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

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