# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

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

### Tom 182 / 2007

Studia Mathematica 182 (2007), 215-226 MSC: 42A20, 42A32. DOI: 10.4064/sm182-3-3

#### Streszczenie

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.

#### Autorzy

• Dansheng YuDepartment 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
e-mail
• Ping ZhouDepartment of Mathematics,
Statistics and Computer Science
St. Francis Xavier University
Antigonish, Nova Scotia Canada, B2G 2W5
e-mail
• Songping ZhouInstitute 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
e-mail

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