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 Müntz'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 $L_w^p(A)$, 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 $L_w^2(A)$, admits the $\mathbf{Fourier}-\mathbf{type}$ series representation $f(x)={\sum }_{n=1}^{\infty }⟨f,r_n{⟩}_{w,A}{x}^{{\lambda }_n}$ 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 $L_w^2(A)$. As a result, we show that the system $\{x^{{\lambda }_n}{\}}_{n=1}^{\infty }$ is a $\mathbf{Markushevich}\mathbf{basis}$ for $\overline{\text{span}}\{x^{{\lambda }_n}{\}}_{n=1}^{\infty }$ in $L_w^2(A)$. Furthermore, we consider a $\mathbf{moment}\mathbf{problem}$. 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 $\mathbf{hereditarily}\mathbf{complete}$ in $\overline{\text{span}}\{x^{{\lambda }_n}{\}}_{n=1}^{\infty }$ in the space $L_w^2(A)$. As a result, a general class of compact operators on the closure is constructed that admit spectral synthesis.

21 pages