Discrete Clifford analysis: the one-dimensional setting

Abstract

In a higher dimensional setting, there are two major theories generalizing the theory of holomorphic functions in the complex plane, namely the theory of several complex variables and Clifford analysis. Discrete Clifford analysis is a discrete counterpart of the latter, studying the null functions of a discrete Dirac operator, which are called discrete monogenic functions. In this contribution, we give several new results in the one-dimensional case. We focus on the basic building blocks of discrete functions, namely discrete delta functions δ j , in relation to the discrete vector variable operator ξ. We introduce discrete distribution theory, in particular discrete delta distributions δ j and define a Fourier transform for discrete distributions. Finally, a comparison is made between discrete delta functions and distributions.