Cesàro Means of Weighted Orthogonal Expansions on Regular Domains

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 Rd. Our theorems extend previous results only for specific reflection groups. Precisely, we consider the weight function hκ(x):=∏ν∈R+|x,ν|κν,x∈Rd 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 Lp space for δ above the corresponding index. We also establish similar results for the corresponding estimates on the unit ball and the simplex.

Fourier Series Approximation in Besov Spaces

“…If T n (f; x) converges to s(x), then the series in ( 1) is said to be T-summable to s(x). For diferent orthogonal expansions, we get a number of special cases for T. Te most famous of all is the Cesáro matrix [2,3].…”

“…where p n 􏼈 􏼉 ∞ n�0 is a sequence of real numbers such that P n : � 􏽐 n k�0 p k and lim n→∞ P n � ∞ [4]. Similarly, there are many other summability techniques such as Nörlund summability, Abel summability, de La Vallée-Poussin summation, and Hausdorf summability [2][3][4][5][6].…”

Editorial for the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”

“…In the paper [23], Cesáro means are investigated for the weighted orthogonal polynomial expansions on spheres with weights being invariant under a general finite reflection group on R d . The theorems in the paper extend previous results only for specific reflection groups and weight functions on the unit sphere.…”