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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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##### References:
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