On the density of multivariate polynomials with varying weights

Kroó, András and Szabados, József (2023) On the density of multivariate polynomials with varying weights. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 151. pp. 1921-1935. ISSN 0002-9939

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Abstract

In this paper we consider multivariate approximation by weighted polynomials of the form w γ n ( x ) p n ( x ) w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x}) , where p n p_n is a multivariate polynomial of degree at most n n , w w is a given nonnegative weight with nonempty zero set, and γ n ↑ ∞ \gamma _n\uparrow \infty . We study the question if every continuous function vanishing on the zero set of w w is a uniform limit of weighted polynomials w γ n ( x ) p n ( x ) w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x}) . It turns out that for various classes of weights in order for this approximation property to hold it is necessary and sufficient that γ n = o ( n ) . \gamma _n=o(n).

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