Discrete Clifford Analysis

Uwe Kähler () and Frank Sommen ()
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Uwe Kähler: University of Aveiro, CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics
Frank Sommen: Ghent University, Clifford Research Group, Faculty of Sciences

Chapter 64 in Operator Theory, 2026, pp 2013-2039 from Springer

Abstract: Abstract This survey is intended as an overview of discrete Clifford analysis and its current developments. Since in the discrete case one has to replace the partial derivative with two difference operators, backward and forward partial difference, one needs to modify the main tools for a development of a discrete function theory, such as the replacement of a real Clifford algebra by a complexified Clifford algebra or of the classic Weyl relations by the so-called S-Weyl relations. The main results, like Cauchy integral formula, Fischer decomposition, CK extension, and Taylor series, will be derived. To give a better idea of the differences between the discrete and continuous cases, this chapter contains the problem of discrete Hardy spaces as well as some discrete objects which do not have an equivalent object in continuous Clifford analysis, such as the CK extension of a discrete Delta function.

Keywords: Discrete Dirac operators; Discrete Clifford analysis; Discrete monogenic function; Discrete Dirac operator; S-Weyl relation; Discrete CK extension; Discrete Hardy spaces; Primary: 39A12; Secondary: 30G25, 31C20, 30G35 (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16356-1_18

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