Approximation and Commutator Properties of Projections onto Shift-Invariant Subspaces and Applications to Boundary Integral Equations

The main purpose of the present paper is to prove approximation and com-mutator properties for projections mapping periodic Sobolev spaces onto shift-invariant spaces generated by a nite 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 pseudodiierential 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.