1. The problem

C O N T R I B U T I O N S T O H A R M O N I C A N A L Y S I S

In Memory of the School of Analysis of H. Hahn, E. Helly, J. Radon, at the University of Vienna

By

H. R E I T E R 1

University of Durham, King's College, Newcastle upon Tyne

1. The problem

Wiener's approximation theorem was the starting point of m a n y developments in harmonic analysis. Carleman, in his proof of the theorem [1], introduced a new method which is of considerable generality and leads to the formulation and solution of new approximation problems. These problems are of the following type:

I n the space ~ L 1 (G) of integrable functions on a locally compact abelian group G a closed linear subspace I is given which is invariant under "translations", i.e., which con- tains with a function /o(X) also all functions /o(ax), for arbitrary a E G.

I t is required to find, for a given function / (x) in L 1 (G), the number

inf f l i (x) /o (x) l d x

foel

which indicates how closely /(x) m a y be approximated, in the metric of L 1 (G), by means of the functions belonging to I. Using geometrical language, this number is called the distance o f / ( x ) from the linear subspace I and denoted by dist {/, I}.

As is well known, this distance is the norm in the quotient-space L I (G)/I (cf. [3], Theorem 22.11.4; since LI(G) is a (commutative) Banach algebra, with convolution as multiplication, and I an ideal in L 1 (G), L 1 (G)/I is actually a quotient-algebra). The exact calculation of the distance makes it possible to determine explicitly the structure of L ' ( G ) / I .

1 The a u t h o r w i s h e s to a c k n o w l e d g e w i t h t h a n k s t h e o p p o r t u n i t i e s for r e s e a r c h accorded to h i m a t t h e U n i v e r s i t y of R e a d i n g w h e r e t h i s p a p e r w a s w r i t t e n d u r i n g the t e n u r e of a t e m p o r a r y l e c t u r e s h i p in 1955-56.

N o t a t i o n a n d t e r m i n o l o g y are as usual; of., e.g., [4]. I n p a r t i c u l a r , dx d e n o t e s t h e H a a r m e a s u r e , a n d i n t e g r a t i o n e x t e n d s o v e r t h e whole g r o u p G unless o t h e r w i s e specified.

254 H. REITER

This was carried out before for two classes of invariant subspaces (cf. [4], Theorem 1.3 and [5]). In the present paper the distance is obtained in a more general case which includes the previous ones; the study of the corresponding quotient-space will be left for a future communication.

2. G e n e r a l m e t h o d o f s o l u t i o n

If I is an arbitrary closed linear subspace of L 1 (G), not necessarily invariant under translations, and /(x) any given function in L 1 (G) then dist {/, I} m a y be found as follows

(cf. [1], Chap. I I I and [4], pp. 402403):

1"/there is no bounded, measurable ]unction q~ (x) satis/ying the conditions

f / o ( X ) ~ V ( x ) d x = O l o r a l l t o e I ,

(1)

f / (x) ~ (x) d x = 1, (2)

then / E I.

I[ there are bounded, measurable/unctions q~ (x) satis/ying

(1)

and

(2),

then

dist {/, Z} = 1/inf H~l[~,

the greatest lower bound being taken/or all such/unctions q~.

Condition (1) is expressed by saying that q~ is

orthogonal to I.

We assert t h a t

i / I is invariant then already the continuous, or even the unilormly con- tinuous /unctions orthogonal to I su//ice /or calculating the distance,

i.e., we m a y replace

"measurable" by "continuous" or even "uniformly continuous". This will be i m p o r t a n t for the applications.

T o p r o v e this assertion w e s h o w that if there is a b o u n d e d , m e a s u r a b l e function

~o(x) satisfying (i) a n d (2) t h e n there is also a b o u n d e d , (uniformly) continuous function (x) which satisfies them and which is such that I[ ~ ll~ exceeds II W li~ by as little as we

please.

