On a generalized moment problem. II

Abstract:

Recently, we have extended the well-known Müntz-Szász theorem by showing that if $f(z)$ is absolutely continuous and $|f’(x)| \geqslant k > 0$ a.e. on $(a,b)$, where $a \geqslant 0$ and if $\{ {n_p}\}$ is a sequence of positive numbers tending to infinity and satisfying $\sum _{p = 1}^\infty 1/{n_p} = \infty$, then the sequence $\{ f{(x)^{{n_p}}}\}$ is complete on $(a,b)$ if and only if $f(x)$ is strictly monotone on $(a,b)$. We now apply Zarecki’s theorem to improve the condition "$|f’(x)| \geqslant k > 0$ a.e. on $(a,b)$" by the condition $f’(x) \ne 0$ a.e. on $(a,b)$". Furthermore, we extend some well-known theorems of Picone, Mikusiński, and Boas.