ANALYTIC FUNCTIONS OF ABSOLUTELY CONVERGENTGENERALIZED TRIGONOMETRIC SUMS

BY R. H. CAMERON

1. Introduction. It has been shown by Wiener that a nowhere vanishingperiodic function with an absolutely convergent Fourier series has a reciprocalwhose Fourier series also converges absolutely. Lvy has pointed out thatthis result can be extended from reciprocals to general analytic functions.Thus if f(x) is periodic and never zero and has an absolutely convergent Fourierseries, it follows that F[f(x)] also has an absolutely convergent Fourier seriesprovided that F(z) is analytic and single valued whenever z f(x). One ofthe results of this paper (Theorem I) shows that these results are true in nor even l0 dimensions. This is accomplished by carrying through Wiener’sproof with the necessary modifications to take care of dimensionality.One might reasonably ask whether this result can be extended from periodic

to almost periodic functions. A partial answer to this question has been givenby Bochner, who has shown that reciprocals of trigonometric polynomials whichare bounded away from zero on the real axis have absolutely convergent Fourierseries. It is shown in the present paper that the theorem is true not only fortrigonometric polynomials, but also for absolutely convergent infinite trigo-nometric sums. No further hypothesis is required; so the exponents are alto-gether unrestricted and may be any countable set of real numbers. Moreoverthis result is true not only for reciprocals, but for all analytic functions; and itholds in n or even N0 dimensions. Thus the final result of the paper isTHEOREM II. Let f(xl x2, be an almost periodic function with an abso-

lutely convergent Fourier series, and let R be the closure of its set of values. Thenif F(z) is a function analytic over an open set S containing R, it follows thatF[f(xl, x2, )] is an almost periodic function with an absolutely convergentFourier series.

Received July 12, 1937.N. Wiener, Tauberian theorems, Ann. of Math., (2), vol. 33 (1932), pp. 1-100; p. 14.p. LSvy, Sur la convergence absolue des sries de Fourier, C. R. Acad. Sci., Paris, vol. 196

(1933), pp. 463-464.S. Bochner, Beitrag zur absoluten Konvergenz fastperiodischer Fourierreihen, Jahres-

bericht der Deutschen Math. Ver., vol. 39 (1930), pp. 52-54.After this paper had been submitted for publication, the author learned that his main

theorem (without the extension to analytic functions or to more than one dimension)has been proved independently by H. R. Pitt. Apparently Pitt’s work was done somewhatearlier than the author’s, though it was not submitted for publication until about the timethe present paper was accepted for publication. It will appear in an early issue of theJournal of Mathematics and Physics, Massachusetts Institute of Technology.

682

ABS(LUTELY CONVERGENT GENERALIZED TRIGONOMETRIC SUMS 683

2. Absolute convergence a local property for periodic functions. Beforeproving Theorem I we shall need to extend to infinitely many variables Wiener’slemma that a periodic function has an absolutely convergent Fourier series ifin the neighborhood of every point it is equal to a function having an absolutelyconvergent Fourier series. Such a generalization naturally depends on the typeof neighborhoods we use, and the appropriate neighborhoods in this case aredefined as follows. We consider the space whose points P(xl, x,... areunrestricted sequences xl, x,.., of real numbers. Then corresponding toeach point P(xl, x, ), each e > 0 and each positive integer n we define the(e, n)-neighborhood of P to be the set of all points xl, x, satisfying

[x. x.] < (mod 2:} (j 1, n).

Jessen has shown that for such neighborhoods in which all but the first nvariables are unrestricted the Heine-Borel theorem holds for the whole space.This fact enables us to extend Wiener’s proof to infinitely many variables andobtainLEMMA 1. Let f(P) f(x x ...) be periodic of period 2 in each variable.

