Inclusion of Hamburger's power moment problem in the spectral theory of the canonical systems

Hamburger's power moment problem (shortly HPMP) that has a solution with infinitely many points of increase is shown to be the problem of finding all spectral functions for some canonical system of the linear differential equations of phase dimension 2 and with Hamiltonian of special class. A rule for construction of this Hamiltonian using the data of HPMP is given. In this connection the Hamburger criterion for the uniqueness of a solution of HPMP acquires a “natural form”, and some results of the classical HPMP theory receive simple proofs.