investigations in harmonic analysis h. j. reiter...

Investigations in Harmonic Analysis

H. J. Reiter

Transactions of the American Mathematical Society, Vol. 73, No. 3. (Nov., 1952), pp. 401-427.

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INVESTIGATIONS IN HARMONIC ANALYSIS(1) BY

H. J. REITER

This paper is concerned with the theory of ideals in the algebra L1 of integrable functions on a locally compact abelian group.

After some preliminaries an analytical proof is given of the known theorem that an analytic function of a Fourier transform represents again a Fourier transform (p. 406). Then, in part I , the continuous homomorphisms of closed ideals I of L' upon C, the field of complex numbers, are studied. Any such homomorphism is given by a Fourier transforin and, if lois its kernel, the quotient-algebra I/Io,normed in the usual way, is not only algebraically isomorphic, but also isometric with C (Theorem 1.2). Another result states that homomorphic groups have homomorphic L1-algebras and that a cor-responding property of isometry holds (Theorem 1.3).

In part 11, which may be read independently of part I , a theorem of S. Mandelbrojt and S. Agmon, which generalizes Wiener's theorem on the translates of a function in L1, is extended to groups (Theorem 2.2). Several generalizations of Wiener's classical theorem have been published in the past few years; references to the literature are given on p. 422. The rest of part I1 is devoted to some applications (pp. 422-425).

In conclusion it should be said that the work is carried out in abstract generality, with the methods, and in the spirit, of analysis, which is then applied to algebra.

To Professors S. Mandelbrojt, B. L. van der Waerden, and A. Weil I owe my mathematical education. The inspiration which I have rcccivcd in their lectures, in letters, and above all in personal contact, is a t the base of this work; may I here express my gratitude.

Groups and Fourier transforms. The standard work of reference is A. Weil's book [14]; cf. also [4].

For the definition of a topological group see 114, p. 91. The additive group of real numbers (the "real axis") is denoted by R1, p-dii~lensional space by Rp.The group operation will always be denoted multiplicatively (it is hoped that this will not cause confusion in the case of R1). The identity (unit element) of the group is denoted by e.

We shall deal only with locally compact abelian groups G. By G we denote

Presented to the Society, December 28, 1951; received by the editors January 12, 1952. (1) This paper was submitted to the Faculty of the Rice Institute as a doctoral disserta-

tion. The author wishes to express his thanks for the generous award of a fellowship.

401

402 H. J. REITER [November

the dual group, by (x, x) the value for x E G of the character defined by xEG. For R l the symbol (x, x) may be regarded simply as an abbreviation of exp(2~ixx) and R1 is the same as R1.

The measure of a set S C G (SCG) is denoted by m(S) (m(S)). L1, L2, Lm denote, respectively, the space of complex-valued functions on G which are integrable, of integrable square, measurable, and essentially bounded. The last notation, a slight departure from Weil's, is used in [4]. Functions in L1, L2, Lm will be denoted, respectively, by f(x), . . . , F(x), . . . , +(x), . . . . We use a dot to indicate the conjugate of a complex number and we write f*(x), +*(x) for f'(x-l), +'(x-l).

The Faltung of two functions in L1 is

We shall also use the Faltung f*+(x), + E L m . Sometimes we shall write f(x) * + or even f(x) * +(x), according to convenience. If + l=f * +, then +T=f* * +*.

The Fourier transform of f EL1 is(2)

This is the definition adopted in [4, p. 871 rather than that used by Weil 114, p. 1121-the difference is irrelevant. For G =R1 it represents a slight, but convenient, deviation from the classical definition.

Banach spaces. For the definition and properties of Banach spaces B over the field of complex numbers, and the notation, we refer to Hille's book 181.

L1 and Lm are Banach spaces, the norms being, respectively,

The dual space of L1 is Lm; we shall write the general bounded linear functional on L1 in the form

The following "distance theorem" is a reformulation of a well known result 18, Theorem 2.9.41:

Let L be a closed linear subspace of B and dejine for x l E B

(2) Thus a function in Ll is indicated by a letter in boldface type and its Fourier transform by the same letter in italic type. This notation, considerably different from that used by French authors, was dictated by technical necessities of printing. Likewise for the use of the dot instead of the customary bar.

19.521 INVESTIGATIONS IN HARMONIC ANALYSIS 403

dist ( X I , L ] = inf 11x1 - xoli (XOE L).

If there is no bounded Linear functional x*(x) satisfying

for all xo E L,

then x l E L If there are bounded linear functionals satisfying condition (*), then

dist {xl , L ] = l/min IIx*II,

where x* ranges over all these functionals. This is an immediate consequence of Theorem 2.9.4 in [8] and the fact

that if x* satisfies (*) then x*(xl -xo) = 1, and hence IIx*II 2 l/IIxl -xol/. This theorem will be applied, in the space L1, to calculate the distance

in some cases (cf. 13, chap. 1111). The function +(x) is called orthogonal to f(x) : + l f , if Jf(x)+'(x)dx=O.

I t is called orthogonal to the linear subspace I C L 1 : +1I,if 4-l-f for all f E I . If 91L1 , then 9 vanishes almost everywhere.

The L1-algebra. L1 is not only a Banach space, but also a commutative Banach algebra [8], multiplication being defined by the Faltung.

A (closed) ideal in the algebra L1 is defined as a (closed) linear subspace of L1 which contains, with a function f , all products f JF g, for any g€L1. Closed ideals will be denoted by I.

A closed linear subspace is an ideal if and only if it is ((invariant under translation," i.e., contains, with a function f(x), all "translates" f(xy-I), for any (fixed) y E G . For the sufficiency see [6 , p. 1271. The necessity is proved here, for the sake of completeness.

