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# On the convergence of double Fourier integrals of functions of bounded variation on ℝ2

Studia Scientiarum Mathematicarum Hungarica
Authors: Bhikha Lila Ghodadra and Vanda Fülöp

We investigate the pointwise and uniform convergence of the symmetric rectangular partial (also called Dirichlet) integrals of the double Fourier integral of a function that is Lebesgue integrable and of bounded variation over ℝ2. Our theorem is a two-dimensional extension of a theorem of Móricz (see Theorem 3 in ) concerning the single Fourier integrals, which is more general than the two-dimensional extension given by Móricz himself (see Theorem 3 in ).

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# A note on the uniform convergence and boundedness of a new class of sine series

Analysis Mathematica
Author: László Leindler
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# Piecewise monotonie and rational approximations and uniform convergence of Fourier series

Analysis Mathematica
Author: Е. Севастьянов

## Abstract

Let

\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$R_n [f] = \inf \left\{ {\mathop {\max }\limits_{ - \pi \leqq x \leqq \pi } \left| {f(x) - \frac{{P(x)}}{{Q(x)}}} \right|} \right\}$$ \end{document}
, whereP andQ denote polynomials (algebraic or trigonometric) of degree ≦n. Theorem 2a. If for a continuous 2π-periodic function f 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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{n = 1}^\infty {\frac{1}{n}R_n [f]< \infty }$$ \end{document}
holds, then the Fourier series of f converges to f(x) uniformly. Theorem 2b. Let {β n} be a non-increasing sequence of positive numbers such that
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{n = 1}^\infty {\frac{1}{n}\beta _n = \infty }$$ \end{document}
Then there exists a continuous 2π-periodic function f0 for which Rn[f0]≦βn for all n=1 and yet the Fourier series of f0 diverges at x=0.R n [f]may be replaced in these theorems byM n [f], whereM n [f] is the minimal uniform deviation off(x) from piecewise monotonie functionsМ n (х) of order ≦n.

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# Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with Non-Zero Angles

Acta Mathematica Hungarica
Author: F.G. Abdullayev
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# Local uniform convergence of the eigenfunction expansion associated with the Laplace operator. II

Acta Mathematica Hungarica
Author: M. Horváth
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# Local uniform convergence of the eigenfunction expansion associated with the Laplace operator. I

Acta Mathematica Hungarica
Author: M. Horváth
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# On the uniform convergence of trigonometric interpolating polynomials to functions in the classC T * (ω)

Acta Mathematica Hungarica
Author: J. Klippert
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# On the Uniform Convergence and Boundedness of a Certain Class of Sine Series

Analysis Mathematica
Author: L. Leindler
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# On summability of weighted Lagrange interpolation. II

Acta Mathematica Hungarica
Authors: László Szili and Péter Vértesi

## Abstract

The aim of this paper is to continue our investigations started in , where we studied the summability of weighted Lagrange interpolation on the roots of orthogonal polynomials with respect to a weight function w. Starting from the Lagrange interpolation polynomials we constructed a wide class of discrete processes which are uniformly convergent in a suitable Banach space (C ρ, ‖‖ρ) of continuous functions (ρ denotes (another) weight). In  we formulated several conditions with respect to w, ρ, (C ρ, ‖‖ρ) and to summation methods for which the uniform convergence holds. The goal of this part is to study the special case when w and ρ are Freud-type weights. We shall show that the conditions of results of  hold in this case. The order of convergence will also be considered.

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# Remarks on the uniform and L 1-convergence of trigonometric series

Acta Mathematica Hungarica
Author: P. Kórus

## Abstract

Chaundry and Jolliffe  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  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  on the L 1-convergence of Fourier series.

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