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On L1-Convergence of Fourier Series under the MVBV Condition

Published online by Cambridge University Press:  20 November 2018

Dan Sheng Yu ,
Ping Zhou  and
Song Ping Zhou
Dan Sheng Yu
Affiliation:
Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036 China e-mail: [email protected] Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, NS B2G 2W5 e-mail: [email protected]
Ping Zhou
Affiliation:
Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, NS B2G 2W5 e-mail: [email protected]
Song Ping Zhou
Affiliation:
Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, NS B2G 2W5 e-mail: [email protected] Institute of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018 China e-mail: [email protected]
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Abstract

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Let $f\in \,{{L}_{2\pi }}$ be a real-valued even function with its Fourier series $\frac{{{a}_{0}}}{2}\,+\,\sum _{n=1}^{\infty }\,{{a}_{n}}\,\cos \,nx$, and let ${{S}_{n}}\left( f,x \right)$, $n\,\,\ge \,\,1$, be the $n$-th partial sum of the Fourier series. It is well known that if the nonnegative sequence $\{{{a}_{n}}\}$ is decreasing and ${{\lim }_{n\to \infty }}\,{{a}_{n}}\,=\,0$, then

$$\underset{n\to \infty }{\mathop{\lim }}\,{{\left\| f-{{S}_{n}}\left( f \right) \right\|}_{L}}=0\text{ifanyonlyif}\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}\log n=0.$$

We weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper, which gives the ${{L}^{1}}$-convergence of a function $f\in {{L}_{2\pi }}$ in complex space. We also give results on ${{L}^{1}}$-approximation of a function $f\in {{L}_{2\pi }}$ under the MVBV condition.

Keywords

42A2541A50complex trigonometric seriesL1 convergencemonotonicitymean value bounded variation
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 4 , 01 December 2009 , pp. 627 - 636
Copyright
Copyright © Canadian Mathematical Society 2009

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