First, the fact that I is a n invariant s u b s p a c e implies that

~o(y-lx)

is orthogonal to I for e a c h (fixed) y E G. It follows that for a n y u (x) 6 L I (G) the function

~ v ~ ( x ) = f u ( y ) w ( y l x ) d y

is orthogonal to I. Moreover,

q~u(X)is

uniformly continuous 1 and I[~v=[[~ ~< ][W[[~" I[ u[[x.

Indeed,

~u(X)=fu(xy)'q,(y-1)dy

and

f [u(=~yl-~,(,y)ldy=f I~,<~y)-**<y)ldy<~ ,or

zE u~ (ef. [8], p. 4U.

C O N T R I B U T I O N S TO I-IAI%MONIC A N A L Y S I S 255 Secondly, we m a y write

f /(x)w(y lx)dx=l +~(y),

where I e (y) ] < e for y 6 U~, for the left-hand side is a continuous (even uniformly continuous) function of y, by an argument similar to t h a t used before. Now let u~ (x) be a real, non- negative function in

LI(G)

vanishing outside the neighbourhood U~ and such t h a t

f u~(x)dx

= 1. Then

f u~(y)dy f /(x)tv(y-~ x)dx= l +u,

where I UI <e" Thus, letting

f~(x)= ~ u~ (y) ~o (y-1 z) dy,

we have a uniformly continuous function q). satisfying conditions (1) and (2); moreover, 1

which completes the proof.

Remark.

Assertion and proof are valid for general locally compact groups if we re- place "invariant" b y "left invariant" and "uniformly continuous" by "left uniformly continuous" throughout.

3. A theorem on bounded, continuous functions and some applications

Let G be the dual group of the locally compact abelian group G. The closed sub- groups of G and G are in one-to-one correspondence, in such a way t h a t if g c G and F ~ G are corresponding subgroups then the dual group of g is G/F and the dual of F is

Gig

(cf. [8], pp. 108-109).

For a bounded, measurable function T (x) on G, the

spectrum

is defined as the (closed) set of all elements of the dual group for which the Fourier transform of every function ] (x) 6 L 1 (G) satisfying

f /(yx)9(x)dx=O

^{for all}

y6G

vanishes (this is equivalent to the usual definition, [2], pp. 128-130).

If 9(x) is given, we m a y consider ~(xs), for fixed

x6G,

as a function of s on a (closed) subgroup g c G. The spectrum of ~(x) is in G, while

q~(xs),

as a function of

sCg,

^{has its}

spectrum in ~ / F where F is the subgroup of G corresponding to g. The relation between the two spectra, for

continuous

9, is as follows:

1 7 - 563802. A c t a mathematica. 96. I m p r i m 6 le 31 d@cembre 1956.

256 H. REITER

T H E o R ~ M 1. Let G be a locally compact abelian group, g a closed subgroup o / G and I' the subgroup o / G corresponding to g. Let q)(x) be a bounded, continuous/unction on G and

~ c ~ the spectrum o[ q).

Then q)(xs), considered as a /unction o/ s on the subgroup g (xQG being/ixed), has a spectrum which is contained in the closure o/the image o[ s resulting/tom the homomorphism

O - , O / p .

This t h e o r e m is the basis of the paper.

L e t 2' be an a r b i t r a r y element of G / F outside t h e closure of t h e image of f2~; we h a v e to show t h a t 2' is n o t in t h e s p e c t r u m of ~v (xs).

Take a closed n e i g h b o u r h o o d 0 ' of 2', with t h e same p r o p e r t y as 2', a n d a f u n c t i o n /(s) E L l(g) such t h a t its Fourier t r a n s f o r m

[(~')=f/(s)(s, ~'~ds (~'eb/P)

g

vanishes off ~ ' a n d [(2') =~ 0; 2' being t h e image of ^ xEG, [(2') ^ is also a periodic f u n c t i o n [(2) on 0 (constant on each eoset of P).

F o r a r b i t r a r y h(x) E L 1 (G), t h e function

/1 (x) = f / (s) h (s -1 x) ds (3)

g

is in L I(G) a n d has the Fourier t r a n s f o r m

[~(~) =f(~).~(:'~) (~e~).

