We consider discrete analogues of pseudo-differential operators and related discrete equations and boundary value problems. Existence and uniqueness results for special elliptic discrete boundary value problem and comparison for discrete and continuous solutions are given for certain smooth data in discrete Sobolev–Slobodetskii spaces.

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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.

Рассматриваются дискретные аналоги псевдодифференциальных операторов и уравнений в дискретных пространствах Соболева-Слободецкого. Используя подходящие дискретные граничные условия, мы доказываем однозначную разрешимость дискретной краевой задачи.