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The aim of this paper is to establish density properties in $L^p$ spaces ofthe 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 isalmost never achieved, we further investigate the density of powers and amodulation of powers $\{\psi^\lambda,\psi^\lambda e^{i\alphat}\,:\lambda\in\Lambda\}$. Finally, we establish a M\"untz-Sz\'asz Theorem fordensity 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 thatdensity is equivalent to a M\"untz-Sz\'asz condition on $\Lambda$ and weconjecture that those arithmetic restrictions are not needed.Some links arealso established with the recently introduced concept of Heisenberg UniquenessPairs.

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