On the absolute logarithmic summability of a Fourier-series

Abstract

1. Let/(t) be L integrable in (--z~, ~) and periodic with period 2z~ and let oo oo (t.t) ](t) ~-~ 89 sinnt)= 89).1 1 ~(t) = ~{/(x + t) +/(x -t) -2/(x)),Let ~= ~(w) be continuous, differentiable and monotonic increasing in (0, oo) and let it tend to infinity as w-+ oo. A series ~ a n is The object of the present paper is to prove the followingREMARK. The order of summability in the conclusion of the above theorem can not be replaced by I as ] R, log w, t[ summability of a Fourier series is a non-local property of its generating function [3] and therefore can not be ensured by the hypothesis of the theorem, which is obviously a local one.2. In the proof of the theorem we require the following inequalities for ( w)k-lcos the kernel h (w, t) = ~ log n log n t. n<W (2.t) h (w, t) = o {t-1 log w (log w t)k-~), (2.2) h (w, t) --0 (~ (log w)k).The proof of (2.2) is trivial. For the proof of (2.t) we require the following lemmas. LEMMA t. dfsin([