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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  

The problem of approximation of continuous functions by Cesàro (C,α)-means, −1 < α < 0, in terms of L p and C-modulus of continuity is studied.

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Abstract

We study L p-integrability (1<p<∞) of a sum ϕ of trigonometric series under the assumptions that the sequence of coefficients of ϕ belongs to the class . Then we discuss the relations between the properties of ϕ and the properties of the sequence (λ n)∊GM(β,r), and deduce an estimate for modulus of continuity of ϕ in L p norm.

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The behaviour of the Cesàro means of trigonometric Fourier series of monotone type functions in the space of continuous functions is studied.

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We analyze the relationships of four classical and three recently defined classes of numerical sequences.

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By employing new ideas and techniques, we will refigure out the whole frame of L 1-approximation. First, except generalizing the coefficients from monotonicity to a wider condition, Logarithm Rest Bounded Variation condition, we will also drop the prior requirement fL 2π but directly consider the sine or cosine series. Secondly, to achieve nontrivial generalizations in complex spaces, we use a one-sided condition with some kind of balance conditions. In addition, a conjecture raised in [9] is disproved in Section 3.

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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  

We introduce a new class of sequences called NBVS to generalize GBVS, essentially extending monotonicity from “one sided” to “two sided”, while some important classical results keep true.

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Abstract  

Let f: R NC be a periodic function with period 2π in each variable. We prove suffcient conditions for the absolute convergence of the multiple Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative in case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to multiple Fourier series.

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Abstract  

Chaundry and Jolliffe [1] proved that if a k is a nonnegative sequence tending monotonically to zero, then a necessary and sufficient condition for the uniform convergence of the series Σk=1 a k sin kx is limk→∞ ka k = 0. Lately, S. P. Zhou, P. Zhou and D. S. Yu [4] generalized this classical result. In this paper we propose new classes of sequences for which we get the extended version of their results. Moreover, we generalize the results of S. Tikhonov [2] on the L 1-convergence of Fourier series.

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