# Density of the span of powers of a function à la Müntz-Szasz

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1 IMB - Institut de Mathématiques de Bordeaux 2 Institute of Mathematics and Informatics, University of Pécs

Abstract : The aim of this paper is to establish density properties in $L^p$ spaces of the span of powers of functions $\{\psi^\lambda\,:\lambda\in\Lambda\}$, $\Lambda\subset\N$ in the spirit of the M\-untz-Sz\-asz Theorem. As density is almost never achieved, we further investigate the density of powers and a modulation of powers $\{\psi^\lambda,\psi^\lambda e^{i\alpha t}\,:\lambda\in\Lambda\}$. Finally, we establish a M\-untz-Sz\-asz Theorem for density of translates of powers of cosines $\{\cos^\lambdat-\theta 1,\cos^\lambdat-\theta 2\,:\lambda\in\Lambda\}$. Under some arithmetic restrictions on $\theta 1-\theta 2$, we show that density is equivalent to a M\-untz-Sz\-asz condition on $\Lambda$ and we conjecture that those arithmetic restrictions are not needed.Some links are also established with the recently introduced concept of Heisenberg Uniqueness Pairs.

Autor: Philippe Jaming - Ilona Simon -

Fuente: https://hal.archives-ouvertes.fr/

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