Cesàro Means of Weighted Orthogonal Expansions on Regular Domains

Feng, Han; Ge, Yan

doi:10.3390/math10122108

Open AccessArticle

Cesàro Means of Weighted Orthogonal Expansions on Regular Domains

by

Han Feng

^†

and

Yan Ge

^*,†

Department of Mathematics, City University of Hong Kong, Hong Kong, China

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Mathematics 2022, 10(12), 2108; https://doi.org/10.3390/math10122108

Submission received: 24 May 2022 / Revised: 15 June 2022 / Accepted: 15 June 2022 / Published: 17 June 2022

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

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Abstract

In this paper, we investigate Cesàro means for the weighted orthogonal polynomial expansions on spheres with weights being invariant under a general finite reflection group on

R^{d}

. Our theorems extend previous results only for specific reflection groups. Precisely, we consider the weight function

h_{κ} (x) : = \prod_{ν \in R +} {| ⟨x, ν⟩ |}^{κ_{ν}}, x \in R^{d}

on the unit sphere; the upper estimates of the Cesàro kernels and Cesàro means are obtained and used to prove the convergence of the Cesàro

(C, δ)

means in the weighted

L^{p}

space for

δ

above the corresponding index. We also establish similar results for the corresponding estimates on the unit ball and the simplex.

Keywords: spherical h-harmonics; Cesàro means; Christoffel functions

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MDPI and ACS Style

Feng, H.; Ge, Y. Cesàro Means of Weighted Orthogonal Expansions on Regular Domains. Mathematics 2022, 10, 2108. https://doi.org/10.3390/math10122108

AMA Style

Feng H, Ge Y. Cesàro Means of Weighted Orthogonal Expansions on Regular Domains. Mathematics. 2022; 10(12):2108. https://doi.org/10.3390/math10122108

Chicago/Turabian Style

Feng, Han, and Yan Ge. 2022. "Cesàro Means of Weighted Orthogonal Expansions on Regular Domains" Mathematics 10, no. 12: 2108. https://doi.org/10.3390/math10122108

APA Style

Feng, H., & Ge, Y. (2022). Cesàro Means of Weighted Orthogonal Expansions on Regular Domains. Mathematics, 10(12), 2108. https://doi.org/10.3390/math10122108

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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