On a necessary condition for $L^p$ $(0 < p < 1)$-convergence (upper boundedness) of trigonometric series

Xh. Z. Krasniqi

Abstract


In this paper we prove that the condition $\sum_{k=\left[\frac{n}{2}\right] }^{2n}\frac{\lambda _{k}(p)}{(|n-k|+1)^{2-p}}=o(1)\, \left(=O(1) \right),$ is a necessary condition for the $L^{p} (0<p<1)$-convergence (upper boundedness) of a trigonometric series. Precisely, the results extend some results of A. S. Belov.


Keywords


trigonometric series, $L^{p}-$convergence, Hardy-Littlewood's inequality, Bernstein-Zygmund inequalities

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