An analytical parameterization for all solutions of the two-dimensional moment problem under Carleman-type conditions

2025/08/14 by Zagorodnyuk, Sergey M.
#44A60 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.10823

Abstract

The two-dimensional moment problem consists of finding a positive Borel measure μ in ℝ² such that ∫_ℝ² t₁^m t₂ⁿ 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^∞ is positive semi-definite, and the following Carleman-type conditions hold: ∑_k=1^∞ \frac1 √[2k] s_2m,2k + s_2m+2,2k = ∞, 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.

Citations

• The Nevanlinna-type parametrization for the operator Hamburger moment problem
• Generalized resolvents of symmetric and isometric operators: the Shtraus approach
• On the two-dimensional moment problem
• The two-dimensional moment problem in a strip
• Devinatz's moment problem: a description of all solutions

Related

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• The two-dimensional moment problem in a strip
• On the two-dimensional moment problem
• On the two-dimensional moment problem
• On a criterion for the determinate-indeterminate dichotomy of the moment problem