On Certain Criteria for Fourier Constants of L Integrable Functions

396

MA THEMA TICS: C. N. MOORE

PROC. N. A. S.

ON CERTAIN CRITERIA FOR FOURIER CONSTANTS OF L INTEGRABLE FUNCTIONS By CHARLES N. MOORE DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CINCINNATI

Communicated April 14, 1932

The relationship between functions whose squares are L integrable and their corresponding Fourier constants is extremely simple and elegant. A combination of the theorems of Parseval and Riesz-Fischer tells us that the convergence of the series Z(a' + b') is both a necessary and sufficient condition that there should be a function of integrable square from which these constants may be obtained. For the more general case of L integrable functions we have no such simple result. A necessary condition is lim lim a. = 0 = bn, but this is not sufficient. Recently Salem has considered the case where the formal integral of the trigonometric series whose coefficients are a given set of constants converges to a continuous function, and he has established necessary and suficient conditions that the set of constants should be the Fourier constants of an L integrable function.' While these conditions are undoubtedly of theoretical importance, it would appear to be difficult to apply them directly as criteria to a particular set of constants. The purpose of the present note is to obtain certain sufficient conditions that a set of constants (a") should not be in the first instance, and should be in the second instance, the Fourier constants of an even L integrable function. It will be shown by simple examples that in a number of cases these criteria can be used directly to test the nature of a given set of constants. We shall obtain first the criterion that a set of constants should not be the Fourier constants of an L integrable function. Let f(x) be an even function of bounded variation in (- 7r, 7r) such that at least one out of every k successive Fourier coefficients is positive and we have for these positive coefficients the inequality ( 1n where a. > ) p

Represent the subscripts of the successive positive coefficients by ml, M2f ... Jin", .... Then we have the inequality (2) mn Mr, where r is some positive integer, that is monotonic increasing and becomes infinite with q, and is such that

VOL. 18, 1932

MA THEMA TICS: C. N. MOORE

q r

397

q(p (qk)(3

is a divergent series. Then the trigonometric series co

E

1

(4)

Cos mxx

(P(mn) is not the Fourier series of an L integrabke function. n=r

For suppose it were such a series, and represent by F(x) the corresponding L integrable function. Form

g5(X)

=

'/2ao + a, cos,x + a2cos 2x + ... + ams cos mx,

(5)

where the a's are the Fourier coefficients of f(x). Then, F(x) being L integrable and gs(x) of bounded variation, we have

f

F(x)g. (x)dx

=

_T~~~~~~In=r

(6)

(p(M.)

Since f(x) is of bounded variation, gs(x) remains bounded for all s and the left-hand side of (6) is likewise bounded for all s. But the right-hand side, in view of inequalities (1) and (2) and the divergence of (3), becomes infinite with s. Hence the coefficients of (4) are not the Fourier constants of an L integrable function. If the mn are equally spaced, the series (4) can be shown to converge at all except a finite number of points of the interval (-7r, 7r). In the neighborhood of one or more of these points the function to which the series converges becomes infinite in such a manner as to cause its integral to diverge. When (p(n) is a sufficiently simple function of n, we can actually sum the series by means of a contour integral and thus obtain in explicit form the function defined by the series. This procedure is feasible in the case of the following senres: cos 3x cos 7x cos llx I+ log 7 + log 11 +...... log 3

cos 20x cos 7x cos 13x +_ + log 20 . log 13 log 7

These examples are of interest because the series E (cos nx)/log n is n=2

known to represent a function which is L integrable over any finite interval. They are obtained from our theorem by choosing as f(x) the functions whose cosine developments are, respectively,2

MA THEMA TICS: C. N. MOORE

398

PROC. N. A. S.

+ 1/scOS 3x - 1/6cos 5x + cos x - 1/6 cos 5x + 1/7 cos 7x - 1/n1 cos lix + We turn next to the criterion that a set of constants should be the Fourier constants of a function that is L integrable. It may be stated as follows: THEOREM: If the set of constanis 1/2ao, a,, a2, . .. satisfy the conditions

1/47r -

cos x

(A)

log n I

a,

(B)

E

n=m

nn|A2a

(n 2

< M

Im

),

where M and K are positive constants then they are the Fourier constants of an L integrable function. If we set s,,(x) = 1/2 +

S.(x)

COS X

+ ... +

so(X) + s1(x) + an(x)= 1/2ao + a, cos x we have the identity =

COS

nx,

s.(x), . .. + a,n cos nx,

... +

n =m-1

crm(X) n=O

S. (x) A2a, + Sm(x) Aam + sm(x)am+i [ff- 1 sin2

^/

Jt0

1/2(ta + l)x 2a5 sin2 1/2

sin2

1/2(m + l)x Aa+ sin2 1/2x

sin 1/2(2m + l)x (7) +( sin 1/2 x We readily infer from conditions (A) and (B) that as in becomes infinite each term on the right-hand side of (7) approaches a limit uniformly in the interval (O < 5 _ x < ir); hence the same is true for the left-hand side. Moreover, we have from (7) I(X) I dx .