A geometric approach to the multivariate Müntz problem

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.