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Clear All  # Strong approximation by Fourier--Laplace series on the unit sphere S n-1

Acta Mathematica Hungarica
Authors: Gavin Brown, Feng Dai, and Ferenc Móricz
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# О линейных методах суммирования рядов Фурье-Лапласа. II

Analysis Mathematica
Author: Tang Ping

## Abstract

Let Σn be the unit sphere inR n for somen≥3 with centre at the origin, L(Σn) the space of all functions integrable on Σn. We prove a theorem on the representation of functions by singular integrals at double Lebesgue points, which is analogous to a theorem by D. K. Faddeev in the one-dimensional case. On the basis of this theorem, we give necessary and sufficient conditions for the fulfillment of the relation

\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} $$\mathop {\lim }\limits_{x \to \infty } U_N (f,x,\Lambda ) = f(x)$$ \end{document}
for an arbitrary integrable functionf at its double Lebesgue pointsx, where byU N (f, x Λ) we denote the linear means of the Fourier-Laplace series off defined by means of the triangular matrix
\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} $$\Lambda = \left\{ {\lambda _k^{(N)} :N = 0,1,...;k = 0,1...,N + 1;\lambda _k^{(N)} = 1,\lambda _{N + 1}^{(N)} = 0} \right\}$$ \end{document}

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