Clifford-Gegenbauer polynomials related to the Dunkl Dirac operator

We introduce the so-called Clifford-Gegenbauer polynomials in the framework of Dunkl operators, as well on the unit ball B(1), as on the Euclidean space $R^m$. In both cases we obtain several properties of these polynomials, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the Jacobi polynomials on the real line. As in the classical Clifford case, the orthogonality of the polynomials on $R^m$ must be treated in a completely different way than the orthogonality of their counterparts on B(1). In case of $R^m$, it must be expressed in terms of a bilinear form instead of an integral. Furthermore, in this paper the theory of Dunkl monogenics is further developed.Comment: 19 pages, accepted for publication in Bulletin of the BM

Fractional fourier transforms of hypercomplex signals

An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained. For the case of dimension 2, also an explicit expression for the kernel is given

Fourier transforms of hypercomplex signals

An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained

The fractional Clifford-Fourier transform

In this paper, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters. A series expansion for the kernel of the resulting integral transform is derived. In the case of even dimension, also an explicit expression for the kernel in terms of Bessel functions is obtained. Finally, the analytic properties of this new integral transform are studied in detail.Comment: 17 pages, accepted for publication in Complex Anal. Oper. T

Cauchy-Kowalevski extensions and monogenic plane waves using spherical monogenics

Clifford analysis may be regarded as a direct and elegant generalization to higher dimensions of the theory of holomorphic functions in the complex plane, centred around the notion of monogenic function, i.e. a null solution of the Dirac operator. This paper dealswith axial and biaxialmonogenic functions containing sphericalmonogenics. They are constructed bymeans of two fundamentalmethods of Clifford analysis, namely the Cauchy-Kowalevski extension and monogenic plane waves

Veronese representation of projective Hjelmslev planes over some quadratic alternative algebras

We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative algebras. These planes are related to affine buildings of relative type Ã_2 and respective absolute type Ã_2, Ã_5 and Ẽ_6