = M T 1 .0Then

-Jf*T- [Jf(x)dx]Tj 1< E,

and (t) = T1(pt) is still I near the origin.

1.9. Theorem. Let fc (Rn) and t0 ER n Then, given

C >0, there is a function Tt0 in 91(Rn) such that

(i) U (t) = 1, for t near t0 '

(ii |(f - f^(Y0) < 00

Proof. For the special case when t0 = 0, this is

merely 1.8 rewritten for 7(Rn). For general t0 c R n let

= Lt . Now, applying 1.8 to g, we can find a(R

such that 7 is 1 near the origin, and

(9 - 20?| |(L- t 0I-"L -t 0 (O)^T||< C.o 0

Put T = Lt Y. Then

?(0=)f(),(o))t I0 0

Lt [(L-t -L-t ' (O))?]||0 0 0

(Lt? -L- (

< .

8

Since T is 1 near the origin, therefore Tt0(t) =(t-t0

is 1 near to.

1.10. Theorem (Wiener-L6vy theorem). Let fG9l(Rn

Let K be a compact set in Rn and denote ?(K) as the image of

K under ?. Suppose A(z) is an analytic function on an open

neighborhood of f(K) in C. Then, there is a c ,1 (R n) such

that g(t) = A(T(t)) for all t c K.

Proof. The theorem will be proved in two steps.

(i) We first prove the following: let t0 be any point

of K. Then there is a function cg t 79(R n) such that

gt0 (t) = A(?(t)) for all t near t0 '

Write z0 =(t0 ). Then, say for all | z - z01< b, we have

A(z) = A(zo) + F Ck(zzO)kk=1

Choose Tt according to 1.9 so that T2

Consider the series of functions

A((tt + f CkF(-(t))t IkO .k=l 0

Then,

|lA(C(to)) t || + )l|Ck[(f(to) t k jo k=l 0

0

||A^f

t ItI||+ CkI 11|( -(to))t k

o k=l 0

Therefore the series A(?(t T t + E Ck(Fl- 1(t\T kcon-o k=1 to

verges to a function in Jl(Rn). Let us denote this function

by gt . Since Tt0(t) = 1 for t near to, therefore

9

gt0 (t) = A( (to)) + EOCk(-(t) -(t 0 )k0 k=1

= A(zo) + 2ECk("(t) -zo)kk=I

for all t near t0 0

(ii) Next, we extend the "local?" result (i) to K. For

each t0 c K, we have

(a) by (i), there is a g ct E (Rn) and a neighborhood

Uto of t0 such that to(t) = A('(t)) for all tCEUt 0(b) by 1.3, there is a function ^T ' 1 (Rn) such that

St' vanishes outside Ut and equals 1 in some neighborhoodt0 0

Utof t'

Now, there are finitely many points (tk)l5kN in K such

that the corresponding neighborhoods U= U' cover K. Wetk

also write Uk, k' k for U T t , respectively. Letk k k

= ',and (if N !2) 12 = ^ ~()' eN =N-'''

(l-?'). Then, k is in 71(Rn), 1!5 k 5 N, and vanishes out-

side Uk. Moreover, by induction, we haveN

N k=l k~2)' N

Hence, E k(t) = 1 for all t cK. Now consider the functionk=l

N= Ck k 71(Rn). If t E K, then for 1 5 k 5 N,

k=l

^ek= k(t)A(f(t)),if tcEUk

=)(t)kk(t)),0 = e k( t)A(f(t)), if t /U k"

10

Therefore for all t e K,

N N9 k( = t)k( = ek(t)A(f(t))

k=1 k=lN

= Z Ee k(t)JA(P(t))k=l

=A('f(t)).

Thus the proof of the Wiener-L6vy theorem is complete.

1.11. Theorem. Let c 7 (Rn). If A(z) is an analytic

function on an open neighborhood of P(Rn) U (0) in C, and

A(0) = 0, then there is a 9 71(Rn) such that g(t) = A( (t))

for all t eRn

kProof. Since A(z) = 0, we have A(z) = Ea z fork=1

jz< <c, say. By 1.7, there is a 7 17l(Rn) with compact sup-

port, say K, and fj-f||<cE. Consider the series

ak - af) . Then this series converges to a function ink=1

(Rn), since |alk k<c)kjj . Let us denote thisk=1k11(

function by g0. Then O(t) = A(P(t)) for all t ERn - K.

Find a 9 E:Ci(Rn) which is 1 on K, and has a compact support,

say K'. By 1.10, there is a g 1 c 71(Rn) such that

gl(t) = A(P(t)) for tcE:K'. Let

9 =(1-^)^0 + le

Then g c71(Rn), and ^(t) A(P(t)) for all tCERn

1.12. Theorem (Wiener's approximation theorem). Let

f L 1(Rn). The linear combinations of translates of f,

11

Ek ak EC, a k c Rnk k

are dense in L1(Rn) if and only if (t) / 0 for all tcERn

Proof. (A) We prove the necessity first. Suppose

f (to)= 0 for some t0 0E . Given c > 0, we can find a

g eLI(Rn) such that |12(t 0 )j> C. Now, by assumption, there

is a finite sum Z Ak Lakf such that llg-Z3Ak L akf IIIHence,

(g-~Ak Laf)(t)J ||g-1 Ak L afItl<k Kk k 1

It follows that

Ig(t0 - n<-aMnkaKtO >^(tO)1 =|(to)I<E,k

which is a contradiction.

(B) The sufficiency will be proved in four steps.

(i) First we prove the following: Let fCELl(Rn), and

f(t) /O for all tERn. Let goEL(Rn) and 90 EC0 0 (Rn

Then there is an hCELl (Rn) such that g0 = h*f.

Let the compact support of ^o be Ko. Applying the

Wiener-L6vy theorem for A(z) =1, there is an f ELl1 (Rn)

such that on Ko. Let h = go *fl. Then (h*f)^A

(g0* K * 1 g = c k on Rn. Therefore, g0 = h*f.

