Summability of Fourier series by triangular matrix transformations

Hille and Tamarkin have proved a result for the Norlund summability of the Fourier series of f(t) at t = x, under the hypothesis (i) φ(t) = {fix + t) + f(x- t)- 2f(x)}/2 = o(l), t- ^ 0, which includes as a special case the corresponding result for the Cesaro summability. However, under the lighter condition S t φ(u)du = o(t), t — » 0, Astrachan has proved a theorem forq the Norlund summability which does not cover the correspond-ing Cesaro case. The object of the present paper is to prove theorems for the Norlund summability and another triangular matrix method of summability which are subtler than Astra-chan's theorem in the sense that they include as a special case the corresponding result for the Cesaro summability. 1 * Definitions and notations * Let Σ~=o v n be a given infinite series with the sequence of partial sums {s n}. We shall consider sequence-to-sequence transformation of the type oo (1.1) u n = Σ d nksk in which the elements of the matrix D — ((d nk)) are real or complex constants and d nk — 0 for k> n. The sequence {un} is said to be the sequence of Z)-means of {s n}. If lim^. ^ u n exists and is equal to u then we say that the series Σ~=o v n or the sequence {s