Bulletin des Sciences Mathématiques ( IF 1.241 ) Pub Date : 2020-11-12 , DOI: 10.1016/j.bulsci.2020.102933
Philippe Jaming; Ilona Simon

The aim of this paper is to establish density properties in ${L}^{p}$ spaces of the span of powers of functions $\left\{{\psi }^{\lambda }\phantom{\rule{0.2em}{0ex}}:\lambda \in \mathrm{\Lambda }\right\}$, $\mathrm{\Lambda }\subset \mathbb{N}$ in the spirit of the Müntz-Szász Theorem. As density is almost never achieved, we further investigate the density of powers and a modulation of powers $\left\{{\psi }^{\lambda },{\psi }^{\lambda }{e}^{i\alpha t}\phantom{\rule{0.2em}{0ex}}:\lambda \in \mathrm{\Lambda }\right\}$. Finally, we establish a Müntz-Szász Theorem for density of translates of powers of cosines $\left\{{\mathrm{cos}}^{\lambda }\left(t-{\theta }_{1}\right),{\mathrm{cos}}^{\lambda }\left(t-{\theta }_{2}\right)\phantom{\rule{0.2em}{0ex}}:\lambda \in \mathrm{\Lambda }\right\}$. Under some arithmetic restrictions on ${\theta }_{1}-{\theta }_{2}$, we show that density is equivalent to a Müntz-Szász condition on Λ and we conjecture that those arithmetic restrictions are not needed. Some links are also established with the recently introduced concept of Heisenberg Uniqueness Pairs.

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