Matrix summability of Fourier series based on inclusion theorems

Abstract

Letdenote the Fourier series expansion of a function. Féver's celebrated theorem states that the series is summable (C, 1) at each point of continuity off, where (C, 1) denotes the Cesàro method of summability of order 1. Riesz extended this result to (C, α) for each α > 0. Since then many authors have established sufficient conditions on various methods of summability to guarantee similar results. Over the years the pattern has been to strive for weaker conditions on the matrix, and to replace the condition of continuity on the function by a less stringent one. Theorems have been proved not only for the summability off, but the summability of the derived series, and other series, related tof. Beginning with the work of Hille and Tamarkin[13], many of the theorems have been extended to absolute summability.