(ii) Next, we prove that the functions g0 cLl(Rn),

whose Fourier transforms have compact support, are dense

in L (Rn

Given that c>0 and f E L (Rn), by 1.7, there is a

Sc L (Rn) such that CCo(Rn) and ||fo7*f - f|Il< c. Let

go = CY* f. Then g0 E L1 (Rn) and go = E Coo(Rn

12

(iii) Now we prove the following: Let hf be in L (Rn)

and E> 0 be given. Then there is a finite sum Z? AkL f such

that lh*f - ?k L a fI< lc9(A k EC. a kCR n.kk k

First we prove the case for hCC0 0 (Rn). Suppose the

support of h is K. There is a 5> 0 such that |ILyf-fl|< Eiflyj <5. For each xcEK, let Ux = ty:|y-x| <5). Then there

are finitely many points (ak1lk N in K such that the open

balls Uk = Ua cover K. Let A1 = KAU1 . If N >1, let

k-lAk = (K- .U A)" Uk for 2sksN. Let Ak = h(x)dx. Then

i=1k

N Nh*f - E Ak La f = Z fh(x)[Lxf-Laf]dx.

k=l k k=lAk k

Hence,N N

Ilh*f 7-jk Lf '1 =fSh(x)[Lxf(t)-L f(t)]dxldtk=l kkk=lA xAkk

AkkN

f h(x)JIL f(t)-La f(t)|dx dtk=l Ak a k

N

< f h(x)|I ||L Xf-L af||1,dxk=l A x k

N

<E * Zf h(x)|dxk=l A

=c - ||h l.

Next, suppose hcEL1 (Rn). Then there is a gce cOO(Rn) such

that ||h-g|| 1 < . For this g, we can find a finite sum

k Lakf such that ||g*f - kLkLak fil< E. Hence,k k k k

h*fD-Z \Ak Lfa 1l 1511(h-g)*f|1|,+|g*f - \A L f||k k k k ak

5 ||h-g||i- + E

< e.

(iv) Finally, let g E L(Rn) be given. Choose g0 EL1(R n)

such that go E Coo(Rn) and I|g-g 0 1 < . Let h E L (Rn) such0 2that go = h*f, and find a finite sum Z kLakf such that

fh*f - kLf 11< L. It follows then that

0 fjgk Lak1-ggj| +Jflh*f - Z)kLaf

1.13. Definition. A linear subspace I of Ll(Rn) is

said to be invariant under translation, or simply invariant,

if f E I implies Laf EI for all a e Rn.

1.14. Theorem. The closed invariant subspaces of

L (R n)coincide with the closed ideals.

Proof. 1.12 (iii) in fact says that every closed in-

variant subspace of L1(Rn) is an ideal. We prove now that

every closed ideal of L (Rn) is an invariant subspace. Let I

be a closed ideal. Let f EI and acERn be given. Given E> 0,by 1.7, there is a Gc7L1(Rn) such that |la*f..fi||< e. Since

(Laa)*f = La( a*f) = a *(Laf), therefore

1* -f f =111 ILa(a*f) -L 1f

!(Laa)*f - Laf 1

< E.

Since Lao * f FI and E is arbitrary, hence Laf E I.

14

1.15. Theorem. Let S c L (Rn) If (f:f S has no

common zero, then the closed invariant linear subspace spann

by S is L1(Rn) itself, and the converse is true, also.

Proof. Let I be the closed invariant subspace spanned

by S. Then by 1.14, I is a closed ideal. For every compact

set K, there are finitely many functions (fklskiN in I such

that at every point of K at least one of the f'5s is nonzeroN

ed

there. Let f = Z)fk*fk, where f(x) = f(-x). ThenN k=1

f k= Ifk|. Hence, f eI and f?(t) > 0 for all tcE:K. Sup-k=l

pose is a function in L (Rn) with g vanishing outside K.

Then by the Wiener-Levy theorem, there exists $ E ,(Rn) such

that i(t) = 1 for all t eK. It follows that goI

since g0 (t) = (g0*h*f5 (t) for all tE Rn. Thus, I contains

the dense set of all functions in L (Rn) whose Fourier trans-

forms have compact support. Therefore I = Ll(Rn), since I

is closed. The converse part is similar to 1.12 (A).

CHAPTER BIBLIOGRAPHY

1. Rudin, Walter, Real and Complex Analysis, New YorkMcGraw-Hill Book Company, Inc., l96X

2. Stein, Elias, M., and Guido Weiss, Introduction toFourier Analysis on Euclidean spaces, Princeton, N. J.,Princeton University Press, 1971,

15

CHAPTER II

FUNCTION ALGEBRAS AND GENERALIZATION

OF WIENER'S THEOREM

Throughout this chapter, X will always denote a locally

compact Hausdorff space, and i(X) will denote an algebra of

complex-valued continuous functions on X with the ordinary

pointwise algebraic operations.

2.1. Definition. (X) is called a standard function

algebra, or standard algebra for short, if it has the fol-

lowing two properties.

(i) If f c 6(X) and f(a) # 0 at a point a EX, then there

is a g E (X) such that g(x) =f() for all x in some neighbor-

hood of a.

(ii) For any closed set E cX and every point aEX - E,

there is an f E a(X) vanishing on E and f(a) / 0.

Remark. 7 (Rn) is a standard function algebra. This

follows from 1.~3 and 1.10, with A(z) = }1. 1(Rn) will

serve as a model of our development in this chapter.

2.2. Theorem. Let a(X) be a standard function algebra.

For any a c X and any neighborhood U a of a, there is a function

Ta Ea(X) such that Ta is 1 near a, and supp Ta C Ua

16

17

Proof. Since X is locally compact, there is an open

neighborhood Va of a with compact closure such that

a c Va CV acUa. By 2.1 (ii), there is an f E 6(X) such that

f(a) / 0 and f(vc) =0. By 2.1 (i), then, there is a g c(X)

such that g(x) = near a. Define Ta = f - gc7(X). Then

Ta is 1 near a and supp Tac Vac Ua

2._1. Theorem. Let a(X) be a standard algebra. If K

is a compact set in X and U is a neighborhood of K, then

there is a function f in c(X) such that f(K) = 1, and

supp f C U.

Proof. For each a E K there is, by 2.2, a Ta c 6(X) and

an open neighborhood Va of a such that Ta(x) = 1 for all

X E Va, and supp T a CU. (Va acK is an open covering of K.

Therefore there are finite points (ak)1l kN of K such that

N

K c k= ak Let us denote Tk Vk for Tak.9Va , 1<kSN,k~l kk k

respectively. Now define el = T1, and (if N2) e2 = T2 (-T),,*.,

eN TN(l TN-1)''(l Tl). Then, (i) ek c (X), lk5N; and

N N(ii) ZDek 1 (1- Tk), as can be shown by simple induc-

k=l k=l

Ntion. Define f = Zek E 6(X). If x E K, then x E Vk for some k,

k=1

hence Tk(x) = 1. It follows that f(x) = 1 for all x cK.