Suppose further it is known that corresponding to each point P’(x, x,...there exist , > 0 and n, and a function f,(P) f,(x x. which equalsf(P) throughout the (, n,,,)-neighborhood of P’ and has an absolutely convergenIourier series

fi,,(P) A’’ - A(" exp i ’Pn,i=1

Then it follows that f(P) has an absolutely convergent Fourier series

f(P A0 + i

For by ghe Heine-Borel gheorem here are a finite number of pointsP, P., P, such ghag every poing P is eongained in N + Ne, + -t- N,,where Ne is ghe (1/2e, nei)-neighborhood of P. and we shall show how o fig

ogeher absolutely eonvergen Fourier series in hese neighborhoods go makef(P). For positive values of ( < -, leg T(x) be periodic of period 2- in z, andleg ig be defined by ghe equagion T(z) max [1 [z I/, 0] in one period-_<x=<.Obviously this function consists of equally spaced isosceles peaks of height 1

with horizontal lines of height zero in between. Again, let

T}.,(x, x, T}(x,)T}(x2) T}(xm),

and note that T}.,(x, x2,... vanishes outside of the (}, m)-neighborhoodof the origin and has a peak (or infinite dimensional edge) of unit height a,

N. Wiener, loc. cit., p. 10.Jessen, The theory of integration in a space of an infinite number of dimensions, Acta

Math., vol. 63 (1934), pp. 249-323; p. 256.

684 R. It. CAMERON

Xl 0, x 0,..., x, =- 0. Finally, if e min (,..., ee.) and nmax (n,, n,), and if X is an integer so great that 2-x < /(2r), let

and note that2X+l--I

U,,,....,,(P) =-- 1,In

and hence that for all P,2X+I--I

(1) f(e) _, S,l,...,,,,(P) f(P).I ," ",ln==O

Now if in each term of this sum we replace f(P) by a function which equalsf(P) except when the coefficient of f(P) is zero, the equation will still be true.But such functions can be found with absolutely convergent Fourier series. For

2-x -xQ 2-xrg, r#, 2 r/ 0,0,-.-

is contained in one of the neighborhoods NP1 Npt,, say Ne, and it followsthat the (2-x, n)-neighborhood of Q is contained in the (ee,, ne,)-neighbor-hood of P,. Thus for all P

U,I,... ,,. (P)f(P) =- U,,,... ,,. (P)fe,(P)and since T(x) has an absolutely convergent Fourier series, so do T(x, x2,and U,I,....,,(P) and U,,,....,,(P)f(P). Consequently, it follows from (1) thatf(P) has an absolutely convergent Fourier series and the lemma holds.

3. Fourier series of small absolute value sum. Again following the courseof the Wiener argument, we prove theLEMMA 2. Let f(x x2 ) have period 2- in each variable, and let it have an

absolutely convergent Fourier series, the sum of the absolute values of the coeifi-cients other than the constant term being K. Then if F(z) is a function analyticinside and on the boundary of the circle z f(O, O, ) <= 2K, it follows thatF[f(x, x2, )] has an absolutely convergent Fourier series.For F(z) can be expanded in a power series about z0 f(0, 0, ), and the

Fourier series of f(x, x, can be formally substituted for z in this powerseries. The sum of the absolute values of the terms arising from (z z0)"will be less than or equal to (2K), and hence the whole series with all paren-theses removed will be absolutely convergent.

4. General periodic functions with absolutely convergent Fourier series. Weare now in a position to proveTHEOREM I. Let f(x, x2, be a function of period 2- in each variable,

and let the sum A of the coecients of its Fourier series==O

(2) f(x, z,.-.) Ao + A, exp i ,

ABSOLUTELY CONVERGENT GENERALIZED TRIGONOMETRIC SUMS 685

be absolutely convergent. Then if R is the closure of the range of f(xl, x, )and the function F(z) is analytic in an open set S containing R, it follows thatF[f(xl x )] also has an absolutely convergent series"

(3) F[f(xl, x., -’)] B0+Bexp iq.x+=1

On account of Lemma 1, we need only show that corresponding to each pointP (x, x., there exists an ee, > 0 and a positive integer he, and a func-tion g,,(x, x.,... which has an absolutely convergent Fourier series andwhich equals F[f(x, x,... )] throughout the (ee,, ne,)-neighborhood of P’.For under these circumstances Lemma 1 establishes the existence and absoluteconvergence of the series (3). And since the function fp,(x, x,...f(xl x, x x., ) satisfies the same hypothesis as f(x, x, ), itfollows that we need only consider the origin and show that there is a functiongo(x, x,... which equals F[f(x, x.,... )] throughout some neighborhoodof the origin and has an absolutely convergent Fourier series.Now for any function g let a(g) denote the sum of the absolute values of the

