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

Xh.Z. Krasniqi

doi:10.15330/cmp.7.1.83-90

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

Authors

Xh.Z. Krasniqi University of Prishtina, Mother Teresa str., 10 000 Prishtine, Prishtina, Kosovo

https://doi.org/10.15330/cmp.7.1.83-90

Keywords:

trigonometric series,

L^{p} -

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

Published online: 2015-07-03

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.