Since supp ek C U, l k N, therefore, supp f C U.

2.4. Definition. Let I be an ideal in a function

algebra c(X) and g be a (complex-valued) function on X. We

18

say that g belongs locally to I at the point aE X if there

is a function fa c I coinciding with g near a. If X is not

compact, we say that g belongs locally to I at infinity if

there is an f E I coinciding with g outside some compact set.

2.5. Theorem. Let 6(X) be a standard algebra and I be

an ideal of 67(X). Suppose g is a function on X belonging

locally to I at every point of X, and also at infinity if X

is not compact. Then g e I.

Proof. There is an f0 c I coinciding with g outside some

compact set K cX (if X is compact, let f0 = 0 and K = X).

For every a E K there is an fa c I coinciding with g in a neigh-

borhood Ua of a. There is also a Ta c 6(X), by 2.2, such that

Ta is 1 in another neighborhood Va of a, and supp T ac U.

Since K is compact, there are finite points (ak)l k<N of K

Nsuch that Kck=V ak. Let Vk'fk, Tk stand for Vakf k Tak

respectively. Now define e1 = T1 , and (if N 2)

e2 = 2~(1-Tl)5''e'N = TN(l~ TNl).(l1Tl). Then, as in 2.~5,

Neach ek c(X) and Z ek is 1 on K. Consider the function

k=l

N Nh = (1 - k= e)f +k k k E I. Notice that efk = ekg, since

k=l k=1ek(x) / 0 implies X EUk, and therefore fk(x) = g(x). Sim-

N Nilarly, (1 - Z ek)f (1 - Ze)g. Hence,

k=l k=1N N

h = (1 - Z ek)g + ( Z ek)g = g E I.k=l k=l

19

2.6. Theorem. If c(X) is a standard function algebra

and f E :(X) is non-zero on a compact set K c X, then there isa g E 6(X) such that g(x) = 1 for all x EK.

Proof. For each a c K, there is an open neighborhood U aof a such that f(x) / 0 for all xc U. Since X is locally

compact, there is an open Va having compact closure such

that aE Va c Va CUa. There are finite points (a k)l k5NsuchN Nk k

that K ck= Vak. Let K '= U V . Then K c K', K'is compact,k=l k

and f (x) / 0 for all x E K'. By 2.3, there is an f1 c(X)which is 1 on K and supp f1 c K'. Define g by g(x) - f(x) iff(x) / 0, g(x) = 0 if f(x) = 0. Therefore, if xfEK'c )ng(x) = 0. If ac K', then f(a) / 0, and there exists an

h c a(X) such that h(x) = 1 in some neighborhood of a.

Hence g(x) f(x) 1 (x) . h(x) in some neighborhood of a.

Therefore, g belongs locally to d(X) at every point of X andat infinity. The proof is then completed by 2.5 with I = c7(X).

2.7. Definition. Let fcE6I(X). The set of all zeros off is called the cospectrum of f, denoted by cosp f. Let S

be a subset in c7(X). The set of points of X where all func-tions in S vanish is called the cospectrum of S.. We denote

it by cosp S.

2.8. Theorem. If I is an ideal in a standard function

algebra 6(X), then a function f E a(X) belongs locally to I

at every point in X-cosp I.

20

Proof. If ac X-cosp I, then there is an hE I such

that h(a) ' 0. By 2.1 (i) , there is an h1 1 C6(X) such that

h() =hx in some neighborhood Ua of a. Then hh f eI,

and h h1f coincides with f on Uae

2.9. Theorem. An ideal I in a standard function

algebra 6(X) contains every function in 6(X) that has compact

support disjoint from cosp I. In particular, if cosp I = 0,

I contains all functions in a(x) A 000(X).

Proof. If f E (X) A 000(X) having compact support K

disjoint from cosp I, then by 2.8, f belongs locally to I at

every point of X and at infinity. It follows by 2.5 that

f G I.

2.10. Theorem. Let 67(X) be a standard algebra and let

E be a closed set in X. The ideal of all functions in 7(X)

with compact supports disjoint from E is the smallest ideal

of 67(X) with cospectrum E.

Proof. In view of 2.9, we need only to show that I =

IE = (f If c (X) n c00 (X) and supp f n E = 0} is an ideal,

and cosp IE = E. Since supp (f + g) c supp f U supp g, and

supp f - g c supp f n supp g, therefore, if f,g E I, c c C, and

hcE7(X), then f +gc-I, cf EI, and hf E I. This proves that

I is an idealof a(X). If ac E and fcEIE, then a/ supp f

so that f(a) = 0. Hence, Ec cosp IE. Suppose a / E,, then

there is an open set Va having compact closure such that

21

a ev CV a c Ec. By 2.2, there is a TaCE 6(X) such that Ta is 1

near a and supp Ta cV a. Therefore, Ta CIE and a /cosp IE'We have proved that cosp IE = E.

2.11. Definition. Let c(X) be a function algebra and

suppose that c(X) is endowed with a certain topology T. Then

6(X) is called a topological function algebra (with respect

to T) if it has the following properties.

(i) 6(X) is a topological vector space.

(ii) The map (f,g) - fg from a(X) x a(X) into 6(X) is

continuous in f for each fixed g.

(iii) For each acEX, the linear functional f-4 f(a) on

a(X) is continuous.

2.12. Definition. a(X) is said to be a normed function

algebra if the following holds:

(i) 6(X) is a normed vector space, with norm J| J|;(ii) |Jfg |J : ||f ||||1 g||;

(iii) for each a EX, the linear functional f-+ f(a) on

c7(X) is continuous.

It is clear that every normed function algebra is a

topological function algebra.

2.13. Definition. A topological function algebra and

a normed function algebra are called topological standard

algebra and normed standard algebra, respectively, if they

have the properties 2.1 (i), (ii).

22

2.14. Theorem. Let 7(X) be a topological standard

algebra. Let E be a closed set and IE E(Xf I f Ea(x) n c00(X)and supp ffnlE = 0). Consider J = IE. Then,

(I) JE is a closed ideal,

(ii) cosp JE = cosp IE = E,

(iii) JE is the smallest closed ideal with cospectrum E.

Proof. (i) Let gGJE = E, and fc 67(X). We need to

show that fg c JE. Let fg } be a net in IE converging to g.