Consider any f E I and any g E L 1 . Then, by hypothesis, f * g E I and hence if .ff(x)+'(x)dx = O for all f G I , then

Since g(y) is arbitrary, i t follows that

Since this holds for any +II,it follows that f(xy-I) €I,for any (fixed) y E G. The intersection of the zeros of the Fourier transforms of all functions in


I,which is a closed set in G, is called the co-spectrum of I by I,. Schn~artz [12]. We shall denote it by Zr.

By I(x, x)' we shall mean the "rotated" ideal obtained by replacing all f(x)€1 by f(x) (x, z) ' , x being any (fixed) element of G.

Let S be a compact, symmetrical neighborhood of the identity e E G and denote by S(x) the characteristic function of S :

This will be used as a standard notation throughout, and neighborhoods S,Sf,S,, etc., will always be supposed to have the above properties (without explicit mention).

Define now

Since S-l= S, S(x)is real and S(x-l) =S(x). If G =R1 and S= [-6/2, 6/21, S(x) = (sin mx) /TX; if G is compact and S= ( e 1 , then S(x) =1.

Consider now the Faltung

S * S(x) = m(xS A S).

By the theorem of Plancherel-Weil

In the applications we shall "normalize" the above Faltung by nieails of the factor l /m(S). The function thus obtained taltes values between 0 and 1 ; it is 1 for x = e and vanishes outside the neighborhood S2.

The function is non-negative and has the followi~~g p rope r t i~s (~ ) :

This follows immediately from the theorem of Plancherel-Weil, applied to S(x) and S(x).

Given E > O , for any (fixed) y E G

(ii) r n - l ( ~ ) / l S ( ~ - l x ) ~- s ( x ) ~ / ~< E ,

(3) We shall denotc the norm in L2 (and in the corresponding space L2) by l l ~ i l ~ = ( J I F(x)I 2 d ~ ] 1 / 2 .

19521 INVESTIGATIONS IN HARMONIC ANALYSIS 405

provided S is small ( S C U, a neighborhood of e depending on E and y ) . By Schwarz's inequality

m-l(~)lI 1~(y- 'x ) + S(x) ) { ~ ( y - l x )- S(x) ) I j 1 5 m-l(~)l j~(y- 'x)+ s(x)jj zjjS(y-lx) - s(x)lj2.

By the theorem of Plancherel-Weil this is equal to

m-112(~)lI~(x) l ) l j 21 (Y, x)' + 1 ) l12m-l12(s)lIs(x) 1(Y, x)* -

provided SCU, where the neighborhood U of e is such that I (y, x) -11 < ~ / 2for all xEU and the given y .

Take now two neighborhoods S and S' (cf. the convention concerning the notation on p. 404). Then the product SS' is a neighborhood of the same kind.

Consider the characteristic functions

S(x) and SS1(x)

(the use of two letters to denote a function is perhaps somewhat unusual, but convenient here). The normalized Faltung of these functions,

1%-I (S)S a SS1(x) = m-I (S)m(xS A SS'),

takes values between 0 and 1 ; it is 1 for xES1, vanishes outside S2S',and is the Fourier transform of

m-I (5') S(x) .SS'(x) . We shall need later the following estimate:

(iii) IIs(x)ssl(x) - B ( ~ ( s s ' ) -s ( x ) ~ ~ ~ I m(S) ]'12rn1/2(~).

Indeed,

I s (x ){s s l (x )- S(X)] [ I IB s (x)~~~I~ss ' (x)- s(x)lIz =

IlS(x)llzllSSf(x)- S(x)llz = m1I2(S) ( ~ ( s s ' ) - m(S) ) 112.

We shall also need a stronger form of property (ii): Given any (fixed) compact set C C G , then for any E > O

(ii') m - 1 ( ~ ) l l ~ ( y - 1 ~ ) 2S ( X ) ~ ~ I ~- < e

for all y € C, provided S is small ( S C U, a neighborhood of e which depends on E and C).

Referring to the proof of (ii), it is required to sholrr that there exists a neighborhood U of e such that 1 (y, x) -11 < ~ / 2 for all y E C and all xEU (this is obvious in the case G =R1; in the general case it is precisely the defini-

406 H. J. REITER [n'ovember

tion of the topology of the dual group given by Pontrjagin). The character (y, x) is a continuous function on G X G [14, p. 101]. Thus,

given E>O,, for each point y , E C there are neighborhoods V , of y , and U, of e such that I (y,x) - I I = I (y, x) - (y,, e ) I < ~ / 2 for all yE V,and all x E U,. Since C is compact, a finite number of the neighborhoods V, already cover C, and we may take for U any neighborhood of e contained in the intersection of the corresponding neighborhoods U,. Thus (ii') is proved(4).

Another estimate to be applied later is the following: Let C C G be a compact set and let E > O be given. Then

(iv) m-'(s)I I S(~-'X) SS ' (~-~X) ~ ( x ) < e(m(SS1)/m(S))'I2- SS'(X) 1 1 for all y E C , provided both S and S' are chosen small (namely, such that SS'C U, where U is some neighborhood of e depending on E and C).

The proof is analogous to that of properties (ii) and (ii'), U being such that I (y, % ) - I [ < ~ / 2 for all y E C , x E U .

We also have

( 4 m - l ( s ) l l ~ ( x ) ~ ~ ' ( x ) j j5 (m(SS')/m(S))1'2.

ANALYTIC TRANSFORMSFUNCTIONS OF FOURIER

THEOREM.Let f(x) be a funct ion in L1 and let CCG be a compact set. Denote by I' the set of points z , in the complex plane, of the form z =f(x), x E C . Let n o w A(z) be a (single-valued) analyt ic funct ion, regular at all points of r. T h e n there i s a funct ion in L1 whose Fourier transform i s equal to A(f(x)) for all x E C .