T h u s [i (2) vanishes on an open set containing f ~ , n a m e l y the inverse image of the comple- m e n t of 0 ' . H e n c e b y a t h e o r e m of G o d e m e n t [2], Th6or6me C, a corollary of W i e n e r ' s theorem,

f f l ( y x ) c f ( x ) d x = O ( y e a ) .

L e t t i n g y = e, r e p l a c i n g / 1 (x) b y (3) a n d changing the order of integration twice, we g e t

f h (x) [ f / (s) ~o (xs) ds] dx = 0

g

Since h (x)E L 1

(G)

is a r b i t r a r y , it follows t h a t

f / (s) ~v (xs) d s = 0

(4)

g

a l m o s t e v e r y w h e r e on G. N o w the left-hand side is a continuous f u n c t i o n of x (since ~ (x) is

CONTRIBUTIONS TO HARMONIC ANALYSIS 257 a b o u n d e d f u n c t i o n , u n i f o r m l y c o n t i n u o u s on a n y c o m p a c t s e t ) a n d h e n c e ( 4 ) h o l d s for all xEG. T h u s 4' is n o t in t h e s p e c t r u m of q~(xs) w h i c h p r o v e s t h e t h e o r e m . 1

T h e n e x t two t h e o r e m s a r e a p p l i c a t i o n s of T h e o r e m 1.

T ~ E o ~E M 2. Let q~ (x) be a bounded, continuous/unction on G such that its spectrum is contained in a closed subgroup F o/ the dual group G.

Then q~ (x) is periodic with respect to the subgroup g E G corresponding to F.

A c c o r d i n g t o T h e o r e m 1 t h e s p e c t r u m of ~ (xs), as a f u n c t i o n of s C g, c o n t a i n s a t m o s t one e l e m e n t , n a m e l y t h e n e u t r a l e l e m e n t , of G / F . I t follows, for e v e r y (fixed) x E G, t h a t q~(xs) is c o n s t a n t on g (cf. [4], p. 422) a n d for s = e we h a v e q~(xs) = F ( x ) w h i c h p r o v e s t h e t h e o r e m . ~

T H E O RE M 3. Let F be a closed subgroup o/ G and suppose that A ' is a closed, denumer- able subset o/ G/F consisting o/ independent elements. Let (2 be the inverse image o/ A' in G, and A any representative system (mod. F ) o/g2. a

Then every bounded, uni/ormly continuous/unction q) (x) with spectrum in ~2 has the/orm

(x) = ~ A ~ (x) (x, ~). (5)

The "coe//icients" q~ (x) are uni/ormly continuous/unctions, periodic with respect to the sub- group g c G corresponding to F, and

(6)

I/, in particular, F is a discrete subgroup o/ ~, then every bounded, uni/ormly continuous /unction with spectrum in (2 is almost periodic.

B y T h e o r e m 1 t h e s p e c t r u m of ~ (xs), as a f u n c t i o n of s C g, is c o n t a i n e d in A ' . H e n c e i t follows f r o m t h e ] e m m a in [5] t h a t

(xs) = ~ a~, (x). (s, ~') (7)

3/r

a n d f r o m K r o n e c k e r ' s t h e o r e m (of. loc. t i t . ) t h a t

y la ,(x)l= sup (S)

~." e A" s e g

1 T h e a u t h o r is obliged to t h e referee w h o g a v e a p r o o f b o t h s i m p l e r a n d m o r e general t h a n t h e original one w h i c h needlessly r e s t r i c t e d q0 (x) to be u n i f o r m l y c o n t i n u o u s . The p r o o f a b o v e is b u t a slight m o d i f i c a t i o n of the p r o o f of t h e referee.

2 T h e o r e m 2 w a s originally s t a t e d o n l y for u n i f o r m l y c o n t i n u o u s f u n c t i o n s . The u n i f o r m i t y of t h e c o n t i n u i t y is n o t required, h o w e v e r , as p o i n t e d o u t b y t h e referee (cf. t h e p r e c e d i n g f o o t n o t e ) . a I.e., a s u b s e t of f~ w h i c h c o n t a i n s e x a c t l y one e q u i v a l e n t element (rood. F) to e v e r y e l e m e n t of f2. F o r the definition of i n d e p e n d e n t e l e m e n t s of a n abelian g r o u p , cf. [5].

