2019/07/25 by A. S. Mikhaylov, Mikhaylov, Alexander, Victor Mikhaylov +2
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Matrix Theory and Algorithms #Numerical methods in inverse problems #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1907.11153
openalex publication_date 2019/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the Hamburger, Stieltjes and Hausdorff moment problems, that are\nproblems of the construction of a Borel measure supported on a real line, on a\nhalf-line or on an interval (0,1), from a prescribed set of moments. We\npropose a unified approach to these three problems based on using the auxiliary\ndynamical system with the discrete time associated with a semi-infinite Jacobi\nmatrix. It is show that the set of moments determines the inverse dynamic data\nfor such a system. Using the ideas of the Boundary Control method for every\nN\∈ \ℕ we can recover the spectral measure of a N\× N block of\nJacobi matrix, which is a solution to a truncated moment problem. This problem\nis reduced to the finite-dimensional generalized spectral problem, whose\nmatrices are constructed from moments and are connected with well-known Hankel\nmatrices by simple formulas. Thus the results on existence of solutions to\nHamburger, Stieltjes and Hausdorff moment problems are naturally given in terms\nof these matrices. We also obtain results on uniqueness of the solution of\nmoment problems, where as a main tool we use the Krein-type equations of\ninverse problem.\n