This theorem is proved in two steps: first it is proved locally, i.e., for a sufficiently small neighborhood of any (fixed) point xo€ C; then the extension to the whole set C is carried out. The proof is modelled after that given by Carleman for the case G =R1 [3, chap. IV] (j).

Let xo be any (fixed) point of C and expand A(z) in a power series about zo=f(xO) .We may write, for x in some (small) neighborhood of xo.

m

A(f(x)) = A(f(x0)) + C cn(f(x) - f(x0))". n= 1

Consider now the Fourier transform (cf. p. 405)

S(X)= m-l(S)S(x) * SS1(x).

If both S and S' are small, the function

(4) The reader may now turn immediately to part I (p. 409) or part I1 (p. 417). (6) Professor A. IVeil has directed my attention to Segal's paper [13]. Theorem 3.8 in [13]

practically contains, as a special case, the theorem given here.

19521 INVESTIGATIOKS I N HARMOKIC ANALYSIS

exists for all x E G and coincides with A( j (x ) ) for x E x o S t The product

is the Fourier transform of

where

S ( X ) = m-l (S )S(x )SS1(x ) .

Write now

Hence

Given E >0 , let KC G be a compact set such that

By property (iv), pp. 406-408, the neighborhoods S , S' may also be so chosen that

for a l ly EK. Keeping S fixed, we may take S' so small that m ( S S 1 )<2 m ( S )(9). Under these conditions

5 ~{2l1~11fll l (cf. p. 406).+ 2(2)ll2j

Define now

q n ( ~ )= 4 * qn-~(x)r n > 1, q l ( x ) = q ( x ) .

Then

(6) This follows from the regularity of Haar measure (cf. [ 7 ] ) and the fact tha t for any open set 0 containing S there is an S' such tha t SS'CO.

H. J. REITER [November

llqnlll s llqll~l n 2 I.

By a suitable choice of the neighborhood., involved, me can thus obtain a function q(x) such that llqill <R, where R is the radius of convergence of the power series x,",,c,zn. Then the series

converges in L1-norm and its Fourier transform coincides with A(ffx)) for all xExoS'.

The theorem has now been proved locally; to extend it to the set C as a whole, we proceed as follows:

To every point x , E C there corresponds a (small) neighborhood x, TT,,V, being a neighborhood of e , such that there exists a function f , E L 1 whose Fourier transform f,(x) coincides with A Cf(x)) for all x ~ x , Vf. The season for considering first x,V, and then x,V: will appear shortly. Since C is compact, a finite number of those neighborhoods, say xl VI, xz Vz, . . . , x . ~VX, already cover C:

Take a neighborhood S so small that S C V, (1 5 r 5N) ; then

Define now a partition of the set CS as follows

n-1

cs1= cs nxlv:, CS, = cs n X,V; nc u cs, ( 2 5 n 5 x).

Thus 2

CS, c x , ~ , . , css c xrv,3 (1 5 r 5 N ) .

Consider the functions f r (x)EL1 (1 S r s N ) whose Fourier transforms f,(x) coincide "locally" with A(f(x)), i.e., for xEx,V:, respectively.

Let CS,(x) be the characteristic function of the set CS, (1 I r IN). Then the Faltung

is the Fourier transform of

(the notation being the usual), and vanishes outside CSrS. Moreover the sum

INVESTIGATIOKS IN HARMONIC AKALYSIS

N

m-l(S)S(x) * CSr(x) r=l

is just

m-l(S)S(x) * CS(x) = m-l(S)m(xS nCS)

and hence is 1 for x E C. Now the Fourier transform

coincides with ACf(x)) for all xEC. Indeed, if x E C , then in the product

either the Faltung in the bracket vanishes or else, if it does not, fr(x) coincides with ACf(x)), since CS,SCX,V~.

Thus, for all x E C , Z(x) =A(f(x)) . 1 and the theorem is proved. The following particular case should be especially noted: I f f (x )EL1 i s such that f(x) # O on the compact set CCG, then there i s a

.function g(x)EL1 such that

g(x) = l/f(x) for all x E C.

This theorem is a t the base of tile results that follow. For that reason it seems worthwhile to have a purely analytical proof; other proofs have been given by Godement [6, ThCor2me A] and Segal 113, Theorem 3.81.

COROLLARY.Let f(x), g(x), be in L1. I f f(x) vanishes outside some compact set and the zeros of g(x) are all interior to the set of zeros of f(x), then there i s a function h(x) EL1 such that

LEMMA1.1.1. I f f EL1 i s such that f ( e ) # 0 , then every solution +oELmof the integral equation

has the following property:

for any constant y, unless +O i s zero almost everywhere.

Consider the function ~ - ' ( S ) S ( X ) ~ . If S is small, we have, by the corollary above since f(x) # O in some neighborhood of e,


so that

rn - l (S )S (~ )~ = 0.$ +:(X)

Now

rn- l (S)S(~)~* 1 y + +o(x) I 2 S ess. sup I y + +o(x) 1 2 . The left-hand side may be written(')

rn- l (S)S(~)~* i: 1 y l 2 -/- 2~ [ys+o(x) 1 + / +o(x) 1'1 = 1 Y l 2 + ~ R [ ~ . ~ - ~ ( s ) s ( x ) ~* + o ] + rn - l (S )S (~ )~* I + o 12.

But the middle term vanishes and hence, unless + O is zero almost everywhere,

I y l 2 < ess. sup I y 4 +o(x) 1 2 , which was to be proved.

