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

Tom 182 / 2007

Dansheng Yu, Ping Zhou, Songping Zhou 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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