Nikol'skii inequality for algebraic polynomials on a multidimensional Euclidean sphere
type: Item
instance of: scholarly article
title: Nikol'skii inequality for algebraic polynomials on a multidimensional Euclidean sphere
zbMATH Open document ID: 1315.41003
DOI: 10.1134/S0081543814020023
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2014-12-17T00:00:00Z
review text: Let \(\mathcal P_{n,m}\) be the set of algebraic polynomials of degree at most \(n\) in \(m\) variables with real coefficients. In this contribution the authors analyze the following Nikol'skii type inequality \[ || P_n||_{C(Sn-1 ) } \leq C(n, m)_q ||P_n||_{L_q (\mathbb S^{n-1} )}, \quad P_n \in \mathcal P_{n,m}, \] with the least constant \(C(n, m)_q\) between the uniform norm \[ ||P_n||_ {C(\mathbb S^{n-1} )} = \max\{|P_n (x)| : x \in \mathbb S^{n-1} \} \] and the \(L_q\) norm \[ ||P_n||_ {L_q(\mathbb S^{n-1} )} = \left(\int_{\mathbb S^{n-1}} |P_n (x)|^q\right)^{1/q}, 1 \leq < \infty, \] of polynomials of a given degree \(n \geq 0\) on the unit sphere. The authors show that the above multidimensional inequality can be reduced to two extremal problems for algebraic polynomials in one variable (see Theorem 2). Also, the authors discuss the uniqueness of an extremal polynomial associated with one of these extremal problems for algebraic polynomials in one variable (cf. Theorem 1).
reviewed by: Yamilet Quintana
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 6381060
zbMATH Keywords: multidimensional Euclidean sphere
zbMATH Keywords: algebraic polynomials
zbMATH Keywords: Nikol'skii inequality
zbMATH Keywords: polynomials that deviate least from zero
author: Vitalii Arestov
author: Marina Valer'evna Deikalova
MaRDI profile type: Publication
cites work: On the best approximation of Dirichlet kernels
cites work: Q4429483
cites work: Q3254471
cites work: Q4861743
cites work: Q4348446
cites work: Direct and converse theorems of the theory of approximation in the metric of \(L_p\) for \(0<p<1\)
cites work: Some extremal properties of polynomials and inverse inequalities of approximation theory
cites work: Inequality of different metrics for trigonometric polynomials
cites work: ON INTEGRAL INEQUALITIES FOR TRIGONOMETRIC POLYNOMIALS AND THEIR DERIVATIVES
cites work: Integral inequalities for algebraic and trigonometric polynomials
cites work: Sharp integral inequalities for fractional derivatives of trigonometric polynomials
cites work: Sharp inequalities for trigonometric polynomials with respect to integral functionals
cites work: The taikov functional in the space of algebraic polynomials on the multidimensional Euclidean sphere
cites work: On the sharp Jackson-Nikol'skii inequality for algebraic polynomials on a multidimensional Euclidean sphere
cites work: Q5645344
cites work: Q4779788
cites work: The Markov brothers inequality in \(L_0\)-space on an interval
cites work: The sharp Markov-Nikol'skii inequality for algebraic polynomials in the spaces \(L q\) and \(L _{0}\) on a closed interval
cites work: Sharp Markov brothers type inequality in the spaces \(L_{p}\) and \(L_{1}\) on a closed interval
cites work: Complexity of approximation problems
description: scientific article; zbMATH DE number 6381060