关于正Lebesgue测度集$A\subset [0,\infty)$上加权$L^2(A)$空间中其闭张成空间的Markushevich基$\{x^{\lambda_n}\}_{n=1}^{\infty}$、遗传完备性及矩问题
On Markushevich bases $\{x^{λ_n}\}_{n=1}^{\infty}$ 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
AI总结：
本文研究加权$L^2_w(A)$空间中幂函数系$\{x^{\lambda_n}\}_{n=1}^{\infty}$的性质，证明其为闭张成空间的Markushevich基，探讨相关矩问题，并在权函数有界且集合含特定区间时证明其遗传完备性，构造可谱综合的紧算子类。
摘要：
受Borwein和Erdelyi \cite{BE1997JAMS}关于Müntz定理推广工作的启发，我们研究了系统$\{x^{\lambda_n}\}_{n=1}^{\infty}$在加权$L^p(A)$空间（记为$L^p_w(A)$，其中$p\ge 1$）中的性质，满足：(I) $A$是实半直线$[0,\infty)$的可测子集，具有正Lebesgue测度；(II) $w$是定义在$A$上的非负可积函数；(III) $\{\lambda_n\}_{n=1}^{\infty}$是严格递增的正实数序列，满足$\inf\{\lambda_{n+1}-\lambda_n\}>0$且$\sum_{n=1}^{\infty}\lambda_n^{-1}<\infty$。我们证明，Hilbert空间$L^2_w(A)$中$\overline{\text{span}}\{x^{\lambda_n}\}_{n=1}^{\infty}$内的函数$f$，在$A$上几乎处处可表示为傅里叶型级数$f(x)=\sum_{n=1}^{\infty} \langle f, r_n\rangle_{w,A} x^{\lambda_n}$，其中$\{r_n\}_{n=1}^{\infty}$是$L^2_w(A)$中$\overline{\text{span}}\{x^{\lambda_n}\}_{n=1}^{\infty}$内$\{x^{\lambda_n}\}_{n=1}^{\infty}$的唯一双正交族。由此，我们证明该系统是$L^2_w(A)$中其闭张成空间的Markushevich基。此外，我们考虑了一个矩问题。最后，若存在正数$m,M$使得在$A$上$m\le w(x)\le M$，且$A$包含区间$[a, r_A]$（其中$a\ge 0$，$r_A$是$A$的本性上确界），我们证明该系统在$L^2_w(A)$的闭张成空间中是遗传完备的，并由此构造了一类定义在该闭包上的、可进行谱综合的紧算子。
英文摘要：
Inspired by the work of Borwein and Erdelyi \cite{BE1997JAMS} on generalizations of Müntz's theorem, we investigate the properties of the system $\{x^{λ_n}\}_{n=1}^{\infty}$ 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) $\{λ_n\}_{n=1}^{\infty}$ is a strictly increasing sequence of positive real numbers such that $\inf\{λ_{n+1}-λ_n \}>0$ and $\sum_{n=1}^{\infty}λ_n^{-1}<\infty$. We prove that a function $f$ in $\overline{\text{span}}\{x^{λ_n}\}_{n=1}^{\infty}$ in the Hilbert space $L^2_w (A)$, admits the $\bf{Fourier-type}$ series representation $f(x)=\sum_{n=1}^{\infty} \langle f, r_n\rangle_{w,A} x^{λ_n}$ a.e on $A$, where $\{r_n\}_{n=1}^{\infty}$ is the unique biorthogonal family of $\{x^{λ_n}\}_{n=1}^{\infty}$ in $\overline{\text{span}}\{x^{λ_n}\}_{n=1}^{\infty}$ in $L^2_w (A)$. As a result, we show that the system $\{x^{λ_n}\}_{n=1}^{\infty}$ is a $\bf{Markushevich\,\, basis}$ for $\overline{\text{span}}\{x^{λ_n}\}_{n=1}^{\infty}$ 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 $\{x^{λ_n}\}_{n=1}^{\infty}$ is $\bf{hereditarily\,\, complete}$ in $\overline{\text{span}}\{x^{λ_n}\}_{n=1}^{\infty}$ 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.
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