On $(J, p_n)$ summability of fourier series

Abstract

In this paper we prove the following two theorems for $| J, p_n |$ summability of fourier series, which generalizes many previous result:

Theorem 1.  If

$$\Phi (t) = \int_t^{\pi} \frac{\phi (u)}{u} du = o \{ p (1- \frac{1}{t} ) \} ~~~~ (t \to 0)$$

then the Fourier series for $t = x$ is summable $(J, p_n)$ to sum $s$.

Theorem 2.  If

$$G(t) = \int_t^{\pi} \frac{g(u)}{u} du = o \{ p(1-\frac{1}{t}) \} ~~~~ (t \to 0)$$

then the differentiated Fourier series is summable $(J, p_n)$ to the value $C$.