De Ridder, HildeDe Schepper, HennieSommen, Franciscus

Discrete Clifford analysis is a discrete higher-dimensional function theory which corresponds simultaneously to a refinement of discrete harmonic analysis and to a discrete counterpart of Euclidean Clifford analysis. The discrete framework is based on a discrete Dirac operator that combines both forward and backward difference operators and on the ...

Baaske, FrankaBernstein, SwanhildDe Ridder, HildeSommen, Franciscus

The main purpose of this paper was to study solutions of the heat equation in the setting of discrete Clifford analysis. More precisely we consider this equation with discrete space and continuous time. Thereby we focus on representations of solutions by means of dual Taylor series expansions. Furthermore we develop a discrete convolution theory, a...

We study the boundary behavior of discrete monogenic functions, i.e. null-solutions of a discrete Dirac operator, in the upper and lower half space. Calculating the Fourier symbol of the boundary operator we construct the corresponding discrete Hilbert transforms, the projection operators arising from them, and discuss the notion of discrete Hardy ...

De Ridder, HildeDe Schepper, HennieKaehler, UweSommen, Franciscus

In this paper we construct the main ingredients of a discrete function theory in higher dimensions by means of a new "skew" type of Weyl relations. We will show that this new type overcomes the difficulties of working with standard Weyl relations in the discrete case. A Fischer decomposition, Euler operator, monogenic projection, and basic homogene...

De Schepper, HennieSommen, FranciscusVan de Voorde, Liesbet

A basic framework is derived for the development of a higher-dimensional discrete function theory in a Clifford algebra context. The concept of a discrete monogenic function is introduced as a proper generalization of the discrete holomorphic, or monodiffric, functions introduced by Isaacs in the 1950s. A concrete model is provided for the definiti...

Brackx, FredDe Schepper, HennieSommen, FranciscusVan de Voorde, Liesbet

We develop a discrete version of Clifford analysis, i.e., a higher-dimensional discrete function theory in a Clifford algebra context. On the simplest of all graphs, the rectangular Z(m) grid, the concept of a discrete monogenic function is introduced. To this end new Clifford bases are considered, involving so-called forward and backward basis vec...