Müntz-Szász type theorems for the density of the span of powers of functions

Abstract

The aim of this paper is to establish density properties in Lp spaces of the span of powers of functions {ψλ:λ∈Λ}, Λ⊂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 {ψλ,ψλeiαt:λ∈Λ}. Finally, we establish a Müntz-Szász Theorem for density of translates of powers of cosines {cosλ⁡(t−θ1),cosλ⁡(t−θ2):λ∈Λ}. Under some arithmetic restrictions on θ1−θ2, 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.