258 H. RnI~ER

The "coefficients" a~, (x) are u n i f o r m l y continuous flmctions of x: az, (x) is t h e m e a n value of ~v

(xs). (s, 2')

over t h e subgroup g a n d t h u s

[ CL).. (Xl) -- a), (x2) ] ~ SUp [ 9) (X 1 8) -- ~ (~'2 8)[ ~-~ ~ 8Eg

for

x 1.x~16 U~,

b y t h e u n i f o r m c o n t i n u i t y of cf (x).

A n o t h e r p r o p e r t y of

a~,(x)

is obtained b y s u b s t i t u t i n g

xt (tE9)

for x in (7):

q~(xts)= ~ ax,(xt).(s, ~').

)~" ~ A "

:Moreover, replacing s b y

ts

in (7) we h a v e

qz(xts)= ~ az,(x)'(t, 2')'(s, 2').

A'eA"

Since t h e coefficients of (s, 2') are uniquely determined, it follows t h a t

ax, (xt)=az, (x).

(t, 2').

:Now we use the representative s y s t e m A m e n t i o n e d in the s t a t e m e n t of t h e theorem. There is a one-to-one correspondence between A a n d A ' , so we m a y write

a~,(x)

instead of

a~,(x),

being the element of A corresponding to 2' EA'. W e define now functions ~vz(x)0.eA) b y t h e relation

a~. (x) = cf~ (x). (x, 2).

These functions d e p e n d in an obvious w a y on the choice of t h e representative s y s t e m A.

E a c h gv~(x) is u n i f o r m l y continuous, a n d periodic with respect to g, i.e.,

~v~(xs)=

~v~ (x) (s 6 g). Assertion (5) of the t h e o r e m follows from (7) for s = e, while (8) implies assertion (6) or m o r e precisely

l ~ ( x ) I = sup I~ (xs) [,

8Eg where the s u m m a t i o n extends over all 2 6 A .

To prove t h e last p a r t of t h e t h e o r e m we observe t h a t if ~v (x) is a bounded, u n i f o r m l y continuous f u n c t i o n of x, t h e n so is sup ]~v(xs) l which is periodic with respect to g a n d

S E g

hence a (uniformly) continuous function on

Gig.

Thus t h e sum of the series El~v~(x) l is continuous. I f now F is discrete t h e n

Gig

is c o m p a c t a n d it follows from Dini's t h e o r e m t h a t

~1 ~v~. (x) l converges u n i f o r m l y on

G/g,

a n d t h u s on G itself. B u t then E~v~ (x). (x, 2) converges u n i f o r m l y on G; hence its s u m is almost periodic.

Remark 1.

T h e o r e m 3 holds in a s o m e w h a t m o r e general form: the set A ' m a y contain, besides i n d e p e n d e n t elements, also t h e neutral element of G/F, a n d need only be reducible instead of denumerable. This is due to t h e fact t h a t the l e m m a in [5] holds in a corresponding

CONTRIBUTIONS TO IIARMONIC ANALYSIS 259 more general form as m a y readily be shown. On each c o m p a c t subset of Gig at m o s t count- ably m a n y "coefficients" ~ ( x ) can assume values different f r o m zero.

Rernarlc 2. If in T h e o r e m 3 A ' is assumed to be discrete, t h e n e v e r y bounded, continuous function with spectrum in ~ is of t h e form (5), t h e coefficients 7~(x) being t h e n also continuous. For, if /(s)CLl(g) is such t h a t [(2') vanishes outside a n e i g h b o u r h o o d of ); CA' which contains no other p o i n t of A ' , t h e n b y (7)

f / (8) ~ (S -1 X) d s = ] ( ~ ' ) . a ~ , ( x ) . g

N o w t h e l d t - h a n d side is a continuous function of x (d. p. 256); as we m a y suppose ](~') * 0, t h e assertion folIows.

