Density of the span of powers of a function \`a la M\"untz-Szasz

TL;DR

This paper investigates the density of powers of functions and their modulations in $L^p$ spaces, extending M"untz-Szász Theorem to various function sets and exploring conditions for density related to arithmetic restrictions.

Contribution

It establishes new density results for powers and modulated powers of functions, including a M"untz-Szász type theorem for translates of cosine powers, with connections to Heisenberg Uniqueness Pairs.

Findings

Density of powers is characterized by M"untz-Szász conditions.

Density of modulated powers depends on arithmetic restrictions.

A M"untz-Szász theorem for cosine translates is proved.

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 $\{\cos^\lambda(t-\theta\_1),\cos^\lambda(t-\theta\_2)\,:\lambda\in\Lambda\}$. Under some arithmetic restrictions on $\theta\_1-\theta\_2$, we show that density is equivalent to a 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.