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

Abstract

The aim of this paper is to establish density properties in $L^p$ spaces of the span of powers of functions $\{{\psi }^{\lambda }:\lambda \in \Lambda \}$, $\Lambda \subset \mathbb{N}$ 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 $\{{\psi }^{\lambda },{\psi }^{\lambda }{e}^{i\alpha t}:\lambda \in \Lambda \}$. Finally, we establish a Müntz-Szász Theorem for density of translates of powers of cosines $\{{\cos }^{\lambda }(t-\theta \_1),{\cos }^{\lambda }(t-\theta \_2):\lambda \in \Lambda \}$. Under some arithmetic restrictions on $\theta \_1-\theta \_2$, we show that density is equivalent to a Müntz-Szász 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.

References in corpus (1)

• Report on some recent advances in Diophantine approximation