Then, by 2.11 (ii)(, fg } converges to fg. Since fg aeiE9therefore fg c IE =E'

(ii) Clearly, cosp JE C cosp IE* If a /cosp JE, then

there is an f e JE such that f(a) / 0. It follows that there

must be some g E:IE such that g(a) / 0. For there exists a

net (g } in IE converging to f, hence, by 2.11 (iii),

g (a) -+ f(a). If g (a) = 0 for all a, then f(a) = 0, a

contradiction.

(iii) If I' is any closed ideal with cospectrum E,then IE c I', by 2.10. Therefore, JE EB c ' =C',

2.15. Theorem. In a topological standard algebra 67(X),the closure of the ideal of all functions in 67(X) with com-

pact supports is the smallest closed ideal with empty co-

spectrum.

Proof. This is simply an application of 2.14 with E =0.

2.16. Definition. A topological standard algebra 67(X)is called a Wiener algebra if 6(X) n Coo(X) is dense in 67(X).

235

2.17. Theorem. In a Wiener algebra i(X), a closed ideal

I coincides with 6(X) if and only if cosp I = s.Proof. Let I be any closed ideal in c(X) with empty

cospectrum. Then c(X) nc0 0 (X) c I. Therefore, I = I = a(X)Conversely, if I = a(X), then cosp I = cosp 7(x) = , by

2.1 (ii).

Remark. 2.17 is the abstract form of Wiener's approxi-

mation theorem.

2.18. Definition. We say that a topological function

algebra 6(X) possesses approximate units if for every f c a(x)and every neighborhood V 0 of 0 in a(X) there is a T c 6(X)

such that TfG f +U(0

2,19. Theorem. Let Q(X) be a topological function

algebra possessing approximate units. If a(X) A C00 (X) is

dense in c(X), then 67(X) also possesses approximate units

with compact support.

Proof. Let f E 7(X) and W be any neighborhood of 0 in

67(X). Let 1 ' be another neighborhood of 0 in 67(X) such that

0 + 0c 40 . Choose ucy c(X) so that af E f+7%' . Since

0 - f =0 c C', there is a neighborhood 'Y of 0 in 67(X) such

that 0- f Choose r c a(x) n0( such that T C 9 + V0r0 ?/0 T6(XA 0 0 (X) 0Then Tf = af + ( T-a)f C f + + - f c f+ +%c f +('U

2.20. Theorem. Let c7(X) be a Wiener algebra with approx-

imate units. If I is a closed ideal of 6(X), and f c 67(X)

belongs locally to I at all points of cosp I, then f E I. In

particular, I contains all functions in a(X) that vanish

near cosp I.

Proof. Let WO be any neighborhood of 0 in c(X). By

2.19, we can find a T E 6(x) n co(X) such that rf E f +V 00Let K be the support of T. Clearly, Tf belongs locally to I

at infinity. If a czK n(cosp I)c, then Tf belongs locally to

I at a, by 2.8. If a cKfncosp I, then f belongs locally to

I at a, by assumption; hence, iTf belongs locally to I at a.

Therefore, Tf belongs locally to I at every point of X and

at infinity. By 2.5, if c I. This shows that I n (f +V0) #for any neighborhood 40 of 0 in 6(x). Since I is closed,

therefore f E I.

2.21. Theorem. Let 6(X) be a Wiener algebra possessing

approximate units. If E is a closed subset in X, then the

smallest closed ideal JE in 67(X) with cospectrum E is the

closure of the ideal of all f c c(X) vanishing near E.

Proof. Consider I = (f c 67(X) : f vanishes near E}. Then

I is an ideal, and cosp I= E (the proof is similar to 2.10).

Also, cosp I = cosp I = E (the proof is similar to 2.14).

The theorem then follows from 2.20.

2.22. Definition. A topological standard algebra 67(X)is said to satisfy the condition of Wiener-Ditkin if for

every point acX the following holds: for any function

f c 67(X) vanishing at a, and any neighborhood'U0 of 0 in O(X),there is a T EO(X) such that (i) Tis 1 near a, and (ii) fT-E 00

25

Remark. The function T in the above definition can be

chosen even, in addition to satisfying (i) and (ii),

vanishing outside any pre-assigned neighborhood Ua of a.

For, by 2.2, there is a Ta vanishing outside Ua and equal

to 1 near a. Then, given f and ?70 in (X), there is a

T ' E 6(X) such that T' is 1 near a and (f Ta) TE Vo. Thus,we can take T = Ta T

2.23. Theorem. Let I be a closed ideal in a topological

standard algebra 6(X) satisfying the condition of Wiener-

Ditkin. Suppose f E :(X) vanishes on cosp I, and let P(f,I)

be the set of all points a in X such that f does not belong

locally to I at a. Then P(f,I) is a perfect set contained

in (Bdr cosp f) n (Bdr cosp I).

Proof. Clearly, f belongs locally to I at every in-

terior point of cosp f (and therefore at every interior

point of cosp I, since cosp f D cosp I). Also, f belongs

locally to I at every point of X-cosp I = (cosp I)c, by 2.8.

Therefore, if a E P(fI), then a/ (Int cosp f) U (cosp I)cSince

X = (Int cosp I) U (Bdr cosp I) U (Int (cosp I)c)

and

X = (Int cosp f) U (Bdr cosp f) U (Int (cosp f)c)

and notice that (cosp f)c c (cosp I)c, therefore

P(fI) c (Bdr cosp f) n (Bdr cosp I). Furthermore, if

b cX-P(f,I), then f belongs locally to I at b. Hence,

26

there is an cf E I and an open neighborhood Ub of b such that

fb coincides with f in Ub. Thus, Ub c X-P(fI). This proves

that X-P(f,I) is open and P(f,I) is closed. It remains to

show that P(f,I) has no isolated points. Suppose

a c(Bdr cosp f) n (Bdr cosp I), and there is a neighborhood

Ua of a such that f belongs locally to I at every point of

Ua - (a). Let Ta c E(X) be such that Ta is 1 near a, and has

a compact support contained in Ua. Since a c Bdr cosp f c cosp f,

f Ta vanishes at a. Given ?40, a neighborhood of 0 in a(X),there is, by the condition of Wiener-Ditkin, a T E 6(x) whichis 1 near a and (fTa) T E %. It follows that fTa (fTa)T

vanishes near a and outside supp Ta. Since f belongs locally

to I at every point of Ua - [a), so does fTa (fra)T. There-

fore f Ta - (fTa)T belongs locally to I at every point of X

and at infinity. By 2.5, fTa~ (fTa)T EI. Moreover,

fTa fTaT + ffTa fTaTE fTa -f T}a T+7V0.Since V 0 is arbitrary and I is closed, hence fT c I. Since

Ta is 1 near a, therefore, f belongs locally to I at a.

