On an application of the Boundary control method to classical moment problems
Alexander Mikhaylov, Victor Mikhaylov
TL;DR
The article develops a boundary-control approach to classical moment problems by embedding Hamburger, Stieltjes, and Hausdorff data into discrete-time inverse problems for Jacobi matrices and their de Branges spaces. It constructs a dynamic framework with response and connecting operators, establishes Fourier representations and a de Branges space $B_J^N$ whose inner product matches the moment form, and shows how truncated moment problems can be solved by a generalized spectral problem $S^N_1 g_k=\lambda_k S^N_0 g_k$ to recover the spectral measure directly. The work provides explicit procedures to obtain $d\rho^N(\lambda)$ from moments, including normalization steps and determinant criteria for determinacy, and connects reproducing kernels to the spectral data through Hankel-structured transformations. Overall, it offers a rigorous link between moment problems, boundary-control methods, and spectral theory, with a concrete route to compute Dirichlet spectral data and assess uniqueness.
Abstract
We establish relationships between the classical moments problems which are problems of a construction of a measure supported on a real line, on a half-line or on an interval from prescribed set of moments with the Boundary control approach to a dynamic inverse problem for a dynamical system with discrete time associated with Jacobi matrices. We show that the solution of corresponding truncated moment problems is equivalent to solving some generalized spectral problems.