4. On certain invariant subspaces

The cospectrum of an i n v a r i a n t subspace I ~ L ~ (G) is defined as t h e (closed) set of all those elements of t h e dual g r o u p G for which the Fourier t r a n s f o r m s of all functions in I are zero [7]. I f a bounded, measurable function is o r t h o g o n a l to I its s p e c t r u m is contained in the cospectrum of I .

The preceding t h e o r e m s yield results a b o u t some classes of i n v a r i a n t subspaces.

T H E O~EM 4. Let I be a closed, invariant subspace o~ LI(G) such that the cospectrum o/ I is a subgroup F o/ G.

Then I consists o / a I 1 ]unctions in L I (G) the Fourier trans/orms o/which vanish on F.

As shown in w a f u n c t i o n / 6 L 1 (G) will belong to I if there is no bounded, continuous function ~(x) orthogonal to I which satisfies (2).

N o w Theorem 2 is applicable to ~; it follows t h a t T (x) is a function on Gig. Hence (2) m a y be written l

Gig

f

g

T h u s I1 [[oo-flf/(xs)dsldx' l.

Gig g

1 W~e d e n o t e the } I a a r m e a s u r e on G / g b y d x ' ; the m e a s u r e s are a s s u m e d to be so n o r m a l i z e d t h a t

f ( ez

Gig g

2 6 0 H. REITER

I t follows a t once t h a t t h e r e c a n be no b o u n d e d , c o n t i n u o u s f u n c t i o n q (x) s a t i s - f y i n g c o n d i t i o n s (1) a n d (2) if

f lfl(xs)dsldx'=O,

G i g g

i . e . , if t h e f u n c t i o n

f' (x')= f/(xs)ds

g

v a n i s h e s a l m o s t e v e r y w h e r e o n Gig. T h i s will h a p p e n if ( a n d o n l y if) t h e F o u r i e r t r a n s f o r m o f / ' (x') v a n i s h e s i d e n t i c a l l y on F , t h e d u a l g r o u p of Gig. This F o u r i e r t r a n s f o r m is j u s t t h e r e s t r i c t i o n of t h a t of /(x) t o t h e s u b g r o u p F c ~. H e n c e , if t h e F o u r i e r t r a n s f o r m of / ( x ) v a n i s h e s on F t h e n /(x) belongs t o I (and, of course, conversely).

T h e d i s t a n c e of a n a r b i t r a r y f u n c t i o n / ( x ) E L 1 ( G ) f r o m I is p r e c i s e l y II1'111 =

f I f iCx )dsldx'

G i g g

as m a y be s h o w n b y t h e m e t h o d of w T h i s r e s u l t was p r o v e d a l r e a d y in [4], T h e o r e m 1.3, w i t h t h e a s s u m p t i o n t h a t I is given, in a d v a n c e , as t h e s u b s p a c e of all f u n c t i o n s in L I(G) t h e F o u r i e r t r a n s f o r m of which v a n i s h e s on t h e s u b g r o u p 1 ~ c ~ .

Remark 1. If, b e i n g a closed s u b g r o u p , is e i t h e r a d i s c r e t e o r a p e r f e c t s u b s e t of r I n t h e first case t h e s t a t e m e n t of T h e o r e m 4 is i n c l u d e d in a k n o w n , a n d m o r e g e n e r a l , t h e o r e m (cf. T h e o r e m 2.2 in [4] a n d t h e references t h e r e given), b u t t h e s e c o n d case is of q u i t e a n o t h e r n a t u r e . Consider t h e g r o u p R ~ = / ~ P of t r a n s l a t i o n s of p - d i m e n s i o n a l e u c l i d e a n space, for p > 2. H e r e t h e s t r a i g h t lines, p l a n e s , a n d g e n e r a l l y t h e d - d i m e n s i o n a l closed l i n e a r m a n i f o l d s (1 < d < p) a r e p e r f e c t s u b s e t s c o r r e s p o n d i n g t o p r o p e r s u b g r o u p s . I n t h i s case w h i c h belongs t o classical a n a l y s i s , T h e o r e m 4 s h o u l d b e c o m p a r e d w i t h a well- k n o w n e x a m p l e , d u e to L. S c h w a r t z [7], w h i c h shows t h a t in L I ( R p) t h e r e e x i s t d i f f e r e n t closed i n v a r i a n t s u b s p a c e s h a v i n g the same c o s p e c t r u m , t h e surface of a s p h e r e in p - d i m e n - s i o n a l space, for p > 3.

