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Discrete Clifford Analysis

In: Operator Theory

Author

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  • Uwe Kähler

    (University of Aveiro, Department of Mathematics, CIDMA – Center for Research and Development in Mathematics and Applications)

  • Frank Sommen

    (Ghent University, Department of Mathematical Analysis)

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 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 case, 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.

Suggested Citation

  • Uwe Kähler & Frank Sommen, 2015. "Discrete Clifford Analysis," Springer Books, in: Daniel Alpay (ed.), Operator Theory, edition 127, chapter 56, pages 1609-1630, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-0667-1_18
    DOI: 10.1007/978-3-0348-0667-1_18
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