A convergence criterion for Fourier series

Abstract

for a special xo does not imply convergence of the Fourier series S[f] of f(x) for that xo. The Hardy-Littlewood convergence test [4, p. 63] gives the convergence of S [f] at xo, provided that the condition (1) is satisfied, and the coefficients of Slf], a,, and bn are O(n-0) for some positive 0. The aim of the note is to give a sufficient condition on the function w(x0, h) which will insure the convergence of the S[f] at the point xo. A similar condition was given by the author [3] under some additional assumptions. The first object of this note is to give a direct proof for this result. Following A. Zygmund [4, p. 186] we shall use the following definition: A positive function L(t) defined for 0 0, L(t)t6 is ultimately a decreasing and L(t)t-' increasing function of t for t->O. From this definition we deduce immediately: If L(t) is slowly varying, then