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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the density of multivariate polynomials with varying weights
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by András Kroó and József Szabados
Proc. Amer. Math. Soc. 151 (2023), 1921-1935
DOI: https://doi.org/10.1090/proc/14882
Published electronically: February 2, 2023

Abstract:

In this paper we consider multivariate approximation by weighted polynomials of the form $w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x})$, where $p_n$ is a multivariate polynomial of degree at most $n$, $w$ is a given nonnegative weight with nonempty zero set, and $\gamma _n\uparrow \infty$. We study the question if every continuous function vanishing on the zero set of $w$ is a uniform limit of weighted polynomials $w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x})$. It turns out that for various classes of weights in order for this approximation property to hold it is necessary and sufficient that $\gamma _n=o(n).$
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Bibliographic Information
  • András Kroó
  • Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, 1053 Hungary; and Budapest University of Technology and Economics, Department of Analysis, Budapest, 1111 Hungary
  • Email: kroo.andras@renyi.mta.hu
  • József Szabados
  • Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, 1053 Hungary
  • Email: szabados.jozsef@renyi.mta.hu
  • Received by editor(s): March 21, 2019
  • Received by editor(s) in revised form: September 19, 2019
  • Published electronically: February 2, 2023
  • Additional Notes: The research of both authors was supported by NKFI Grant No. K128922.
  • Communicated by: Yuan Xu
  • © Copyright 2023 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 1921-1935
  • MSC (2010): Primary 41A10, 41A63
  • DOI: https://doi.org/10.1090/proc/14882
  • MathSciNet review: 4556189