# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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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## A geometric approach to the multivariate Müntz problemHTML articles powered by AMS MathViewer

by András Kroó
Proc. Amer. Math. Soc. 121 (1994), 199-208 Request permission

## Abstract:

For a countable set $\Omega \subset {\mathbb {R}^n}$ denote by $P(\Omega )$ the space of polynomials spanned by ${x^\omega }, \omega \in \Omega (x = ({x_1}, \ldots ,{x_n}) \in {\mathbb {R}^n}, \omega = ({\omega _1}, \ldots ,{\omega _n}) \in \Omega , {x^\omega } = \prod _{i = 1}^nx_i^{{\omega _i}})$. In this paper we investigate the question of the density of $P(\Omega )$ in $C(K)$, the space of real valued continuous functions endowed with the supremum norm on compact set $K \subset {\mathbb {R}^n}$. In case $n = 1$ the classical theorem of Müntz gives an elegant necessary and sufficient condition for density. This problem (closely related to the distribution of zeros of Fourier transforms) is much more complex in the multivariate setting. We shall present an extension of Müntz’ condition to the case $n > 1$ which will suffice for density. This, in particular, will enable us to construct "optimally sparse" lattice point sets $\Omega$ for which density holds.
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