Pseudo-Differential Operators and Equations in a Discrete Half-Space

Abstract

We introduce a digital pseudo-differential operator acting in discrete Sobolev--Slobodetskii spaces and consider pseudo-differential equations with such operators in a discrete half-space. The theorem on a general solution of such equations is proved for a special case.

Operator Symbols and Operator Indices

“…In our opinion, such considerations will also be useful for discrete situations in which pseudo-differential operators are defined in functional spaces of discrete variables. Some first considerations in this direction were done, for example, in [24]. Moreover, a discrete situation is more practical, since it permits applying computer calculations.…”

Discrete Pseudo-Differential Operators and Boundary Value Problems in a Half-Space and a Cone

“…If we consider a half-space case then the Fourier image of the operator P D d is evaluated [12,13] and we'll demonstrate it in the following Example 1.…”

“…P 2. MULTIDIMENSIONAL PERIODIC RIEMANN BOUNDARY VALUE PROBLEM For D = R m + we'll remind some author's constructions [12,13] for discrete equations in a half-space. We have B d (z) = cot z 2 , z = (ξ , ξ m + iτ ), ξ = (ξ 1 , · · · , ξ m−1 ), τ > 0.…”

“…Like classical studies [5,6] we want to use a special representation for an elliptic symbol to solve the problem (4). We can easily obtain so-called characteristic singular integral equation associated with multidimensional periodic Riemann boundary value problem (4) like [5,6] (see also [12,13]).…”

“…If D = R m + then the operator (6) is a one-dimensional singular integral operator with periodic Cauchy kernel and a parameter ξ [12,13] 3. PERIODIC WAVE FACTORIZATION…”