Remark 2. T h e o r e m 4 s h o u l d be c o m p a r e d w i t h a r e s u l t p r o v e d in [6] c o n c e r n i n g g e n e r a l l o c a l l y c o m p a c t groups. T h e m a i n p r o b l e m t h e r e is t o s h o w t h a t t w o left i n v a r i a n t sub- s p a c e s of L : (G), d e f i n e d in q u i t e d i f f e r e n t w a y s , a r e a c t u a l l y i d e n t i c a l . N o w if G is a b e l i a n t h e s e t w o s u b s p a c e s h a v e t h e s a m e c o s p e c t r u m w h i c h is a s u b g r o u p of G. This is a n a n a l o g y t o T h e o r e m 4, b u t in t h e g e n e r a l case t r e a t e d in [6] no F o u r i e r t r a n s f o r m s are a v a i l a b l e . 1

: A difficulty in [6] should be pointed out: by the lemma on p. 74, loc. cir., the relation q ( x a -1) =

~(x) holds for every (fixed) gE g almost everywhere on (7. But here the exceptional set of measure zero may depend on g! The assertion that ~(x) is constant on the left cosets of 9 lacks, therefore, sufficient foundation--it would have to be shown that the exceptional null set may be chosen independently of

COi~TICIBUTIONS TO HARMONIC ANALYSIS 2 6 1

T H E 0 ~ ~ M 5. Let F be a closed subgroup o/G and suppose that A' is a closed, denumerable subset o/ ~ / F consisting o/ independent elements. Let ~ be the inverse image o / A ' in ~, and A any representative system (rood. F) o / ~ , as in Theorem 3. Let I be a closed, invariant sub- space o/ L 1 (G) with cospeetrum ~.

Then I consists o/ a l l /unctions in L 1 (G) the Fourier trans/orms o/ which vanish on ~ . For arbitrary ] (x) E L 1 (G)

dist o,.f ^sup lf/(xs)(xs, )dsldx',..

where g denotes the subgroup o~ G corresponding to F.

Given /ELl(G), we m a y calculate dist {/, I} b y means of the uniformly continuous functions ~(x) orthogonal to I(w 2). The spectrum of such a function is contained in f / so t h a t Theorem 3 is applicable. Thus (2) m a y be written

f / (x) { ~ A ~ (x) . (x, ~t)} d x = 1 or, since the functions ~ are periodic,

Gig g

Hence b y (6)

II ll f sup I f/(xs(xs, )dsJdx'>_ (9)

Gig ~eA

[The function sup ] ( / ( x s ) ( x s , ~)ds] is in LI(G/g); in fact, for each

2cA

I f/(xs)(xs, )dsI<_ f[/(xs)[ds

g g

which belongs to L 1 (G/g).]

Relation (9) immediately implies t h a t there is no bounded, uniformly continuous function ~9(x) satisfying (1) and (2) if the integral

d-- f s.p[//(xs)(xs, )dsidx'

Gig 3.cA

vanishes. 1 This will be ~he case if (and only if) each of the functions

a. N o w t h i s d i f f i c u l t y m a y be e n t i r e l y a v o i d e d b y u s i n g o n l y c o n t i n u o u s f u n c t i o n s ~ ( x ) , in a c c o r d a n c e w i t h w 2 a b o v e . I t w a s m a i n l y d i f f i c u l t i e s w i t h n u l l s e t s , e s p e c i a l l y in [6] w h i c h led to t h e c o n s i d e r a t i o n s i n w

1 I t s h o u l d be o b s e r v e d t h a t t h i s i n t e g r a l is i n d e p e n d e n t of t h e p a r t i c u l a r r e p r e s e n t a t i v e s y s t e m A u s e d b e c a u s e ! f ](x s) ( x s , ) . ) d s I d e p e n d s o n l y o n t h e c o s e t of 1 ~ to w h i c h 2 b e l o n g s .