Thus a /P(fI). We have then proved that P(f,I) has no

isolated points.

2.24. Definition. A set A in X is said to be a scat-

tered set if A contains no non-empty perfect subset.

2.25. Theorem. Let a(X) be a topological standard

algebra satisfying the condition of Wiener-Ditkin and let I

27

be a closed ideal of (X). Let f be a function in c7(X)vanishing on cosp I and with compact support. If

(Bdr cosp f)fn (Bdr cosp I) is a scattered set, then f is in I.

Proof. By 2.23, P(fI) = O. Therefore, f belongs

locally to I at every point of X and at infinity. Thus, f E I,

by 2.5.

2.26. Theorem (Generalization of Wiener's approximation

theorem). Let 6(X) be a Wiener algebra with approximate

units and satisfying the condition of Wiener-Ditkin. Then

a closed ideal I of a(X) contains all functions f c i(X)

vanishing on cosp I such that (Bdr cosp f) n (Bdr cosp I) is

a scattered set.

Proof. By 2,23, P(fI) = 0. Therefore, f belongs

locally to I at every point of X, and in particular at every

point of cosp I. Hence, by 2.20, f e I.

Remark. From 2026, we have, in particular, a generaliza-

tion of Wiener's theorem for ,7(Rn) (cf. 1.12).

CHAPTER III

WIENER'S THEOREM IN L 1 (G)

Throughout this chapter, G will always denote a

locally compact Hausdorff abelian group. As a well-known

result, we know that on G there is an "essentially" unique

translation invariant Haar measure which induces a Haar

integral. The set of all complex-valued integrable func-

tions with respect to the Harr measure is denoted by L (G),

and the integral of a function f EL1(G) is denoted by

ff(x)dx, or simply ff(x)dx, if there is no confusion. TheG

L1 -norm on L1 (G) is defined by ||f|| 1 = flf(x)fdx, for eachG

f EL1(G). Similarly, the space LP(G) and LP-norm for l<p< coare defined by the usual way. For our further discussion, a

fixed Harr measure on G will be assumed.

3.1. Definition. Let T be the one-dimensional circle

group, T = (z E C:IzI = 1}.

(i) A continuous homomorphism of G into T is called a

character of G.

(ii) The set of all characters of G, denoted by 8, is

called the dual group of G. The addition in " is defined as:

for Xy^ca, +y(x) = (x).2(x) for each x c G. The identity

28

29

of G is denoted by 0. Instead of using Z(x), we shall let

Kx,_> denote the value of the character x EG at x e G.

3.2. Theorem. -, endowed with the compact-open topology,

is a locally compact Hausdorff abelian group.

Proof. (i) First, we prove that a, endowed with the

compact-open topology, is a Hausdorff abelian group. Since

the compact-open topology is finer than the usual product

topology, and since T is Hausdorff, therefore G c T with

the compact-open topology is Hausdorff. It is clear that 8

is an abelian group. It remains to show that the map

(X,9) - -y is continuous. Let us write, for each compact

K in G and each open U in T, (K,U) = {^ E:x(K) c U). Let

W be a neighborhood of x -Y. Then there is an open neighbor-

hood U = (K,U) of 0 in G, where K is compact in G, and U is

an open neighborhood of 1 in T such that x - y^ +U U cW. Find0

V cG such that V is a symmetric open neighborhood of 1, and

V - V c U. Let V x = +(K,V), and + = +(K,V). Then xcY,,y x

y C V^, andy

V^ - V^ =[ + (KV) ]-[ + (K, V)]x yy

A

= x - y + (KV) + (K, V)

C x - y+ (KU)

c W.

(ii) Next, we prove that ^G is locally compact. It

suffices if we can find a compact neighborhood of 0 E G. For

30

simplicity of the proof, we identify at this moment T as the

quotient group R/'Z = T', where R is the additive topological

group with the usual topology, and Z is the subgroup of in-

tegers. Let us denote, on T/G, k for the compact-open topology,

and p for the point-open topology (i.e., the usual product

topology). Let U be a neighborhood of 0 in G such that U is

compact. Let V = (x E T0' 0:5x <1 or <x <1J. Then V is an0' 3 "JA A

open neighborhood of 0 in T'. Let V = (u,V). Then V is an

open neighborhood of 0 in G. We want to show that V is com-

pact in G. For this, we first show that V is equicontinuous.

Let E> 0 be given. Choose m large enough so that -- < c.3m

Let W be a symmetric neighborhood of 0 in G such thatmE2W.c U, W. = Wj=l 0

for all j = 1,...,m. Hence, if xeW, then jxcU for all

j l.,...,m. Let xV, and write KxX>= y. Then jyEV for

j= l,...,m. This implies that 05y <y, or 1 - <y1.

Therefore, for all xceW and x 1,K|x,2)>-K(,x^)|<i.-< c33m

This proves that V is equicontinuous at 0 E G, and hence atevery point of G. Therefore Vk p c G is also equicon-

tinuous. Since T ' is compact, therefore for each x cG, the

closure of the set T[xi = {(x,> : xcV) is compact in T'.

Hence, by Ascoli's theorem (see [1]), V is k-compact.

3.3. Theorem. Suppose 1l5p<co., and fcELP(G). The map

x-> Lxf is a uniformly continuous map of G into L (a).

Proof. The proof is similar to 1.5.

3.4. Definition. Let fg be two measurable functions

on G. We define their convolution f*g by the formula

(f*g)(x) = If(x-y)g(y)dyG

provided that fJff(x-y)g (y)J|dy < .

We state, without proof, the following basic theorems.

For the proofs, refer

3.5. Theorem. (f*g(x) = g*f(x).

(ii) If fcEL (G)

uniformly continuous.

(iii) If f and g

A and B, respectively,

so that f*g eC00 0(G)

(iv) If 1< p < C,

then f*g CO(G)

to Rudin's book [21.

i) If f*g(x) is defined, then

and g cL*(G), then f*g is bounded and

are in a00 (G) with compact supports

then the support of f*g lies in A +B

1 1,np + q 1, fcELP(G) andgEL()

(v) If f and g are in L (G), then f*g E L (G), and

l f*gll : 11f 11, * 1.

