On a generalized moment problem

Abstract:

The well-known Müntz-Szász theorem asserts that the sequence of powers ${x^{{n_p}}}$ is complete on $[a,b]$, where $a \geqslant 0$, if and only if (1) \[ (1)\quad \sum \limits _{p = 1}^\infty {\frac {1} {{{n_p}}} = \infty ,\quad {\text {where}}\;0 < {n_1} < {n_2} < \cdots .} \] Let $f(x)$ be absolutely continuous, $\left | {f’(x)} \right | \geqslant k > 0$, and $f(a)f(b) \geqslant 0$. We prove that under the assumption (1) the sequence $\left \{ {f{{(x)}^{{n_p}}}} \right \}$ is complete on $[a,b]$ if and only if $f(x)$ is monotone on $[a,b]$.