September  2007, 6(3): 549-567. doi: 10.3934/cpaa.2007.6.549

A theoretical framework for wavelet analysis in a Hermitean Clifford setting

1. 

Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000 Gent, Belgium, Belgium, Belgium

Received  January 2006 Revised  September 2006 Published  June 2007

Hermitean Clifford analysis focusses on monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Here monogenicity is expressed by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a representation of the unitary group. In this paper we have further developed the Hermitean theory by introducing so-called zonal functions and by studying plane wave null solutions of the Hermitean Dirac operators. Moreover we have defined new Hermite polynomials in this Hermitean setting and expressed them in terms of the former Clifford-Hermite polynomials and of the one-dimensional Laguerre polynomials. These Hermitean Hermite polynomials are good candidates for mother wavelets in a Hermitean continuous wavelet transformation theory yet to be developed.
Citation: Fred Brackx, Hennie De Schepper, Frank Sommen. A theoretical framework for wavelet analysis in a Hermitean Clifford setting. Communications on Pure and Applied Analysis, 2007, 6 (3) : 549-567. doi: 10.3934/cpaa.2007.6.549
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