Clarkson-Erdős theorem for many variables (Q1569763)

\textit{J. A. Clarkson} and \textit{P. Erdős} [Duke Math. J. 10, 5-11 (1943; Zbl 0063.00919)] have proved: Let \(\lambda_n\in \mathbb{N}\) with \(\sum^\infty_{n =1} \lambda_n^{-1} <\infty\) be given. Then the closed span of the \(x^{ \lambda_n }\) in \(C[0,1]\) is the set of those functions which possess an analytic extension onto the open unit disk. This result complements the known Müntz theorem [cf. \textit{R. A. DeVore} and \textit{C. G. Lorentz}, Constructive Approximation (1993; Zbl 0797.41016)]. In this paper, the author generalizes the Clarkson-Erdős theorem to multivariate polynomials.