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

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

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

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

Theorem 2.  If

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

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

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