  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.

Classical Analysis and ODEs;Functional Analysis