(vi) (f*g)*h = f*(g*h) for fg,h E L1 (G).

The well-known fact that L (G) is a Banach space com-

bining with 3.5 (i), (v), and (iv), and some other simple

properties give the following theorem.

3.6. Theorem. L (G) is a commutative Banach algebra

with convolution as multiplication.

3~2

3.7. Definition. The Fourier transform of a function

f Ll (G) is defined as the complex-valued function ? on the

dual group 8 such that

= ff(x)K<-xi >dx.G

3.8. Theorem. Let f,gcL (G).

(i) 1 |ff||1.

(ii) (af+Pg) =a + $9, for a, P E C.

(iii) (f*g) = . ^.

(iv) (Laff"(2) = K-a, Xf(X), for aEG.

(v) If a E G, then ( p f)^ = LJf, where cp,(x) = Kx,^>.

(vi) f = f, where f(x) = f(-x).

Proof. The proofs are simple, and we shall prove (iii)

only.

(f *g) (X)= f(f*g)(x) K-x, >dxG

f Jf(x-y)g (y) dyK<-x,2> dxGG

= fg (y)-yb2>dy fK-x+y,X> f (x-y) dxG G

=2(^)?(x^).

We shall state, without proof, the following important

theorems which can be referred to in Rudin's book [21.

3.9. Theorem (Riemann-Lebesgue lemma). The Fourier

transform of a function feL1 (G) vanishes at infinity. That

Ais, given E > 0, there exists a compact set K c G such that

f(x)l < E for all x c K^R

133

3.10. Theorem (Uniqueness theorem). A function fcEL1 (G)

is uniquely determined a.e. by its Fourier transform. That

is, if fcL (G) and f(^) = 0 for all X^ 0, then f(x) = 0 a.e.

3.11. Theorem (Inversion theorem). There is a normali-

zation of the Haar measure on 0, say d2, such that for all

functions fcEL1 (G) which have a Fourier transform PEL (G)

the following relation holds.

f (x) = 5fx,x> P(2)dx, for almost all x c G.

3.12. Theorem (Plancherel's theorem). There is a

normalization, the same as the one in 3.11, of the Haar mea-

sure on 6 and a bijective linear map f-. f of LI(G) onto

L 2() with the following properties.

(i 1|f||12 1? 2'

(ii) If fcL (G) n L (G), then f^(2) = ff(x)K-x,2)dx, a.e.1l 2 Gon G. If fcLQ() n L (G), then f (x) = Sx, ̂ >( )dA, a.e. on

G .

Remark 1. The function ? which we get from the

Plancherel theorem for-a given f L2 (G) is usually called

the "Plancherel transform," which coincides with the usual

notion of the Fourier transform for fEL1 (G) n lL2 (G). The

symbol ? will be used for a Fourier transform as well as a

Plancherel transform whenever it makes sense.

Remark 2. The set of all Fourier transform of func-

tions in L1 (G) is denoted by 71(1). We define a norm on

91L(a) by Jj1 = 11f11 1, f E:L1 (G). Then, 91(G) is a commutative

Banach algebra with the usual algebraic operations. More-

over, the map f - 9 from L (G) onto 71(t) is an isometric

isomorphism.

13.13. Theorem. If f ,f2 EL (G), then

(I) ff 2(x) dx =f 2 ()^d5

and

(ii) [If-f 2 f1*2'

Proof. (i) follows easily from 3.12 (i), and the rela-

tion between the L -norm and the inner product in L2-space.

(ii) We prove first that (ii) is true for f2 EL (G) L2 (G).Replacing f2 by 72 in (i), we get

Jf(x) f2 (x) dx = Sf 2 (-x) d5.

Again replacing f2 by (Cpa)f2 in the above, where & G_, and

cp (x) = K-x,a>, we obtain-a

f1(X) f2(x) K-x,a> dx = 5f()2(9-x) d^.G .

A ~1 2Therefore,[f-f2( ) f1* 2(a) for f2 cL (G) AL (G).

Now suppose, in the general case, f ,f2 c L (G). Choose a

sequence {g n} cL (G) n L2(G) such that IIg n~f 2 2- 0 as n -co.

Then

1f-ff 4r- 11 If f *I1ACf2 1f'2 2,r- f1* n1 Io+ f~*^gn~ f2

if f2 - n1 o + 1 * (gn -~f2) 1 c

:51f~ 2 -n) ll1 12 ~n f2 12

<If 112 1f2-gn11 2 + 1 n112 12gn-f 212 - 0

as n -+ c. Therefore, [fa. f 21 f 1*if 2

35

Remark 3. Notice that the "^" on the lefthand side of

(ii) denotes the Fourier transform, and the "a" on the right-

hand side denotes the Plancheral transform.

3.14. Theorem. 1( ) consists precisely of the convolu-

tion F1 *F2 , with F1 ,F2 EL (G).

Proof. Suppose F1 ,F2 L(E): . Write F1 = ?F, F2 f2.with f ff2 EL (G). Then by 3.13 (ii), F1 *F2 1* 2

[fw f 2 r C f2 ^ E 1(a)

On the other hand, if f EL (G), then we can write

f = ff 2 with f1 ,f2 E L2 (G). Therefore, f = f f2 1*f2

where f, EL2

2.15. Theorem. Let ^, EL2( ) with supports containedA 2in a compact set K. Let g,hcL (G) be their inverse

Plancherel transform, respectively. If f = g-h, then

(i) f = gA

(ii) IlLff-| 2ax |K<y,> 1XER

Proof. The proof of (i) is simply from 3.13 (ii).

(ii) Write Lyf-f = (Lyg-g)Lyh +g(Lyh-h). By Holder's

inequality, and Plancherel's theorem,

A(L g-g)L h|L | |Lyh|2

(Lyg-g) ||2 |||| 2 = ll(Lyg) ^g||2 I1hJ1 21

= 2 111 1

$() |-y,x>-|2dx^)||||2

:51gJ1 2 1lhll2 max I(y<Y > -XEK

Similarly,

g (L h-h)||11 5 &2 2max | (y,^) 1>XEK

3.16. Theorem. Let , be a compact symmetric neighbor-

hood of O in ^. Let 9 be the characteristic function of 2,

and f, be the inverse Plancherel transform of 0.Let

o(x) =an(x) m(1)[fjx) ]2 , for each xcEG, where m denotes

the normalized Haar measure on 9. Then,

(i) a(x) 0 for all x c G and fa(x)dx = 1,

(ii) |JLa - a 5 2 maxJ| (y,x^>- 1,for y EG.XeS

Proof. (i) By the inversion theoremf(x) =

f x,X> (2)d = f(x,x^>d. Since 8 =

f (-x) = ,fK-x,> O(x)d

K-x,> dx = S x,>d

S A= .x>)dx.

