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Abstract  

We discuss determination of jumps for functions with generalized bounded variation. The questions are motivated by A. Gelb and E. Tadmor [1], F. M�ricz [5] and [6] and Q. L. Shi and X. L. Shi [7]. Corollary 1 improves the results proved in B. I. Golubov [2] and G. Kvernadze [3].

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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  

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  

A theorem of Ferenc Lukács determines the jumps of a periodic Lebesgue integrable function f at each point of discontinuity of the first kind in terms of the partial sums of the conjugate Fourier series of f. The aim of this note is to prove analogous theorems for functions and series, introduced by Taberski ([10], [11]).

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Abstract

We prove that the conjugate convolution operators can be used to calculate jumps for functions. Our results generalize the theorems established by He and Shi. Furthermore, by using Lukács and Móricz's idea, we solve an open question posed by Shi and Hu.

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The present note will present a different and direct way to generalize the convexity while keep the classical results for L 1-convergence still alive.

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Abstract  

We generalize some old and new results on the determination of jumps of a periodic, Lebesgue integrable function f at each point of discontinuity of first kind in terms of the partial sums of the conjugate series to the Fourier series of f in [1], and in terms of the Abel-Poisson means in [2], to some more general linear operators which satisfy some certain conditions. The linear operators in discussion include the Fejér means, de la Vallée-Poussin means, and Bernstein-Rogosinski sums.

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Abstract

A theorem of Ferenc Lukács [4] states that the partial sums of conjugate Fourier series of periodic Lebesgue integrable functions f diverge at logarithmic rate at the points of discontinuity of first kind of f. F. Móricz [5] proved an analogous theorem for the rectangular partial sums of bivariate functions. The present paper proves analogues of Móricz's theorem for generalized Cesàro means and for positive linear means.

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