On $ (J, p_n) $ summability of fourier series - Chandra, Satish (Tamkang Journal of Mathematics, 2001)
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 $$ Φ (t) = \int_t^{π} \frac{φ (u)}{u} du = o \{ p (1- \frac{1}{t} ) \} ~~~~ (t → 0) $$ then the Fourier series for $ t = x $ is summable $ (J, p_n) $ to sum $ s $. Theorem 2. If $$ G(t) = \int_t^{π} \frac{g(u)}{u} du = o \{ p(1-\frac{1}{t}) \} ~~~~ (t → 0) $$ then the differentiated Fourier series is summable $ (J, p_n) $ to the value $ C $.