I a ~ l n ~ ~ ~ 0, then the integral equation 1.1.2. If f EL1 i s such that f ( e )=

f *+*(x) = 1

has no solution +ELw.

Suppose there is a + E L m such that

S ~ ( Y ) + . ( X - ~ Y ) ~ Y for all x E G.= 1

Multiply both sides by W - ~ ( S ) S ( X ) ~ and integrate with respect to x:

Since Jf(y)dy =0, the left-hand side is equal to

Given E > O , there is a compact set CCG such that

(7) R [ z ]=real part of z.

19521 INVESTIGATIONS IN HARRlONIC ANALYSIS

Furthermore, for any y € C,

provided S is small, by property (ii'), p. 405. Thus the last repeated ~211+11 ,+llfll 1 ~ 1 1 + 1 1integral is in absolute value smaller than ,. Since this

may be made arbitrarily small, no solution + E L * can exist. Given a function f E L 1 , denote by (f) the closed ideal generated by f , i.e.,

the smallest closed ideal in L1 containing f. I t may be defined explicitly as the closure of the set of all linear combinations of "translates" of f(x):

where al, az, . . . , a , ~are arbitrary complex numbers, yl, yz, . a . , YN are any elements of GI and N is any positive integer (cf. p. 403).

Define now the ideal (f), as the closure of the set of all linear combinations

where the coefficients b, satisfy the condition

The reason for the notation mill appear later. REMARK.I t is important to observe that (f), may also be defined as the

closure of the set of all linear combinations

with arbitrary coefficients a,.

THEOREM1.1. For any f (x)EL1

where N ranges over all positive integers, bl, bz, . . . , bH range over all complex numbers satisfying EL1bn =0, and yl, yz, . . . , y~ over the elements of G.

(If the equality sign is replaced by '(2 ", the statement becomes trivial.) Making use of the definition of the ideal (f),, the theorem may be stated


concisely thus:

dist I f , ( f ) , ] = ( f (c ) 1 . The proof is an application of the distance theorem (p. 402): me have

dist { f , ( f ) ,) = l/min / + I , , where + ranges over all functions in L* satisfying

(here the remark on p. 411 is used). If no such + exists, then fE(f ) , . Thus we have to investigate the solutions of the integral equation

Suppose first that y =lf(x)dx#0. Then every solution may be written in the form

where +0 satisfies f * +z(x) =O. By Lemma 1.1.1 the function +(x) = l /y ' is the solution with the smallest norm and hence

disi, { f , (f) ,) = j f (e) I If lf(x)dx = O , then, by Lemma 1.1.2, there is no + E L * satisfying

f * +*(x) =1 and hence fE(f) , . Thus the proof of the theorem is complete. Theorem 1.1 may be generalized. Let I be the closed ideal generited by

the functions f , g, . . . , i.e., the smallest closed ideal containing (f), (g), . . . ; we express this by writing

I = (f , g,.. . ) .

We define noxv the ideal I, as the smallest closed ideal containing (f),, (g),, . . . and write

I e = (f, g , . . . )e

Thus I, is the closure of the set of all finite sums of certain linear combina-tions (see p. 411) of the translates of f, g, . . . , respectively.

Then for any k € I

dist { k , I , ) = 1 k ( e ) 1 . Indeed, since k E I , there are, for any E > O , among the generating functions

f, g, . , a finite number, say f ~ ,£2, . . - , f~ such that the function

INVESTIGATIONS I N HARMONIC ANALYSIS

(where the choice of the coefficients a,, and the elements y,,E G depends on E )

satisfies the condition

llk(x) - kc(x) l l~< E. Consider now the function k, and the ideal (k,),: by Theorem 1.1

dist (k., (k.).] = lSk . (x)dxI . From the definition of I, it is clear that

(kt) 0 C I e

and since for all h E I0

it follows that

dist (k,, I.] = 1 Sk.(x)dx 1 This implies that

dist { k , I,) 5 Ilk - k.111 + / Sk.(x)dx 1

Now the inequality dist { k, I,] 2 I Jk(x)dxl is again trivial and hence, since E > O is arbitrary, the result is proved.

We proceed now to the final generalization of Theorem 1.1. If I= (f), we define the ideal I,= (f), as the closure of the set of all linear

combinations

where x is a fixed element of G, N is any positive integer, y,E G (1sn 5N), and the coefficients satisfy the condition


If I = ( f , 8, . . . ), we define I , = ( f , g , . . - ), as the smallest closed ideal containing ( f ) , , (g),, . . We have now (in the notation of p. 404):

dist { k ( x ) ,I , ] = dist { k ( x ) ( x , x ) ' , I , ( x , a ) ' ) .

But the ideal I , ( x , x)' bears to the ideal I ( x , x)' the relation

I,( . , x ) . = ( ~ ( x ) ( x ,x ) ' , g ( x ) ( x 7 x ) . , . . . ) e = I ( x , x)e,

and hence we may apply the first generalization of Theorem 1.1 to the function k ( x ) ( x , x ) ' E I ( x , x)' and the ideal I(x,x ) ; , so that we finally obtain

dist { k , I , ) = I k ( x ) 1 . The results obtained may now be summarized as follows:

THEOREM1.2. Let I be a closed ideal in L1.Dejine, for a j i s e d s E G , the idea2 I , C I a s above. T h e n for a n y funct ion fE I

dist { f , I , ) = I f ( x ) I . Hence I , i s the kernel of the homomorphism

f -+ J(.) of I into C , the Jield of complex numbers. T h u s I , i s a nzasinzal ideal of I or coincides wi th I according a s x E e Z 1 or x E Z I . JIoreover, i f the guotient-algebra I / I , i s considered a s a Banach algebra(8) , then the i somorphism

preserves the norm. Conversely: ginen a cont inuous homomorphism

f -t P ( f )

of I u p o n C , then

where x , E G i s a uniquely determined element of CZr.

