Approximation of functions on the sphere
type: Item
instance of: scholarly article
title: Approximation of functions on the sphere
zbMATH Open document ID: 0728.41021
author: V. M. Fedorov
published in: Moscow University Mathematics Bulletin
publication date: 1990-01-01T00:00:00Z
review text: Let f be a function defined on the unit sphere \(S_ m\) of \(R^{m+1}\) such that \(f\in M_ p(S_ m)\). Using the finite difference of order r, the author considers the modulus of smoothness of f with respect to the \(L_ p\) norm, denoted by \(\omega_ r(f,\delta)_{m,p}\). The main result of the paper is a Jackson type theorem for the approximation of f by multi-polynomials of order at most N. If \(E_ N(f)_{m,p}\) is the best approximation of f in the mentioned circumstances, then it is proved that: \[ E_ N(f)_{m,p}\leq M_ r\omega_ r(f,1/d^{1/m})_{m,p}, \] where \(d=(2N+m)\left( \begin{matrix} N+m\\ m\end{matrix} \right)/(N+m)\), \(M_ r\) is a positive constant and m is an odd number.
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 4202987
zbMATH Keywords: modulus of smoothness
zbMATH Keywords: Jackson type theorem
reviewed by: Ioan Şerb
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4202987