Discrete singular integrals in a half-space

TL;DR

This paper investigates the invertibility of discrete Calderon--Zygmund singular integral operators in a half-space, establishing conditions under which their invertibility matches that of their continuous counterparts, using boundary problem theory.

Contribution

It introduces a novel approach linking discrete singular integrals' invertibility to the solvability of a periodic Riemann boundary problem.

Findings

Discrete singular integral operator invertibility iff its continuous analogue is invertible.

Develops a solvability theory for a periodic Riemann boundary problem.

Establishes a criterion for invertibility in discrete half-space settings.

Abstract

We consider Calderon -- Zygmund singular integral in the discrete half-space $h{\bf Z}^m_{+}$, where ${\bf Z}^m$ is entire lattice ($h>0$) in ${\bf R}^m$, and prove that the discrete singular integral operator is invertible in $L_2(h{\bf Z}^m_{+}$) iff such is its continual analogue. The key point for this consideration takes solvability theory of so-called periodic Riemann boundary problem, which is constructed by authors.