A Tauberian theorem and its application to convergence of Fourier series

Abstract

integer and let 2} bk,, = Bk.,,. Whenever ~. bh, n_ , 9 S~/Bk. . converges to f = O 0 a limit say S as n---~~,, I shall indicate that fact by the symbol S,----~S (N, k) as n---~~,. (1 "1) The main object of this note is the proof of two theorems (I and II) the first of these being a Tauberian theorem on mean convergence or summability (N, k), the second being the summability (N, 1) of the Fourier series of a summable (L) function f ( x ) at a point xb at which, { ( f (x0+ lT)-f(x0)) ' log I h D}-+0 as h -*0 . From these two theorems, which are new, ir is noteworthy that the truth of the convergence criterion, of Hardy and Littlewood (1932) of the Fourier series of a summable function f ( x ) a t a point xo, namely (1 ){ ( f (x0+ h ) f ( x o ) ) log [h 1}~0 as h---~0 (2) tl~e Fourier coefficients of f (x) of order n 0, 91 (1.2) follows at once. Before proceeding further it should be noted that summability (N, k) is more stringent than Cesaro means of any positive order, for it can be established that (N, k) implies (C, r) when k > 0 and r > 0, but not conversely. 2. Statement o f Theorems