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A generalization of an inequality of V. Markov to multivariate polynomials. II. (English) Zbl 0531.41012
The paper is concerned with multivariate generalizations of the classical Markov’s inequalities for polynomial coefficients [see e.g. 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 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)].

MSC:
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A10 Approximation by polynomials
26C05 Real polynomials: analytic properties, etc.
41A63 Multidimensional problems
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