N. De Schepper

Clifford algebra-valued orthogonal polynomials in Euclidean space

In this paper a new method for constructing Clifford algebra-valued orthogonal polynomials in Euclidean space is presented. In earlier research, only scalar-valued weight functions were involved. Now the class of weight functions is enlarged with Clifford algebra-valued functions. The method consists in transforming the orthogonality relation on the Euclidean space into an orthogonality relation on the real axis by means of the so-called Clifford-Heaviside functions. Consequently appropriate orthogonal polynomials on the real axis yield Clifford algebra-valued orthogonal polynomials in Euclidean space. Three specific examples of such orthogonal polynomials in Euclidean space are discussed, viz. the generalized Clifford-Hermite, the Clifford-Laguerre and the half-range Clifford-Hermite polynomials.

The Cylindrical Fourier Spectrum of an L2‐basis consisting of Generalized Clifford‐Hermite Functions

In this paper we devise a new so‐called cylindrical Fourier transform within the Clifford analysis setting by substituting for the standard inner product in the classical exponential Fourier kernel a wedge product of the old and new vector variable as argument. The cylindrical Fourier spectrum of an L2‐basis consisting of generalized Clifford‐Hermite functions is then expressed as a sum of generalized hypergeometric series.

Curved Radon transforms and factorization of the Veronese equations in Clifford analysis

This paper is an updated version of our former article (Sommen, F., 1998, Curved Radon transforms in Clifford analysis. In: Clifford Algebras and their Applications in Mathematical Physics, Fund. Theories Phys., Vol. 94 (Dordrecht: Kluwer Academic Publishers), pp. 369–381). First of all, we study weighted integrals of functions over general surfaces of higher codimension whereby the weights take values in a Clifford algebra. We also introduce and study multi-linear Grassmann and Clifford algebras and apply them to the multi-linear Radon transform. In cases of symmetric multi-linear functions (homogeneous polynomials), we obtain a Clifford analysis generalization of the generalized Radon transform investigated by V.P. Palamodov (Palamodov, V.P., 1994, Radon transformation on real algebraic varieties. In: Gindikin, S. and Michor, P. (Eds) 75 Years of Radon Transform, Lecture Notes in Mathematical Physics, Vol. IV (Boston: International Press), pp. 252–262). †Dedicated to Richard Delanghe on the occasion of his 65th birthday.

The generalized Hermitean Clifford‐Hermite continuous wavelet transform

The one‐dimensional continuous wavelet transform (CWT) is a successful tool in signal and image analysis, with numerous applications (see e.g. [8, 9]). Standard (or orthogonal) Clifford analysis is a higher dimensional function theory which has proven to constitute an appropriate framework for developing higher dimensional CWTs, where all dimensions are encompassed at once, as opposed to tensorial approaches with products of onehyp‐dimensional phenomena; the specific construction of higher dimensional wavelets is based on particular families of orthogonal polynomials, see e.g. [4, 5, 6, 7]. We explicitly mention the generalized Clifford‐Hermite polynomials, introduced in [10] and applied to wavelet analysis in [7]. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, refining the orthogonal case, see [1]. Hermitean Clifford‐Hermite polynomials and their associated families of wavelet kernels ! were constructed in [2, 3]. In this contribution, we introduce generalize...