Fourier series of the function g, and let f(0, 0, ) z0, so that

n--1 j,l

Let ti > 0 be so small that z is in S if z z0] _-< t, and let N be so great that

Thus if

and

g(P) g(xl, ..) 1_A exp i

h(P) , A, exp in:N+l

it follows that f(P) Zo q- g(P) q- h(P) so that a(h)N

define W(P) II V(x,), wherenl

0 if Ixl >= 2

V(x) 2x

if

1 if

Now for 0 <: < 1/2r

(mod 2r),

=< xl _-< 2 (mod 2),

(mod 2r);

and note by actual computation that V(x) (and hence also W(P)) has anabsolutely convergent Fourier series whose absolute value sum is a boundedfunction of . Moreover, hm (r[V(x)(e" 1)] 0; for if 2;’ denotes the

686 n. I-I. CAMERON

sum from n to n W , omitting 0 and p, we have by actual computa-tion for all integers p,

o’[V(x)(e’ 1)]

2 3 cosp cos 2p2r pr.

-t- ’ 2[sin (2n p) sin p sin (2n p) sin p](n- p)

<2 3 2 sin p sin 1/2p2r pr

_1_2(2np p:) sin n sin 1/2n

(n p)n

-b ’ f2 sin 1/2(2n p) 11/2.1sin 1/2PI + 21sin (2n p) t.lsin

2 p. 1/2p

(n- p)2r

212np P sin n].lsin 1/2n I\+ (n 2

(211/2(2n P) 11/2"l 1/2PI -[- 21(2n P) t’lpl7’z_,’\ (n p)r+

=< 61i _[.. I? _,’ (p 1/2(2n(np) I +-)i2p 2n p I

Now consider W(P)g(P), and note that

lim a[W(P)g(P)] O,-0+

since

21/212np P I" 3}(n p)21n -lim (r[W(P)e(’+’++’"+’vx)(e.... 1)] 0/--.0 +

holds for any integers p and implies

lira o-[W,(P)(ei(pxI+px:+’’’+pNN 1)] 0./-,0 +

Finally, choose > 0 so small that

o-[W(P)g(P)] < -,and consider the functions

(P) Zo + W,(P)g(P) + h(P).Since

o,... h(0, 0,... 0,

ABSOLUTELY CONVERGENT GENERALIZED TRIGONOMETRIC SUMS 687

the constant term of ](P) is ](0, 0, z0; and since

[](P) z0] o’[W.(P)g(P) -+- h(P)] <: 1/2,

Lemma 2 applies to ]’(P) and shows that go(P) F[]’(P)] has an absolutelyconvergent Fourier series. But by definition W,(P) --- 1 throughout the(e, N)-neighborhood of the origin, and hence go(P) FIr(P)] in the neighbor-hood, and our proof is complete.

5. Almost periodic functions. We can now pass to the case of generalizedFourier series and prove Theorem II, which has been stated in the Introduction.Let

f(xl, x, ..) Ao + A, exp iX.ix..n==l ’I

The proof can be based on Theorem I in the following way. Let > 0 be sosmall that if zl z21 -< and z e R, then z2 e S; and let R* be the closedset consisting of all points whose distance from R is not greater than . Let N

be so great that ] A < 1/2, and let #1, u, u be an integral basis

for all k,. for which n =< N and j -< p so that

Here the k.i,, are integers, and no integers kl,

If p is the greatest of p,

k except 0, 0 make

pv, consider the functions

Yp, ;N+, N+, "")

A0+ Aexp i k,.,Y,n=l 1 =1

h(Y,, ".., Y,, Y,,, ..., Y. ;+, +, ...) + A,e’.It is clear that g has period 2r in all its arguments, and that

(x, x, ).Putting in exponents arbitrarily as we hve done fter the N-th term changes the

range of the function; while putting them in according to the bsis s Bochner did in hispaper (loc. cit.) nd s we have done in the erlier terms would require the use of limitperiodic functions if it were crried out for all the terms. The author wishes to thnkProfessor Norbert Wiener for suggesting this combination of the two methods.

688 R.H. CAMERON

Moreover, the closure of the range of h is the same as the closure of therange of

N Pn

A0 -F- AN exp i X,.x-.nl

Thus by Theorem I it follows that F(g) has an absolutely convergent Fourierseries

nl j=l 1 j,N-F1

Hence

Fir(x1, x2, ...)]- B0 + lBexp i q,.tx + i r,Xj,,x,j==l ’==1 j=NWI ==1

and Theorem II is proved.

MASSACHUSETTS INSTITUTE OF TECHNOLOGY.