  When the classical Hamburger moment problem has solutions, it has either
exactly one solution or infinitely many solutions. Correspondingly, the moment
problem is said to be either determinate or indeterminate. In terms of Jacobi
operators, this dichotomy translates into the operator being either selfadjoint
or symmetric nonselfadjoint. In this work, we present a new criterion for the
determinate-indeterminate classification which hinges on bases of
representation (in Akhiezer-Glazman terminology) for Jacobi operators so that
the corresponding matrices have a certain structure.