T o establish the converse, let s be any (fixed) element of G. I t is proved in [6, p. 1221 tha t

Pf(.-"x)) = (st ~ , ) . P ( f ) i

where x , is a uniquely determined element of G. Consider now the ideal I,,. I t is the kernel of the (continuous) homo-

morphism

(8) See, e.g. [8, Theorem 22.1 1.41.

INVESTIGATIONS IN HARMONIC ANALYSIS

f -+J f ( x ) ( x , %,)*ax.

Let I, be the kernel of the given homomorphism f -+p( f ) . If we can prove that IzpCI, ,it will follo~v that I zp=I ,and hence that p ( f ) =J f (x ) (x , x,)'dx.

Suppose there is a function g E l , , which is not in I,. Then p ( g ) # O and, for all s EG ,

while

Now p ( f ) , considered as a bounded linear functional on I, may be extended to the whole space L1,by the complex analogue of the Hahn-Banach theorem [a, Theorems 2.9.2 and 2.9.51. Hence

We have therefore for the function g ( x ) :

S s ( ~ - ~ x ) + : ( x ) ~ x (s , XJ .PCS) for all s E G ,=

S g ( x ) ( x , x , ) 'dx = 0.

If we write this in a slightly different form and apply Lemma 1 . 1 . 2 , we see that it is impossible. This completes the proof.

THEOREM the group G' i s homomorphic image of G , then the 1.3. I f a L1-algebra on G' i s a homomorphic image of the L1-algebra on G. The kernel I. of this homomorphism consists of all functions in L1(G) whose Fourier transforms vanish on the dual group G ' C G , and the isomorphism

preserves the norm (the measure on G' being properly normalized).

For any f ( x ) E L 1 ( G ) ,define

where the closed subgroup g C G is such that G 1 = G / g [14, p. 1 1 1 , and x' denotes the coset x g . Then fl(x')EL1(G1)and (with proper normalization of


the measure)

This result (immediate for G=R1) is proved in [I]. We have also

Furthermore, since GI= G/g, the dual group G' is contained in G and (y, x') =1 whenever yEg, x'EG' (cf. [14, pp. 108-1091). Thus for x =xlEG' ,

Since a function in L1(G') is entirely determined by its Fourier transform, it follows that T(f(x)) is determined by the values of f(x) for x =x'EG1. Hence we have

i.e., T is a homomorphism of L1(G) into L1(G'). The kernel of this homomorphism is the closed ideal I. consisting of all

functions in L1(G) whose Fourier transforms vanish on G'CG. We have

inf ((f(x) - fo(x) / (~I(\f '(xl)(\l (fo E 10).

To establish the opposite inequality, we use the distance theorem (cf. p. 402) :

dist ( f , 10)= l/min l l + l l m where

LEMMA.Any + l I o has the property that

almost everywhere (and conversely).

For any k(x) EL1(G), k(sx) -k(x) Elo,i.e.,

19.521 INVESTIGATIONS IN HARMONIC ANALYSIS

Since kEL1(G) and s E g are arbitrary, the assertion follows (the converse follows from the fact that f(x) $ +* =fl(x') * +'*-cf. below).

Thus the function $ 1 1 0 defines a function +'(xl) on G' and we have

Hence

ll+ ' l lml l f ' l l1 2 1

and thus

ll+(x)llm 2 l/llf'll 1.

This gives the opposite inequality. Thus the Banach quotient-algebra L1(G)/Io is not only algebraically iso-

morphic, but also isometric with the image of L1(G) in L1(G') under the homomorphism T. I t remains to show that this image is L1(G') itself.

Since L1(G)/Io is complete, it suffices to show that the functions T(f) are dense in L1(G'). But this follows from a proof given by A. Weil [14, pp. 42-43]. Thus the theorem is established.

11. A GENERALIZATION OF IVIENER'STHEOREM, WITH APPLICATIONS

By Wiener's theorem is meant Theorem IV in [IS]. I t was extended by Godement [6, p. 12.51 to locally compact abelian groups and reads:

Let I be a closed ideal in L1. In order that I= L1, it is necessary and sufficient that for every xoEG there exist an f(x) €I such that f(x0) t ' O .

In the course of the proof of Wiener's theorem a result is established which may be stated as follows:

An ideal I contains all functions fc(x) whose Fourier transforms vanish outside some compact set C C 021.

(Actually this is stated here in a slightly more general form which is precisely what we shall need. The proof is that of Godement.)

For any x l E C there is a function £ , € I such that I fl(x) / 2 1 in some small neighborhood x,Sl of x,. Since C is compact, it may be covered by a finite number of those neighborhoods, x,S, (1 s r 5N). Then the Fourier transform of the function EL,f,(x) * f:(x) €1is strictly positive on C. If we now apply the corollary, p. 409, to fc(x) and this function, the result follows.

We shall suppose from now on that G has a denumerable fundamental system of neighborhoods of the identity (but cf. Appendix). We may then


choose in G a fundamental sequence (S,), n 2 1, of compact, symmetrical neighborhoods of e such tha t Sn+ICS,. Then every subsequence (S,,), r 2 1, will still be a fundamental system.

LEMMA2.1.1. For any function + E L m there exists (at least) one constant M such that, for a certain subsequence (n,),

~- '(S,,)S,,(X)~* + -t M (7 +w ) ,

for all x E G.

Since the numbers m-l(S,)JS,(y)2+(y)dy are all bounded, in absolute value, by l l + l l m , there exists a subsequence (n,) such that m-1(Sn,)J~,,(y)2 .+(y)dy tends to a limit, M, say, as r-+w.