Therefore, f,(-x) = f (x) and f(x) is real. By definition,

G(X) 1 2f(x)2 0 for all x EG. By Plancherel's

theorem, |f9 2 = M(S). Hence,

(X)12Ja(x)dx = 1 2dx = I 2 - I.

m(f) fxm('(ii) Applying 3.15 (ii) with g = T , = Og,K S, we get

L a-c|I : 2j# 1 max | y,9>-1|y 12 m(g) A

= 2 max |xy,5^) -lj.

37

3.17. Theorem. Let 9 be the Fourier transform of

the function a,- obtained in 3.16. Then 0 : 5 1, <() = 1,S S

and supp ^a c 9+.

Proof.

m(X) (0*)()

0() S

dy =m((X^+^)4%

( ) (x+S)n$ m(y

Hence, 0 :5 < 1, and 9Q(6) = 1. Moreover, if o (x) / 0,A Athen (X^+S)n^ / i. Therefore x E S+S, since S is symmetric.

It follows that supp ^ c S+2., since 9 is compact.A12

.Remark. Since G is locally compact, we can take S

arbitrarily small. The functions ag and their translates

show that the algebra 71(G) has the second property of a

standard algebra.

3.18. Theorem. Let 9 and T be two compact symmetric

neighborhoods of 0 in G. Let ,$^ be the characteristicS +T

functions of $ and S+T, respectively. Let fV, f be the

inverse Plancherel transform of - and 0 ^. Define9 9+T

T(X) = T A(x) 1f (x) f ,(x), for each x EG. Then,T m(S) S+T

(i) TEL1(G) and IT1l1m:5) 2

(S)lm(^)

(ii) JJL y Tlll 2 fM ) 2 ^max |(y,x> -11, for ycEG.m(') xE+T

Proof. (i) Since f and f,.^ are in L2 (G), therefore

T cL1(G). Moreover,

11 T 11 11f 12 1 ^fA ̂ 12 12 1-0m(^+T)22m()S S+T m(2) +T m

(ii) Using 3.15, with g = h hK = S + T, we

have

JL T-T|| 5 2 - s12 210max y^> ,m(S) 1$+T

1

= 22 }m xx2 ^1(+T 2malyA>l1m(s) xcS+T

3.19. Theorem. If is the Fourier transform of

the function T obtained in 3.18, then 0 : <1,

T ^() = 1 for all x^c T, and supp c +S+ T.

Proof. The proof is similar to that of 3.17.

Y 4(2) (# * Om( x S nS+T)T( )+T m(()

Hence, 0 T < <1, Y (^)= 1 for all xc ET, andST STsupp T AA c S+ S+ T.

S,TRemark. Theorems 3.16 through 3.19 may be viewed as

the generalized result of 1.3 in Chapter I.

3.20. Theorem. If K c G is compact, then for every

C> 0, there is a non-negative function aE L (G) such that

Sa(x) dx = 1, and iLy a -a |<c for all ycE:K.

Proof. Let 2 be any compact symmetric neighborhood of

0 contained in the open neighborhood (K,U), where

U = tz c T:fz-l|< L). Let a = a.. Then the results follow2S

from 1.16.

39

3.21. Theorem. If a >1, then given any compact set

K C G , there is, for every c > 0, a function T E L (G) suchthat

(i)11 T jl < a, and T is 1 near 0 in

(ii) IL T-T,| for ally c K.Proof. Given aK, and c> 0, let

U = (xEG: yX>-1< 1 (/:/a) for all yEK}.

Let S be a compact symmetric neighborhood of 0 contained in

U. Then we can choose a second such neighborhood ^' satisfying

S+T cU and m(S+T)< am(s^), since U is open and a>l. Define

T = TA as in 3.18. ThenST , 1

L T-T11 5 2 max | )XES+T

21 E< 2(a) g(-. <E.

Furthermore, (X) = 1 for all 2cT, by 1.19.

3.22. Theorem. Let o be a subset of L1 (G) having thefollowing properties.

(i) d is bounded in L (G); that is, there is an M> 0such that ||al|II5 M for all a E d.

(ii) If K is a compact set in G, then for every c> 0,

there is a function a E P such that |ILy a -all < c for all y K.

Then, given any fcEL (G), there exists, for every E> 0, a a c dsuch that |ff* a - [ff(x)dx]a ill < E.

Proof. First let Tcd be arbitrary. Write

(f*T - [f(x)dx]T)(x) = Sf(y) T(x-y)dy- [ff(y)dy]T(x)

= ff (Y) [T(x-y) - T(X).1dy.

Thus,

Ilf*T- [ff(y)dyiTI|| = flff(y)[rT(x-y) - T(x)]dyjdx

<fIf(y) I JJL y T-|TI|dy.

Since f EL (G), there is a compact set K c G such that

f If(y)Idy <'. By property (ii), there is a ac vP such thatKc

maxffLya-a|| < / || .G Thus, for this a,

f*a-[ff(y)dyaH< Sf(y)| I|Lya-aI dy + 5|f(y)| I|Lya- aI|dyK Kc

7 E/ IfI 11) SIf(y)Idy+(2M) 5 |f(y)jdyK Kc

< E +2M E- = .

3.23. Theorem. Given f cL1(G) and E >0, there is a

non-negative function aE L (G) such that a(x)dx = 1, and

f*a - [ff(x)dxial 1 < c, Jf*aj1: |ff(x)dxl +c.Proof. Consider the family K in L1 (G).

4i = (a E:L1(G):a o, fa(x)dx = 13.

According to 3.20, 2 exists and satisfies the conditions (i)and (ii) of 3.22. The conclusion then follows from 3.22.

Notice that ||f* a - [ff (x)dxl a |l1< E implies

I f*a111 <1ff (x)dxf + r, since 11la 1 = 1.

