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# Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps

Acta Mathematica Hungarica
Authors: D. Holý and P. Vadovič

## Abstract

For metric spaces (X, d x) and (Y, d y) we consider the Hausdorff metric topology
\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} $$\tau _{H_\rho }$$ \end{document}
on the set (CL(XY), ρ) of closed subsets of the product metrized by the product (box) metric ρ and consider the proximal topology
\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} $$\tau _{\delta _\rho }$$ \end{document}
defined on CL(XY). These topologies are inherited by the set G(X, Y) of closed-graph multifunctions from X to Y, if we identify each multifunction with its graph. Finally, we consider the topology of uniform convergence τ uc on the set F(X, 2Y) of all closed-valued multifunctions, i.e. functions from X to the set (CL(Y),
\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} $$H_{d_y }$$ \end{document}
) of closed subsets of Y metrized by the Hausdorff metric
\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} $$H_{d_y }$$ \end{document}
. We show the relationship between these topologies on the space G(X, Y) and also on the subspaces of minimal USCO maps and locally bounded densely continuous forms.
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# Numerical solution of linear differential equations by Walsh polynomials approach

Studia Scientiarum Mathematicarum Hungarica
Authors: György Gát and Rodolfo Toledo

coefficient and free term on the close interval [0 , 1]. The second example shows us the uniform convergence of the numerical solution in case of integrable coefficient and free term which are only continuous on the interval [0 , 1[. The third example

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# On the order of magnitude of the uniform convergence of multiple trigonometric Fourier series with respect to cubes on the function classes \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} $$H^l_{p,m} [\omega]$$ \end{document}

Analysis Mathematica
Author: N. Ilyasov
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# An Erdős type convergence process in weighted interpolation. II

Acta Mathematica Hungarica
Authors: L Szili and P. Vértesi

## Abstract

The aim of this paper is to continue the investigation of the second author started in , where a weighted version of a classical result of P. Erdős was proved using Freud type weights. We shall show that an analogous statement is true for weighted interpolation if we consider exponential weights on [-1,1].

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# On summability of weighted Lagrange interpolation. I

Acta Mathematica Hungarica
Authors: László Szili and Péter Vértesi
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# On the summability of weighted Lagrange interpolation on the roots of Jacobi polynomials

Acta Mathematica Hungarica
Author: L. Szili
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# On summability of weighted Lagrange interpolation. III

Acta Mathematica Hungarica
Authors: László Szili and Péter Vértesi

## Abstract

Starting from the Lagrange interpolation on the roots of Jacobi polynomials, a wide class of discrete linear processes is constructed using summations. Some special cases are also considered, such as the Fejr, de la Valle Poussin, Cesro, Riesz and Rogosinski summations. The aim of this note is to show that the sequences of this type of polynomials are uniformly convergent on the whole interval [-1,1] in suitable weighted spaces of continuous functions. Order of convergence will also be investigated. Some statements of this paper can be obtained as corollaries of our general results proved in .

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# Weighted approximation by Meyer-König and Zeller type operators

Studia Scientiarum Mathematicarum Hungarica
Authors: Biancamaria Vecchia and Oktay Duman

In this study we deal with the weighted uniform convergence of the Meyer-König and Zeller type operators with endpoint or inner singularities.

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# Spaces of S Bounded Multiplier Convergent Series

Acta Mathematica Hungarica
Authors: A. Aizpuru and J. Pérez-Fernández

## Abstract

We prove that some results on uniform convergence of sequences of unconditionally convergent series, in Banach spaces, can be generalized to sequences of weakly unconditionally Cauchy series.

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# Cesàro summability of lagrange interpolation on Jacobi roots

Studia Scientiarum Mathematicarum Hungarica
Author: L. Szili

The aim of this paper is to give such weighted function spaces in which the sequence of Cesàro means of Lagrange interpolatory polynomials on Jacobi roots are uniformly convergent.

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