On Markushevich bases $\{x^{\lambda_n}\}_{n=1}^{\infty}$ for their closed span in weighted $L^2 (A)$ spaces over sets $A\subset [0,\infty)$ of positive Lebesgue measure, hereditary completeness, and moment problems

Abstract

Inspired by the work of Borwein and Erdelyi \cite{BE1997JAMS} on generalizations of M\"{u}ntz's theorem, we investigate the properties of the system $\{x^{\lambda_n}\}_{n=1}^{\infty}$ in weighted $L^p (A)$ spaces, for $p\ge 1$, denoted by $L^p_w (A)$, where (I) $A$ is a measurable subset of the real half-line $[0,\infty)$ having positive Lebesgue measure, (II) $w$ is a non-negative integrable function defined on $A$, and (III) $\{\lambda_n\}_{n=1}^{\infty}$ is a strictly increasing sequence of positive real numbers such that $\inf\{\lambda_{n+1}-\lambda_n \}>0$ and $\sum_{n=1}^{\infty}\lambda_n^{-1}<\infty$. We prove that a function $f$ in $\overline{\text{span}}\{x^{\lambda_n}\}_{n=1}^{\infty}$ in the Hilbert space $L^2_w (A)$, admits the $\bf{Fourier-type}$ series representation $f(x)=\sum_{n=1}^{\infty} \langle f, r_n\rangle_{w,A} x^{\lambda_n}$ a.e on $A$, where $\{r_n\}_{n=1}^{\infty}$ is the unique biorthogonal family of $\{x^{\lambda_n}\}_{n=1}^{\infty}$ in $\overline{\text{span}}\{x^{\lambda_n}\}_{n=1}^{\infty}$ in $L^2_w (A)$. As a result, we show that the system $\{x^{\lambda_n}\}_{n=1}^{\infty}$ is a $\bf{Markushevich\,\, basis}$ for $\overline{\text{span}}\{x^{\lambda_n}\}_{n=1}^{\infty}$ in $L^2_w (A)$. Furthermore, we consider a $\bf{moment\,\, problem}$. Finally, if $m\le w(x)\le M$ on $A$ for some positive numbers $m$ and $M$ and the set $A$ contains an interval $[a, r_A]$, where $a\ge 0$ and $r_A$ is the essential supremum of $A$, we prove that the system $\{x^{\lambda_n}\}_{n=1}^{\infty}$ is $\bf{hereditarily\,\, complete}$ in $\overline{\text{span}}\{x^{\lambda_n}\}_{n=1}^{\infty}$ in the space $L^2_w(A)$. As a result, a general class of compact operators on the closure is constructed that admit spectral synthesis.