  This note is devoted to inverse spectral problems for canonical Hamiltonian
systems on the half-line. An approach to inverse spectral problems based on the
use of truncated Toeplitz operators has been especially effective in the case
when the spectral measure of the system is a locally finite periodic measure
(see \cite{MP}). In this note we extend the periodic algorithm to the case of
non-periodic measures by considering periodizations of a spectral measure and
showing that the Hamiltonians corresponding to the periodizations converge to
the Hamiltonian of the original measure.