g

2 6 2 ~ . R ~ I T ~ R

I,~(~')= ~. l(xs)(x~, 2)d~ (,leA),

g

defined on

G/g,

vanishes almost everywhere on

Gig

or, equivalently, if the Fourier trans- form of each

['~(x')

vanishes identically. The Fourier transform of

/j.(x')

coincides with the restriction of the Fourier transform of

[(x) to

the set )l-P ~ ~ . Hence I contains all functions in L I(G) the Fourier transforms of which vanish on g~ (and of course no others).

To show t h a t for arbitrary

[CLI(G)

dist {/, I} is given b y the integral d above, we observe t h a t for d = 0 this has just been proved; moreover, b y the preceding argument,

]r

if d > 0. L e t then rf(x) be a bounded, uniformly continuous function satisfying (1) and (2). Then, b y (9), inf II ~ II~ >

l/d

and it follows (w 2) t h a t dist {/, I} < d.

The opposite inequality is easy to establish. If /(x) and

]o(x)

are in LI(G) we h a v e

f l l ( ~ ) - l o ( ~ ) i d x = f dx' f ll(~8)-lo(X*)Ids

Gig g

and

f I / ( ~ ) -/o (x ~)I d~ ~ sup I f / (x~) (~, ~) d ~ - f Io (x ~) (~,, ~) d, l-

g ~ e A g g

I f now /o(X) belongs to I , then

:]o(xs)(xs,~.)ds

vanishes almost everywhere on

g

G/g,

for each ~tEA. Thus for / 0 C I and / C L I(G)

i.e., dist {/, I} > d, which completes the proof.

Theorem 5 m a y be generalized in the same w a y as Theorem 3 with regard to the con- dition concerning the set A' (el. R e m a r k 1 to Theorem 3, p. 258):

I n this somewhat generalized form Theorem 5 contains Theorem 4 which corresponds to the case where A' contains only the neutral element of G/F; moreover, if F reduces to the neutral element of G one has the result of [5]. Thus Theorem 5 represents the most general ease in which the approximation problem, for invariant subspaees I with non- e m p t y eospeetrum, has been explicitly solved. The formula for the distance which was obtained has significance in connexion with the quotient-algebra L ~(G)/I; this will be studied later.

i If .~_' is n o t d e n u m e r a b l e t h e f o r m u l a for t h e d i s t a n c e is

d i s t { / , I } = AcAc:gsup f ~Asup

I:/(xs)(xs, 2) dsldx',

A denoting denumerable subsets of A.

CONTRIBUTIONS TO HARMONIC ANALYSIS 263

References

[1]. T. CARLEMAN, L'int~grale de _Fourier et questions qui s'y rattache~t. U p p s a l a , 1944.

[2]. R . GODEMENT, Th~or~mes t a u b 4 r i e n s et th@orie spectrale. A n n . F, cole Norm., 64 (1947), 119-138.

[3]. E . HILLE, F u n c t i o n a l analysis a n d senti-groups. Amer. Math. Soc. Colloquium Publications X X X I , N e w Y o r k , 1948.

[4]. H . REITER, I n v e s t i g a t i o n s in h a r m o n i c analysis. Trans. Amer. Math. Soc., 73 (1952), 401-427.

[5]. , On a c e r t a i n class of ideals in t h e L l - a l g e b r a of a locally c o m p a c t a b e l i a n g r o u p . Ibid., 75 (1953), 505-509.

[6]. , U b e r L l - R ~ u m e auf G r u p p e n I. ~lonatsh. Math., 58 (1954), 73-76.

[7]. L. SCRWARTZ, Sur u n e propridt@ de synth~se s p e c t r a l e dans les groupes n o n c o m p a c t s . C.R. Acad. Sci. Paris, 227 (1948), 424-426.

[8]. A, ~VEIL, L'intdgration dans les groupes topologiques et ses applications. 2 e 5d., Paris, 1953.