We want to show that for any (fixed) x E G , ~- ' (S, , )S, , (X)~ * ++M (r+m). I t will be sufficient to prove that

for any (fixed) x E G . Now the integral may be written

and this, in absolute value, will be smaller than e/ j+/ lm,by property (ii), p. 404, if n, is sufficiently large. This completes the proof.

LEMMA2.1.2. Suppose that $11and that the identity e E G is not a limiting point of 21. Then, if the neighborhood S is small ( ~ ~ n z ~ n ~ ( e } =@),

m - l ( s ) S ( ~ ) ~* + = const.,

the constant being independent of S (subject to the above condition).

REMARK.This is of course trivial if G is compact, since then we may take S= ( e l . See also the Appendix a t the end of this paper.

Take a neighborhood S A T of the fundamental sequence of neighborhoods, so small that s ~ s H A z ~ ~ ~ ( ~ )Then the Fourier transform =@.

S * SSN(x)- S * SS,(x), n > N ,

vanishes outside the set S2S~nCS,(cf. p. 405; it may be helpful to plot a figure for the case G =R1). Applying the result on p. 417, we have

I t is our first aim to prove that

S(x) .SS,y(x) - S(x) E I.

19521 INVESTIGATIONS IN HARMONIC ANALYSIS

Indeed, we have

S(X).SSM(X)- S ( X ) ~= { S(X) .SSN(X) - S(X).ssn(x) ] + { ~ ( x ) ~ ( x ) ~ } ..ss*(~) -

The first part is in I (see above); the second is, in L1-norm, less than m(SS,) -m(S) } 1/2m1/2(S)(cf. property (iii), p. 405). But this can be made

arbitrarily small, by taking n sufficiently large(g), and thus the partial result is proved.

On the other hand, if we take S n C S , then

(cf. p. 405) the reader may again find it helpful to plot a graph of the Fourier transforms for the case G =R1).

But

(set S=S, in the partial result just proved) Hence finally

for all S,CS, i.e.,

+ being the function of the hypothesis. Hence we have

Let now (n,) be a subsequence such that

(Lemma 2.1.1). Since the left-hand side of the above equation is independent of n, i t follows that

rn - l (S )S (~ )~*+ = M = const.,

which proves the lemma. I t also entails that M is unique. In this case we write ~11+(x) ) instead of M and call i t the mean value of +(x).

This terminology is justified by the fact that , i f a ( x ) is not only in La,but also almost periodic in the sense of von Neumann, then

( 9 ) This follows from the regularity of the Haar measure (cf. [ 7 ] ) and the fact that, for any open set 0 containing S, the product SS, will be contained in 0 for large n.

420 H. J. REITEK [Kovember

exists and coincides with the usual mean value. Let A be tha t mean value. Then, given E > O , there exist positive numbers cl, 62, . . . , c , ~ ,whose sum is 1, and elements yl, yz, . . . , y x E G such that for all x E G

We may write the difference

in the form

But each term of the first sum tends to 0, as n+m, by property (ii), p. 404, and the second term is in absolute value less than E, for every n. Since E > 0 is arbitrary, the result is proved.

From Lemma 2.1.2 there follows immediately a general result:

THE ORE^^ 2.1. Let I be a closed ideal in L1 and suppose that x lEG is not a limiting point of 21.Suppose that + I I . Then the mean value of +(x)(x, xl)' exists and, if S is suficiently small ( x l S 2 n Z I n e { xl ] =@) ,

Indeed, we may reduce this case to Lemma 2.1.2 by writing +(x)(x, xl). I I ( x , xl)', the right-hand side denoting the "rotated" ideal (cf. p. 404).

Now we are ready to prove a generalization of Wiener's theorem (for groups G whose dual groups have a denumerable fundamental system of neighborhoods of the identity; cf. hon~ever the Appendix) :

THEOREM2.2. Let I be a closed ideal in L1, with co-spectrum Zr (p. 404). If the function k(x) EL1 is such that its Fourier transform k(x) satisjies the following conditions :

(i) Zk>Zr (21,denotes the set of zeros of k(x)) ; (ii) J(Zk) nJ(ZI) is denumerable (J(Z) denotes the frontier of Z) ,

then k(x) EI.

This theorem mas first proved, for the case G =R1, by S. n'fandelbrojt and S. Agmon 111, ThC.ori.rne 111, malting use of Carleman's analytic functions [3, p. 7.51. The reader will see for himself the connection between their proof and that given here.


To prove the theorem, it suffices to show (cf, the distance theorem p. 40.2) that whenever +IIthen +Lk .

Consider a +IIand define the function +l(x) by +l(x) = k*(x) * +(x). If we can prove that +ILL1, then it will follow that k * +*(x) =0 , and + I k .

Let Il be the set of all f E L 1 such that f * + T = O ; this is a closed ideal. We shall prove:

(2) ZI, has no isolated points.

From (1) it follo\vs, under the hypothesis of the theorem, that 21,is denumerable, and from (2) that it is perfect. This implies that ZI,=@, and hence (Wiener's theorem) Il=L1. Thus the theorem will be proved if we can establish (1) and (2).

(1) Since g * +*=O for any g € I , we have

Thus +lII,i.e., ICIl, and hence ZI,CZI. Now we shall show that no interior point of ZI, can belong to 21,.

Indeed, if xo is an interior point of Zk, let S be so small that xoS2CZk, and consider the function (x, ~ o ) S ( x ) ~ . This function is in Illsince [(x, x o ) ~ ( x ) ~ ] * +;= [(x, X O ) S ( X ) ~ ] (observe that * k * +*=O the Fourier transform of [(x, x ~ ) s ( x ) ~ ] * k vanishes identically). Thus xo is not in ZI, (cf. [ l l , Lemme I]).

