Discrete Operators and Equations: Analysis and Comparison

Abstract

We develop a discrete variant of a theory of pseudo-differential equations and boundary value problems in canonical domains which are model situations for manifolds with non-smooth boundaries. Using digitization process for ordinary functional spaces we construct certain discrete functional spaces or spaces of functions of a discrete variable and define discrete pseudo-differential operators acting in such spaces. A main problem in which we are interested is to establish a correspondence between continual and discrete solutions of considered continual and discrete equations and in future boundary value problems. We have illustrated our considerations by certain examples of Calderon–Zygmund operators for which we have some interesting conclusions.