Absolute Riesz summability of Fourier series. I

Abstract

In this paper we prove some theorems on the absolute summability of Fourier series which connect diverse | C , γ | \left | {C,\gamma } \right | results such as Bosanquet’s classical theorem (1936), Mohanty (1952), and Ray (1970) and the recent | R , exp ⁡ ( ( log ⁡ ω ) β + 1 ) , γ | \left | {R,\;\exp ({{(\log \omega )}^{\beta + 1}}),\gamma } \right | result of Nayak (1971). It is also shown that in some sense some of the conclusions of the paper are the best possible.