Discrete Clifford Analysis

This article focuses on discrete Clifford analysis,a theory of discrete monogenic functions on the lattices.Recently,several authors have shown interest in finding an appropriate framework for the development of discrete counterparts of the basic no-tions and concepts of Clifford analysis.There are a lot of contributions already being made on some topics such as skew Weyl relation,discrete Taylor expansion,discrete integral theory,boundary value problems for discrete monogenic functions and others.However,there are some weak points in the established framework of discrete Clifford analysis.One weak point stands out in the further development of this discrete function theory.That is,the discrete Cauchy-Pompeiu formula does not share the same form with the continuous version.To overcome it,we develop a new approach to the dis-crete integral theory based on the concepts "discrete boundary measure" and "discrete normal vector".This new approach also makes it possible to study the convergence relation between discrete monogenic functions and continuous ones on an arbitrary do-main.Moreover,we establish a discrete Taylor expansion theory in the split quaternions based on two Sheffer sequences.This article is organized as follow:Chapter One is the introduction.We show the background of our research field,explain our motivation,and give an outline of our achievement.Chapter Two is devoted to the discrete integral theory based on the the concepts"discrete boundary measure" and "discrete normal vector.We study the fundamental solution to the discrete Cauchy-Fueter operator.In particular,we give an asymptotic expansion of it.Furthermore,based on this fundamental solution and the discrete Stokes formula we establish a discrete Cauchy-Pompeiu formula,which shares the same form with its continuous counterpart.Chapter Three focuses on the boundary behaviour of discrete monogenic functions.We study the solvability of the Dirichlet problem for discrete Cauchy-Fueter system.A serious difficulty arises due to the absence of the notion of the non-tangential limit on lattices.We find out that the restriction of the discrete Cauchy-Bitsadze integral on the inner layer of the discrete boundary plays the same role as the non-tangential limit of its continuous counterpart from the inside of the domain,and the analogous result holds true for the outer layer.Based on this fact,the discrete Sokhotski-Plemelj formula is established over an arbitrary bounded boundary.This allows us to give a criterion for the solvability of the Dirichlet problem via discrete Plemelj projections.In Chapter Four we solve some natural convergence problems in the discrete Clif-ford analysis.By combining the discrete integral theory and our results on discrete ap-proximations to domains in Euclidean spaces,we reveal a convergence relation between the discrete and continuous monogenic functions:a function is monnogenic if and only if it is the scaling limit of some discrete monogenic functions.More remarkably,we show that the scaling limits of all the discrete integral operators appearing in the theory of the boundary behaviour of discrete monogenic functions,including discrete Plemelj projections,are their continuous counterparts.Chapter Five focuses on a natural question in discrete complex analysis,i.e.,2-D discrete Clifford analysis:whether the Taylor series of a discrete holomorphic function is convergent to itself in the whole lattice,we answer this question in the affirmative in the setting of a new kind of discrete holomorphic function on the square grid hZ2(h>0)with values in split quaternions based on the methods of Sheffer sequences.