3.24. Theorem. Let a> 1 be fixed. Then given any

f E 1(G) and c> 0, there is a function TrEL (G) such thatIIT I< a, and the Fourier transform / is 1 near 6 EG, and

1f *T - [ f f (x)dx] T11| < C, IMf*TII 1 < a I f f (x)dxI + c.

41

Proof. The proof is similar to 3.23 by considering

= { T EL (G):1TII< a and T is 1 near 0EcG}. The con-

clusion then follows from 3.21 and 3.22.

3.25. Theorem. The functions in L (G) whose Fourier

transforms have compact support are dense in L (G).Proof. Let f czL1 (G). We can write f = g-h with

2 2g,h cL(G). Let g',he L2(G) be the Plancherel transforms of

g,h, respectively. Hence, by 3.13 (ii), f= ̂ *. Take

n n o (G^() such that ||2-2II2< , |/h-n 2< , for n> l.

Let gn hn EL2 (G) be the inverse Plancherel transforms of

gn' hn, respectively, and let fn =n-hn EL (G). Write

f-fn = g(h-hn) +hnn(g-g). By Hblder's inequality and

Plancherel' theorem,

n 1 I-n1i5 g(h-hn 11+ |1 hnh I2 l 1

5 119112 1h-hn 11 2 <+1hn 2 2

T11g 2 + n 11 2 <T 12+1h/1 2 + i)- 0 as n-c.Since ,n cC(), therefore f n n C E G

3.26. Theorem. In .7(^), the functions with compact

supports are dense (in the usual norm of1( )).

Proof. This is just an equivalent form of 3.25.

3.27. Theorem. 7(6) possesses approximate units.

Proof. Given that f E L1 (G) and E >0, choose a neighbor-

hood U of 0 in G such that |ILyf-f||1 <E for all y cU. Let T be

any function such that JT(x)dx= 1, and supp T c U. Then,

T * f-fil fLyf-f1 1 T(y)dy< E. Therefore, ||f-f^||I < C.

"doom

42

3.28. Theorem (Wiener-L6vy theorem). Let f^E; 1 (^:) and

K be a compact subset of G. If A(z) is an analytic function

defined on an open neighborhood of f^(Z), then there is a

function gc(E ) such that g^() = A(f(x^)) for all ^E R.

Proof. Using the result of 3.24, we see that the

results in 1.8 and 1.9 can be extended to any locally com-

pact abelian group. On account of the result of 3.18 and

3.19, we see also that part (ii) of the proof in 1.10 can

be carried over to i(G). Thus the proof of the Wiener-Levy

theorem for 91(a) is simply a copy of 1.10.

3.29. Theorem. .71(^) is a Wiener algebra.

Proof. The conditions (i) and (ii) in 2.1 are ful-

filled by 3.28 and 3.17 (remark), respectively. Therefore,

'( )is a standard function algebra. It is clear that

1 (6) is a normed function algebra. Hence, 71(^G) is a

topological standard algebra. Therefore, on account of 3.26,

,A(G) is a Wiener algebra.

3.30. Theorem. 7( ) possesses approximate units with

compact support.

Proof. This is simply from theorems 3.27 and 3.29,

and from 2.19.

3.31. Theorem. "'(a) satisfies the condition of

Wiener -Ditkin.

Proof. Suppose f cL1(G) and f(8) = 0. Then ff(x)dx==0.

GLet E >0 be given. Then by 3.24, there is a function TE L (G)

435

such that " is 1 near 0 c a, and flf*T-f ff(x)dx]T||I<cE.

Therefore, 1f9|| = ||f*T1 < c. Let a be any point of G,

and let gc L (G) be such that g(^a) = 0. Consider f = Lg.Then f e 1(G), and f(O) = 0. Therefore, there is a

T ' ( such that T' is 1 near 3, and |f '<cE. Let

T= La T 71(G). Then T is 1 near a, and

gY| = ||_0 ^Y1| = ||1 '||<E.

3.32. Definition. Let S be a subset of L (G). The

cospectrum of S, denoted by cosp S, is the set of all X'%

for which f(x) = 0 for all f c S.

3.33. Theorem. A subset S of L (G) is a closed in-

variant subspace if and only if S is a closed ideal in L (G).

Proof. The proof is similar to 1.14 and 1.12 (iii),

except that the neighborhood U = fy:|y-x| <5} needs to be

replaced by x +UO, where U0 is a neighborhood of 0 such that

IL f-fII 1 < E for all yEUO (cf. page 12).

3.34. Theorem. A closed ideal I of L1 (G) coincides

with L (G) if and only if I has an empty cospectrum.

Proof. Let I = (f:f EI}. Then I is a closed ideal

of i(). Since 7l(^) is a Wiener algebra, therefore I

coincides with 91( ) if and only if cosp I = t, and hence

I coincides with L (G) if and only if cosp I = cosp I =.

In particular, we have the following.

3.35. Theorem (Wiener's approximation theorem). Let

f EL1 (G). The linear combinations of translates of f,

Zk kak CkC, a kcG,

are dense in L (G) if and only if j? () 0 for all E c G.

Proof. The result follows from 3.34 by considering

the invariant subspace S = ) Ak La kf E C, a G) of1 k

L (G).

3.36. Theorem (generalization of Wiener's theorem).

A closed ideal I of 7(G) contains every function c,7i(6)such that f vanishes on cosp 1, and (Bdr cosp ) fn (Bdr cosp *)is a scattered set.

Proof. On account of 3.31, 3.27, and 3.29, the result

follows from 2.26.

CHAPTER BIBLIOGRAPHY

1. Kelley, John L., General Topology, Princeton, N. J.,D. Van Nostrand Company, Inc., 1955.

2. Rudin, Walter, Fourier Analysis on Groups, New York,Interscience Publishers, 192.

45

BIBLIOGRAPHY

Kelley, John L., General Topology, Princeton, N. J.,D. Van Nostrand Company, Inc., 1955.

Reiter, Hans, Classical Harmonic Analysis and LocallyCompact Groups, London, Oxford University Press, 1968.

Rudin, Walter, Fourier Analysis on Groups, New YorkInterscience Publishers, 1962.

, Real and Complex Analysis, New York,McGraw-Hill Book Company, Inc.., 1966.

Stein, Elias M., and Guido Weiss, Introduction to FourierAnalysis on Euclidean Spaces, Princeton, N. J.,Princeton University Press, 1971.