On the absolute Cesàro summability of Fourier series

L. S. Bosanquet [2] proved the following: THEOREM A. If 4a(t)/ta is of bounded variation in (0, 7r), then, for ,y > a, the Fourier series of f(t) at t =0 is evaluable I C, Py , and conversely, if the Fourier series of f(t) at t =0 is evaluable I C, Py then, for a >y +l, dIoa(t) /ta is of bounded variation in (0, 7r). Earlier, L. S. Bosanquet [II proved the following: THEOREM B. If 4)a(t)/ta is of bounded variation in an interval to the right of t = 0, then the Fourier series of f(t) at t = 0 is evaluable (C, a 1) when a > 1. The purpose of this paper is to establish the following: THEOREM. If the Fourier series of f(t) at t = 0 is evaluable I C, a(x to zero, then (t) = O (ta) when a > 1. This theorem was implicitly proved by N. Obreschkoff [4, Satz 2] when a= 1. 2. Preliminary lemmas.