An analytical parameterization for all solutions of the two-dimensional moment problem under Carleman-type conditions
Year: 2025
Type: preprint
Abstract: The two-dimensional moment problem consists of finding a positive Borel measure $μ$ in $\mathbb{R}^2$ such that $\int_{\mathbb{R}^2} t_1^m t_2^n dμ= s_{m,n}$, $m,n=0,1,2,...$, where $s_{m,n}$ are prescribed real constants (moments). We study this moment problem in the case when the sequence $\{ s_{m,n} \}_{m,n=0}^\infty$ is positive semi-definite, and the following Carleman-type conditions hold: $$ \sum_{k=1}^\infty \frac{1}{ \sqrt[2k]{ s_{2m,2k} + s_{2m+2,2k} } } = \infty,\quad m=0,1,2,.... $$ In this case all solutions of the moment problem are parameterized by a class of analytic contractive operator-valued functions. The special case of the determinate moment problem is characterized. We introduce a notion of a generalized resolvent for a pair of commuting symmetric operators. We use basic properties of such generalized resolvents as a main tool in studying the above moment problem.
Source: arXiv (Cornell University)
Author: Zagorodnyuk, Sergey M.
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6979357775
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