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 $.