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Description:
The main purpose of the present paper is to prove approximation and commutator properties for projections mapping periodic Sobolev spaces onto shift-invariant spaces generated by a finite number of compactly supported functions. With these prerequisites at hand and using certain localization techniques, we then characterize the stability of generalized Galerkin-Petrov schemes for solving periodic pseudodifferential equations in terms of elliptic type estimates of the numerical symbol. Moreover, we establish optimal convergence rates for the approximate solutions with respect to the Sobolev norms.