On the Moment Problem and Related Problems

Firstly, we recall the classical moment problem and some basic results related to it. By its formulation, this is an inverse problem: being given a sequence (y(j)) j is an element of N-n of real numbers and a closed subset F subset of R-n, n is an element of {1, 2, ...}, find a positive regular Borel measure mu on F such that integral F-tjd mu = y(j) , j is an element of N-n. This is the full moment problem. The existence, uniqueness, and construction of the unknown solution mu are the focus of attention. The numbers y(j) , j is an element of N-n are called the moments of the measure mu. When a sandwich condition on the solution is required, we have a Markov moment problem. Secondly, we study the existence and uniqueness of the solutions to some full Markov moment problems. If the moments y(j) are self-adjoint operators, we have an operator-valued moment problem. Related results are the subject of attention. The truncated moment problem is also discussed, constituting the third aim of this work.