On the Summability of Fourier Series. Third Note

PHYSICS: HILLE A ND TA MA RKIN

594

PR~oc. N. A. S.

"

This corresponds to Lemaitre's equation (30). The article of Friedman is interesting in considering a number of possibilities as to the past history of the universe, including that of a periodic expansion and contraction, all, however, with the restriction Po = 0. De Sitter also considers a number of such possibilities, all, however, with the restriction dM/dt = 0. " Reference 2, Equation 23. This equation was derived setting the constant g2 0. This is also desired for our present purposes, however, to make our equation (13) agree with Lemaitre's equation (24). 18 Reference 2, § 8 (e). Note that this should read 4r(po + 4po) or 4(poo + Po) instead of 4 (po + Po). 14 Reference 2, §8 (g). "O Reference 2, Equation 28. 16

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ON THE SUMMABILITY OF FOURIER SERIES. THIRD NOTE BY EINAR HILLS AND J. D. TAMARKIN PRINCUTON UNIVERSITY AND BROWN UNIVURSITY Communicated August 8, 1930

.1. In the present note we continue the discussion of the summability of Fourier series by the methods [H, q(u)] of Hurwitz-Silverman-Hausdorff.1 We give necessary and sufficient conditions for effectiveness, and show that the problem is closely connected with the theory of Fourier transforms. We recall that the function q(u) associated with the method [H, q(u)] is supposed to satisfy Hausdorff's conditions: q(u) is of bounded variation in (0, 1) and continuous at u = 0, q(O) = 0, q(l) = 1. Let f(x) be any integrable function of period 2ir. The nth (H, q)-mean of the Fourier series of f(x) is

~r+

H.[f(x), q] = J f(t + x)H,(t) dt,

(1)

where

Hn(t)

= -1

{(j1 -

Y J (1 if"

-

u

+ue')it} 'dq(u).f

(2)

We put

p(t) =f(x + t) +f(x - t) - 2S, A~~(t) = J sp(r)dr, 1'(t) = J Iv(T)IdT.

(3) (4)

We shall use the following terminology. A point x will be called regular or pseudo-regular with respect to f(x) according as there exists an S = S,

VOL. 16, 1930

PHYSICS: HILLE AND TA MARKIN

595

such that lim (p(t) = 0 or c1'(0) = 0. It is well known that almost every t -0 point, and, in particular, every regular point is pseudo-regular. The method (H, q) will be said to be Fej&r or (F) effective, resp. Lebesgue or (L) effective with respect to Fourier series if lim H,, [f(x), q] = S, for every integrable function f(x) at every regular point, resp., at every pseudoregular point of the function. We introduce the following auxiliary functions. We denote the total variation of q(u) in the interval (0, u) by Q(u). Further , l