A generalization of an inequality of V. Markov to multivariate polynomials. II
type: Item
instance of: scholarly article
title: A generalization of an inequality of V. Markov to multivariate polynomials. II
zbMATH Open document ID: 0531.41012
DOI: 10.1016/0021-9045(84)90022-4
author: Heinz-Joachim Rack
published in: Journal of Approximation Theory
publication date: 1984-01-01T00:00:00Z
review text: The paper is concerned with multivariate generalizations of the classical Markov's inequalities for polynomial coefficients [see e.g. \textit{I. Natanson}, ''Constructive Function Theory, Vol. I'' (1964; Zbl 0133.311), p. 56]. Let \(P_{m,r}(x)=\sum_{| k| \leq m}b_ kx^ k\) (where \(b_ k\in {\mathbb{R}}\), \(k\in {\mathbb{N}}^ r_ 0\), \(x^ k=x_ 1^{k_ 1}...x_ 1^{k_ r}\), \(| k| =k_ 1+...+k_ r)\) denote a polynomial in r variables of total degree \(\leq m\). Estimates for the leading coefficients \(b_ k\) with \(| k| =m\) were given by \textit{M. Reimer} in terms of Chebyshev polynomials \(T_ n\), \(n\in {\mathbb{N}}\) [ibid. 23, 65-69 (1978; Zbl 0386.41020)]. Here the following Theorem is proved: Let \(\| P_{m,r}\| \leq 1\) (uniform norm on the unit cube); let k with \(| k| =m-1\in {\mathbb{N}}\) be arbitrary but fixed and denote by \(\bar r\) the number of nonvanishing components of k. Then \(b_ k\) satisfies the estimate \(| b_ k| \leq 2^{m-\bar r-1}\) with equality if \(P_{m,r}(x)=\prod^{r}_{q=1}T_{k_ q}(x_ q)\). It is shown by example that neither products nor any rational functions of coefficients of the \(T_ n's\) are enough to majorize the \(b_ k's\) if \(| k| \leq m-2\). The paper is related to an earlier note of the author [ibid. 35, 94-97 (1982; Zbl 0487.41008)].
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 26C05
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 3842432
zbMATH Keywords: multivariate polynomials
zbMATH Keywords: Markov's inequalities
zbMATH Keywords: polynomial coefficients
MaRDI profile type: Publication
full work available at URL: 0021-9045(84)90022-4
OpenAlex ID: W4254410793
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description: scientific article; zbMATH DE number 3842432