On $L^p$ integrability and convergence of trigonometric series
Volume 182 / 2007
Studia Mathematica 182 (2007), 215-226
MSC: 42A20, 42A32.
DOI: 10.4064/sm182-3-3
Abstract
We first give a necessary and sufficient condition for $x^{-\gamma }\phi ( x) \in L^{p}$, $1< p< \infty $, $1/p-1< \gamma < 1/p,$ where $\phi ( x) $ is the sum of either $\sum_{k=1}^{\infty }a_{k}\cos kx$ or $\sum_{k=1}^{\infty }b_{k}\sin kx$, under the condition that $\{\lambda _{n}\}$ (where $\lambda _{n}$ is $a_{n}$ or $b_{n}$ respectively) belongs to the class of so called Mean Value Bounded Variation Sequences (MVBVS). Then we discuss the relations among the Fourier coefficients $\lambda _{n}$ and the sum function $\phi (x)$ under the condition that $\{\lambda _{n}\}\in $ MVBVS, and deduce a sharp estimate for the weighted modulus of continuity of $\phi (x)$ in $L^{p}$ norm.