Therefore 21,is in ZI and contains no point interior to Zk. Since Z I C Z ~ , the only points that can possibly belong to Zr, are those points of J(ZI) which are in J(Zk). Thus (1) is proved.

(2) Let us suppose that x lE r (Zr )n r (Zk) is not a limiting point of ZI,. Consider first the case xl=e. By Theorem 2.1

e.i., rn - l (S)S(~)~ * k * +* = MS{+l ) . Take now S=S,. By Lemma 2.1.1 there exists a subsequence (n,) such

that rn-l(S,,)S,,(~)~ * +* tends (boundedly) to a constant c ( r + w ) . But then the left-hand side of the last equation tends to cJk(x)dx=c.k(e) =O. Thus M'(+l)=0, i.e., S ( X ) ~ E I I , and hence e is not in 21,(see also Appendix).

The general case may be reduced to the preceding one by considering instead of +l(x) the function

The ideal corresponding to that function is just the "rotated" ideal (cf.


p. 404) Il(x, xl)', with co-spectrum x;'Zr,, and the co-spectrum of k(x)(x, xl)' is x;'Zk. Thus, by the special case just proved, e is not in x;'Zr,, i.e., xl is not in Zr,, and the proof of (2) is complete.

REMARKS.Condition (i) of the theorem is obviously necessary as well; moreover, for compact groups it is also sufficient, condition (ii) being then superfluous, since the dual group of a compact group is discrete. Thus for compact groups G there is a one-to-one correspondence between the closed ideals of L1 and the subsets of G which inverts the inclusion (cf. [lo, p. 3921). On the other hand L. Schwartz has proved [12] that condition (i) alone is not sufficient in general, if G is not compact. This shows the role of condition (ii).

The reader is referred to Mackey's survey [lo, pp. 392-3951 for a discus- sion of previous generalizations of Wiener's theorem. Besides the papers by Mandelbrojt-Agmon [ l l ] and Godement [ 6 ] ,we mention here in particular those of Ditkin [5], Segal [13], and Kaplansky [9](1°).

We shall now apply the preceding results to study the functions + E L m orthogonal to a given ideal I, when the co-spectrum ZI has a particularly simple structure.

If ZI is jinite, then any +IIis a linear combination, with constant coefi- cients, of the characters of G defined by the elements of Zr.

Let ZI consist of xl, xz, . . , XN. If k(x) is any function in L1, then (for S suitably small)

N

k(x) - k(x) * C (x, ~,)rn-l(S)S(x)~ n=l

will be in I (Theorem 2.2), i.e.,

for any +II.By Theorem 2.1 we may then write

k(x) * +*(x) -x ai(x, x,) = 0{ N

1n=l

where an =M(+(x) (x, x,)' ). Since k(x) is an arbitrary function in L1, i t follows tha t

(lo) It should be pointed out that some of these papers contain theorems proved for arbitrary locally compact abelian groups. Since the present paper was submitted for publication, Dr. Henry Helson has kindly sent me a reprint of his paper Spectral synthesis of bounded functions, Arkiv for Matematik vol. 1 (1951) pp. 497-602. This paper contains also a generalization of Wiener's theorem (for arbitrary locally compact abelian groups). The theorem is stated there in a weaker form, but the proof would actually yield the result given here.


almost everywhere. Cf. [6, Hypothkse A, p. 1361 and [9]. If 21is discrete, then any uniformly continuous +IIis almost periodic, witlz

spectrum in Zr (and conaersely). Let kc(x) be any function in L1 whose Fourier transform vanishes out-

side some compact set C. Then, in the same way as before,

kc(x) * +* = C kc(x,)a:(x, x,) (x, E ZI A c).

Now the functions of the type kc are dense in L1 [13, Theorem 2.61. Thus for any k E L 1 the function k * + is almost periodic, with spectrum in 21. In particular, set k(x) =m-l(V)V(x), where V(x) is the characteristic function of a neighborhood V of e. Given E > O , there is a V such thatI m-l(v) ~ ( x ) < E , for all x E G , by the assumed uniform con-*+-+(x)] tinuity of +(x). Hence +(x) itself is almost periodic, with spectrum in ZI (the converse follows from the approximation theorem for almost periodic functions).

For the case G =R1 this was proved by A. Beurling [2]. If now G =R*, let V, be a sequence of (#-dimensional) cubes, with center

a t the origin and sides of length l /n . Then m-l(V,) V,(x) *+ -+ +(x) almost everywhere (72-403). Since m-l(V,) V,(x) * + is almost periodic, there are trigonometric polynomials P,(x), with frequencies in 21,such that /m-l(V,)V,(x)*+-P,(x)/ < l / n , n 2 1 . Thus P,(x) -+ +(x) almost everywhere (n-+co) and 1 P,(x)I <l/+i/m+l G=Rp, <f 21isfor all n. Hence, for discrete, then the set of all +IIconsists of all those fz~nctions in L* which are limits (almost everywhere) of sequences of uniformly bounded trigonometric polynomials with frequencies in Zr.

In the case G =R1 this was proved by Carleman [3, p. 1151, with an addi- tional restriction on 21.

If 21is discrete, then to any +IIthere corresponds (by Theorem 2.1) the "Fourier series"

where a, =M{+(x)(x, x,)' ) and the summation extends over all x ,EZr . The ('sum" is purely formal-no ordering of the terms, or convergence, is implied.

If all the Fourier coefficients are zero, then +(x) vanishes almost everywhere ("uniqueness theorem"), since then + I L 1 .

