Jacobi polynomial estimates and Fourier-Laplace convergence
type: Item
instance of: scholarly article
title: Jacobi polynomial estimates and Fourier-Laplace convergence
zbMATH Open document ID: 0909.42017
DOI: 10.1007/BF02648263
author: Gavin Brown
author: Wang Kunyang
published in: The Journal of Fourier Analysis and Applications
publication date: 1999-03-22T00:00:00Z
full work available at URL: https://eudml.org/doc/59532
review text: The second named author [J. Beijing Norm. Univ., Nat. Sci. 29, No. 2, 143-154 (1993; Zbl 0799.43002)] reduced the convergence problems for Fourier-Laplace series on the sphere to the consideration of a certain equiconvergence operator defined by convolution with certain Jacobi polynomials. In the present paper, the authors derive new estimates for the Jacobi polynomials \(P^{(\alpha,\beta)}_k(x)\) with complex indices \(\alpha\), \(\beta\), where \(\text{Re }\alpha> -1\), \(\text{Re }\beta> -1\). By making use of Stein's interpolation theorem, they obtain \(L^p\) estimates for the corresponding maximal equiconvergence operator. As a consequence, they deduce the following remarkable Theorem 2: If \(n\geq 3\) and \[ \int_{\Omega_n}| f(x)|(1+ \log^2_+| f(x)|)dx< \infty, \] where \[ \Omega_n:= \{(x_1,\dots, x_n)\in \mathbb{R}^n: x^2_1+\cdots+ x^2_n= 1\}, \] then the Cesàro means \(S_N(f)\) at critical index of the Fourier-Laplace series of \(f\) converge almost everywhere.
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 42B08
zbMATH DE Number: 1143801
zbMATH Keywords: convergence
zbMATH Keywords: Fourier-Laplace series
zbMATH Keywords: Jacobi polynomials
zbMATH Keywords: equiconvergence operator
zbMATH Keywords: Cesàro means
reviewed by: Ferenc Móricz
MaRDI profile type: Publication
cites work: Q5640160
cites work: Q4052481
cites work: Q5820721
cites work: Q4278800
cites work: Spherical harmonics
cites work: Sommes De Cesaro Et Multiplicateurs Des Developpements en Harmoniques Spheriques
cites work: Localization and summability of multiple Fourier series
description: scientific article; zbMATH DE number 1143801