The Multivariate Müntz-Szasz Problem in Weighted Banach Space on Rn

Abstract

The purpose of this paper is to give an extension of Müntz-Szasz theorems to multivariable weighted Banach space. Denote by $\{{\lambda }_k=({\lambda }_k^1,{\lambda }_k^2,...,{\lambda }_k^n){\}}_{k=1}^{\mathrm{\infty }}$ a sequence of real numbers in ${\mathbb{R}}_{+}^n$. The completeness of monomials $\{t^{{\lambda }_k}\}$ in $C_{\alpha }$ is investigated, where $C_{\alpha }$ is the weighted Banach spaces which consist of complex continuous functions $f$ defined on ${\mathbb{R}}^n$ with $f(t)$ exp$(-\alpha (t))$ vanishing at infinity in the uniform norm.