On Approximation Methods Generated by Bochner-Riesz Kernels

Abstract

Means and families of operators generated by Bochner-Riesz kernels are studied. Some sharp results on their convergence are achieved. The equivalence of the approximation errors of these methods to smoothness quantities related to the Laplacian is proved.Keywords Bochner-Riesz kernels and means · Families of operators · Necessary and sufficient conditions of convergence · K-functional · Realizations and moduli of smoothness related to the Laplacian Mathematics Subject Classification (2000) 42A10 · 42A15

Moduli of Smoothness Related to the Laplace-Operator

“…Theorem 6.4 extends the result of Theorem 5 in [12] which corresponds to the case m = 1 and which is restricted to p > d/(d + 2). Note that p m,d → 0 if m → ∞.…”

Approximation by families of linear trigonometric polynomial operators and smoothness properties of functions

“…We call the corresponding family Bochner-Riesz family and we denote it by B (α ) n ; λ . It is known that (see, e.g., [15,Theorem 9]) Fϕ α belongs to L p R d if and only if p > 2d/(d + 2α + 1). In particular, Fϕ α belongs to L 1 R d if and only if α > (d − 1)/2.…”

“…We give the details for the sake of completeness and for the convenience of the reader. However, note also that the scheme described in the previous sections in order to study the approximation by families of linear trigonometric polynomial operators is mainly based on the ideas we elaborated in [15]. ψ(·) is valid for any 0 < q ≤ +∞ and for any infinitely differentiable function η with support contained in {| ξ | < 1}.…”

“…The family of linear trigonometric polynomial operators (1.1) has been introduced in [10], [11]. Later on this method of approximation has been further developed in [1], [7], [9], [15] and other papers. For its applications, in particular, for an algorithm of stochastic approximation, we refer to [12].…”

“…For example, the equivalence of the approximation error for the family of generalized sampling series generated by Fejér kernels and the realization functional associated with the Riesz derivative in the non-periodic case has been proved in [2], Theorem 5.2. For results on the approximation by families of linear trigonometric polynomial operators generated by Bochner-Riesz kernels we refer to [15] (see also the comments in Subsection 7.3). In this respect we also want to emphasize that the main results of this paper will enable us to study families of type (1.1) generated by classical kernels and their approximation properties from a unique point of view within a general scheme.…”