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

The aim of this paper is to establish density properties in $L^p$ spaces of the span of powers of functions $\{ψ^λ\,:λ\inΛ\}$, $Λ\subset\N$ 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 $\{ψ^λ,ψ^λe^{iαt}\,:λ\inΛ\}$. Finally, we establish a Müntz-Szász Theorem for density of translates of powers of cosines $\{\cos^λ(t-θ\_1),\cos^λ(t-θ\_2)\,:λ\inΛ\}$. Under some arithmetic restrictions on $θ\_1-θ\_2$, 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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