On $L^p$ integrability and convergence of trigonometric series

Dansheng Yu ¹   ; Ping Zhou ²   ; Songping Zhou ³

¹ Department of Mathematics Hangzhou Normal University Hangzhou, Zhejiang 310036, China and Department of Mathematics, Statistics and Computer Science St. Francis Xavier University Antigonish, Nova Scotia Canada, B2G 2W5
² Department of Mathematics, Statistics and Computer Science St. Francis Xavier University Antigonish, Nova Scotia Canada, B2G 2W5
³ Institute of Mathematics Zhejiang Sci-Tech University Xiasha Economic Development Area Hangzhou, Zhejiang 310018, China and Department of Mathematics, Statistics and Computer Science St. Francis Xavier University Antigonish, Nova Scotia Canada, B2G 2W5

Studia Mathematica, Tome 182 (2007) no. 3, pp. 215-226

Résumé

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.

Détail

DOI : 10.4064/sm182-3-3

Keywords: first necessary sufficient condition gamma phi infty p gamma where phi sum either sum infty cos sum infty sin under condition lambda where lambda respectively belongs class called mean value bounded variation sequences mvbvs discuss relations among fourier coefficients lambda sum function phi under condition lambda mvbvs deduce sharp estimate weighted modulus continuity phi norm