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Abstract  

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

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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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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 [2] to study L convergence case cannot keep the classical theorem working in L 1 convergence case.
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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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