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We consider the Hamburger, Stieltjes and Hausdorff moment problems, that areproblems of the construction of a Borel measure supported on a real line, on ahalf-line or on an interval $(0,1)$, from a prescribed set of moments. Wepropose a unified approach to these three problems based on using the auxiliarydynamical system with the discrete time associated with a semi-infinite Jacobimatrix. It is show that the set of moments determines the inverse dynamic datafor such a system. Using the ideas of the Boundary Control method for every$N\in \mathbb{N}$ we can recover the spectral measure of a $N\times N$ block ofJacobi matrix, which is a solution to a truncated moment problem. This problemis reduced to the finite-dimensional generalized spectral problem, whosematrices are constructed from moments and are connected with well-known Hankelmatrices by simple formulas. Thus the results on existence of solutions toHamburger, Stieltjes and Hausdorff moment problems are naturally given in termsof these matrices. We also obtain results on uniqueness of the solution ofmoment problems, where as a main tool we use the Krein-type equations ofinverse problem.

Dynamic inverse problem for special system associated with Jacobi matrices and classical moment problems