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Applications of MVBV condition in L 1 integrability

Acta Mathematica Hungarica
Authors: M. Wang and S. Zhou

Abstract

We generalize classical results concerning L 1 integrability, and tell a somewhat different story for sine series.

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On L 1-convergence of Fourier series of complex valued functions under the GM7 condition

Acta Mathematica Hungarica
Authors: F. J. Feng and S. P. Zhou

Abstract

Let fL 2π be a real-valued even function with its Fourier series , and let S n(f,x) be the nth partial sum of the Fourier series, n≧1. The classical result says that if the nonnegative sequence {a n} is decreasing and , then if and only if . Later, the monotonicity condition set on {a n} is essentially generalized to MVBV (Mean Value Bounded Variation) condition. Very recently, Kórus further generalized the condition in the classical result to the so-called GM7 condition in real space. In this paper, we give a complete generalization to the complex space.

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A remark on a condition raised by Tikhonov: An example in L 1 convergence of fourier series

Acta Mathematica Hungarica
Author: S. Zhou

Abstract

By constructing an example, the present note will show that the condition
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\overline {GM}$$ \end{document}
raised by Tikhonov  to study L convergence case cannot keep the classical theorem working in L 1 convergence case.
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On the uniform convergence of double sine series with generalized monotone coefficients

Periodica Mathematica Hungarica
Author: Péter Kórus

Abstract

Chaundy and Jolliffe proved their classical theorem on the uniform convergence of sine series with monotone coefficients in 1916. Lately, it has been generalized using classes MVBVS and SBVS2 instead of monotone sequences. In two variables, the class MVBVDS was studied under the uniform regular convergence of double sine series. We shall generalize those results defining a new class of double sequences for the coefficients.

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