Periodic Jacobi operator with finitely supported perturbations: the inverse resonance problem

We consider a periodic Jacobi operator $H$ with finitely supported perturbations on $\mathbb{Z}\mathrm{.}$ We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the scattering data: the inverse of the transmission coefficient and the Jost function on the right half-axis, is one-to-one and onto. We consider the problem of reconstruction of the scattering data from all eigenvalues, resonances and the set of zeros of $R_{-}(\lambda )+1,$ where $R_{-}$ is the reflection coefficient.