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On linear summation methods for Jacobi series in the case of half integer α

Analysis Mathematica
Volume/Issue:
Volume 22: Issue 1
DOI:
https://doi.org/10.1007/bf02342337
Author: с. кАльНЕИ

Abstract  

An upper estimate is proved for the Lebesgue function with respect to Jacobi polynomials

\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} $$P_m^{(\alpha ,\beta )} (x)$$ \end{document}
in the case of half integerα and it is expressed in terms of the matrix coefficients determining the linear summation method. The author also proves the analogue of the well-known theorem by S. M. Nikol'skii on the necessary and sufficient condition for the summability of trigonometric Fourier series.

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On the divergence of linear summation methods of Fourier series

Analysis Mathematica
Volume/Issue:
Volume 17: Issue 2
DOI:
https://doi.org/10.1007/bf01906602
Author: M. A. Skopina

Пусть функция ϕ задан а на [0, 1],f∈L(0,2π),

\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} $$\sigma _{\varphi , m} (f, x) = \sum\limits_{k = - n}^n {\varphi \left( {\frac{{\left| k \right|}}{n}} \right) } \hat f_k e^{ikx} ,$$ \end{document}
гдеf k — коэффициенты Ф урье функцииf. Получе ны условия на ϕ, при котор ых существует такая функцияf∈L(0, 2π), чт о последовательност ь {σϕ, n(f, x)} расходится для почти всехx∈(0,2π).

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On necessary and sufficient conditions of the regularity of summation methods for Laguerre-Fourier series

Analysis Mathematica
Volume/Issue:
Volume 32: Issue 4
DOI:
https://doi.org/10.1007/s10476-006-0013-x
Author: M. D. Burmistrova

Summary  

The problem of convergence of linear means is considered for the Laguerre-Fourier  series of continuous functions. An upper estimate is obtained for the Laguerre-Lebesgue function  in terms  of the entries of the matrix which determines the linear summability method  in question.  This allows us to prove for such series an analogue of the well-known theorem by S. M. Nikol'skii which provides necessary and sufficient conditions for the summability  of trigonometric Fourier series. A theorem on the regularity of the summability methods  is also  established.

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On summability of weighted Lagrange interpolation. I

Acta Mathematica Hungarica
Volume/Issue:
Volume 101: Issue 4
DOI:
https://doi.org/10.1023/b:amhu.0000004943.68968.d2
Authors: László Szili and Péter Vértesi
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On the universal A.S. central limit theorem

Acta Mathematica Hungarica
Volume/Issue:
Volume 116: Issue 4
DOI:
https://doi.org/10.1007/s10474-007-6070-1
Author: S. Hörmann

Abstract  

Let (X k) be a sequence of independent r.v.’s such that for some measurable functions gk : R k → R a weak limit theorem of the form

\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} $$g_k (X_1 , \ldots ,X_k )\xrightarrow{\mathcal{L}}G$$ \end{document}
holds with some distribution function G. By a general result of Berkes and Csáki (“universal ASCLT”), under mild technical conditions the strong analogue
\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} $$\frac{1} {{D_N }}\sum\limits_{k = 1}^N {d_k I\left\{ {g_k (X_1 , \ldots ,X_k ) \leqq x} \right\} \to G(x)} a.s.$$ \end{document}
 is also valid, where (d k) is a logarithmic weight sequence and D N = ∑k=1 N d k. In this paper we extend the last result for a very large class of weight sequences (d k), leading to considerably sharper results. We show that logarithmic weights, used traditionally in a.s. central limit theory, are far from optimal and the theory remains valid with averaging procedures much closer to, in some cases even identical with, ordinary averages.

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On summability of weighted Lagrange interpolation. II

Acta Mathematica Hungarica
Volume/Issue:
Volume 103: Issue 1-2
DOI:
https://doi.org/10.1023/b:amhu.0000028233.68543.40
Authors: László Szili and Péter Vértesi

Abstract  

The aim of this paper is to continue our investigations started in [15], 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 [15] 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 [15] hold in this case. The order of convergence will also be considered.

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