  Necessary and sufficient conditions are presented for a measure to be the
spectral measure of a finite range or exponentially decaying perturbation of a
periodic Jacobi operator.
  As a corollary we can fully solve the inverse resonance problem: given
resonances and eigenvalues we can recover the spectral measure of the Jacobi
operator; we provide necessary and sufficient conditions under which such an
operator exists and is unique; and we show that the inverse resonance problem
is stable under small perturbations.
