Frames of solutions and discrete analysis of pseudo‐differential equations

Abstract

We construct discrete analogs of multidimensional singular integral operators and study their invertibility. Moreover, we give a comparison between continual and discrete case. We give the theory of periodic Riemann problem also, because it is needed for studying invertibility of so-called paired equations.For more general case of pseudo-differential operators, we construct the solvability theory for discrete pseudo-differential equations in discrete analogs of Sobolev-Slobodetskii spaces. Some comparison results for discrete and continuous solutions are given also in appropriate discrete normed spaces.