Skip to Main Content

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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

A geometric approach to the multivariate Müntz problem
HTML articles powered by AMS MathViewer

by András Kroó PDF
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.
References
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 41A30, 41A63
  • Retrieve articles in all journals with MSC: 41A30, 41A63
Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 121 (1994), 199-208
  • MSC: Primary 41A30; Secondary 41A63
  • DOI: https://doi.org/10.1090/S0002-9939-1994-1181170-4
  • MathSciNet review: 1181170