LLBessel'sinequality" holds in the sense that

S , (X)~+(x)I a , 1 _I lim inf mn-l(S,) S 1 1 ~ d x , n-t rn

the set of all a,#O being denumerable. The proof is the same as in the classical case.

The question now arises under what conditions it is true that

H. J. REITER

("Parseval's equation"). Consider the function

which is orthogonal to I. We have, for large n,

But, in view of Theorem 2.1, this is just

by an argument already familiar. The last sum (over all x,EZr) has actually only'a finite number of terms.

Set now S=Snand write correspondingly an(x) , A,(L, K) . We are going to indicate some restriction on 21which will guarantee the absolute convergence of

Suppose 21is such that, for some S,x S n 2 1 =x , for all X E Z I (21 inay then be called "uniformly discrete"). Then we have, for sufficiently large n ,

by the uniqueness theorem. Suppose now that 21is the union of a finite number N of sets, each uni-

formly discrete. Then for sufficiently large n , say n z n o , the following holds:


if one of the indices L, K is kept fixed, then A,(L, K) > O for a t most N values of the other one.

Define now the function T(L, K) by

Then both x,T(L, K) and ELT(L, K) are a t most equal to N. Hence

Hence the series

xr,rAn(~i K)aLa,*

will converge absolutely, and (some order of the terms having been fixed) uniformly with respect to n. By the uniqueness theorem

Thus

M [ I + 1 2 } = lirn W-'(S,) S 1 +(XI 12dxS ~ ( X ) ~ n+ =

exists and

M { I + 121 = x I a1 1 2 .

Thus, i f Zr m a y be decomposed into a finite number of sets each of which i s uniforlnly discrete (p. 424), then "Parseval's equation" holds for all +II.I t follows that the Fourier series of + converges "inthe mean" to +.

For the case G= R1,this means that all + I I are almost periodic B2 (Besicovitch), since in this case M [ I + /2 ) , as defined here, coincides with the mean used by Besicovitch, in view of a theorem of Wiener [16, Theorem 211.

APPENDIX(")

I t is possible to modify the proof of Lemma 2.1.2, avoiding Lemma 2.1.1 entirely and dropping the assumption that the dual group possess a de-numerable fundamental system of neighborhoods of the identity (first axiom of countability). The modification is as follows:

In Lemma 2.1.2 it is proved that the function

(11) Added September 15, 1952.


for S sufficiently small, is independent of S, without any recourse to the first axiom of countability (it is convenient, of course, to change the notation, writing, say, Sf,S" instead of S,, SN,respectively, where SfCS").

To show that the function above is also independent of x, take x =e and set

Now the difference

is equal to

for any S'CS, and hence we have

5 m-'(~~)ljs '(x- 'Y)~- ~~(Y)~ljljj+lj ,

=< ~ l / + / l m ,

by property (ii), p. 404, if Sfis small enough. Thus Lemma 2.1.2 is established by a method which is more general as well as simpler.

The only other place where the first axiom of countability is used is on p. 421 where it is proved that in the equation

the constant M' (41) is zero. Keeping S fixed and setting

we have

k *$" = ~ ' { + l j

and also

Now in Lemma 1.1.2 (part I) i t is proved, without any countability restriction, that the last two equations imply

I t results that the proof of the extension of the theorem of Mandelbrojt- Agmon (Theorem 2.2) is valid for locally compact abelian groups in general.

19521 INVESTIGATIONS IN HARMONIC ANALYSIS 42 7

BIBLIOGRAPHY 1. W. Ambrose, Direct sunz theorenz for Haar measures, Trans. Amer. Math. Soc. vol. 61

(1947) pp. 122-127. 2. A. Beurling, S z ~ r zlne classe de fonctions presgue-pdriodiques, C.R. Acad. Sci. Paris vol.

225 (1947) pp. 326-328. 3. T. Carleman, L'intigrale de Fourier et qz~estions gz~ i s 'y rattachent, Uppsala, 1944. 4. H. Cartan and R. Godement, Thdorie de la dualitd et analyse harnzonigue dans les groupes

abdliens loca2enzent conzpacts, Ann. &ole Norm. vol. 64 (1947) pp. 79-99. 5. V. Ditkin, On the structure of ideals i n certain nornzed rings, cf. Mathematical Reviews

vol. 1 (1940) p. 336. 6. R. Godement, Thiordnzes taz~bk iens et thiorie spectrale, Ann. gcole Norm. vol. 64 (1947)

pp. 119-138. 7. P. R. Halmos, Measure theory, New York, 1950. 8. E. Hille, Functzonal analysis and semi-groups, Amer. Math. Soc. Colloquium Publica-

tions, vol. 31, 1948. 9. I. Kaplansky, Primary ideals i n group algebras, Proc. Nat. Acad. Sci. C.S.A. vol. 35

(1949) pp. 133-136. 10. G. W. Mackey, Functions on locally cotupact groups, Bull. Amer. Math. Soc. vnl. 56

(1950) pp. 385-412. 11. S. Mandelbrojt and S. Agmon, Une gindralisation d z ~ thiordtne tauberien de Wiener,

Acta Szeged vol. 12 B (1950) pp. 167-176. 12. L. Schwartz, Sur zlne propridtd de synthise spectrale dans les grozrpes non compacts, C.R.

Acad. Sci. Paris vol. 227 (1948) pp. 424-426. 13. I. E. Segal, The group algebra o f a locally conzpact group, Trans. Amer. Math. Soc. vol. 61

(1947) pp. 69-105. 14. A. Weil, L'intigration duns les groupes topologigues et ses applications, Paris, 1940. 15. N. Wiener, Taz~berian theorems, Ann. of Math. vol. 33 (1932) pp. 1-100. 16. ---, The Fourier integral and certain of its applications, Cambridge University

Press, 1933.

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