The cylindrical Fourier transform

In this paper we devise a so-called cylindrical Fourier transform within the Clifford analysis context. The idea is the following: for a fixed vector in the image space the level surfaces of the traditional Fourier kernel are planes perpendicular to that fixed vector. For this Fourier kernel we now substitute a new Clifford-Fourier kernel such that, again for a fixed vector in the image space, its phase is constant on co-axial cylinders w.r.t. that fixed vector. The point is that when restricting to dimension two this new cylindrical Fourier transform coincides with the earlier introduced Clifford-Fourier transform.We are now faced with the following situation: in dimension greater than two we have a first Clifford-Fourier transform with elegant properties but no kernel in closed form, and a second cylindrical one with a kernel in closed form but more complicated calculation formulae. In dimension two both transforms coincide. The paper concludes with the calculation of the cylindrical Fourier spectrum of an L2-basis consisting of generalized Clifford-Hermite functions

On a chain of harmonic and monogenic potentials in Euclidean half-space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials is constructed in the upper half of Euclidean space R^(m+1), including a higher dimensional generalization of the complex logarithmic function. Their distributional limits at the boundary R^(m) turn out to be well-known distributions such as the Dirac distribution, the Hilbert kernel, the fundamental solution of the Laplace and Dirac operators, the square root of the negative Laplace operator, and the like. It is shown how each of those potentials may be recovered from an adjacent kernel in the chain by an appropriate convolution with such a distributional limit

Hermitian clifford analysis

This paper gives an overview of some basic results on Hermitian Clifford analysis, a refinement of classical Clifford analysis dealing with functions in the kernel of two mutually adjoint Dirac operators invariant under the action of the unitary group. The set of these functions, called Hermitian monogenic, contains the set of holomorphic functions in several complex variables. The paper discusses, among other results, the Fischer decomposition, the Cauchy–Kovalevskaya extension problem, the axiomatic radial algebra, and also some algebraic analysis of the system associated with Hermitian monogenic functions. While the Cauchy–Kovalevskaya extension problem can be carried out for the Hermitian monogenic system, this system imposes severe constraints on the initial Cauchy data. There exists a subsystem of the Hermitian monogenic system in which these constraints can be avoided. This subsystem, called submonogenic system, will also be discussed in the paper

Hilbert transforms in Clifford analysis

The Hilbert transform on the real line has applications in many fields. In particular in one–dimensional signal processing, the Hilbert operator is used to extract global as well as instantaneous characteristics, such as frequency, amplitude and phase, from real signals. The multidimensional approach to the Hilbert transform usually is a tensorial one, considering the so-called Riesz transforms in each of the cartesian variables separately. In this paper we give an overview of generalized Hilbert transforms in Euclidean space, developed within the framework of Clifford analysis. Roughly speaking, this is a function theory of higher dimensional holomorphic functions, which is particularly suited for a treatment of multidimensional phenomena since all dimensions are encompassed at once as an intrinsic feature

A General Geometric Fourier Transform

The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature. We show which constraints are additionally necessary to obtain certain features like linearity or a shift theorem. As a result, we provide guidelines for the target-oriented design of yet unconsidered transforms that fulfill requirements in a specific application context. Furthermore, the standard theorems do not need to be shown in a slightly different form every time a new geometric Fourier transform is developed since they are proved here once and for all.Comment: First presented in Proc. of The 9th Int. Conf. on Clifford Algebras and their Applications, (2011

Introductory clifford analysis

In this chapter an introduction is given to Clifford analysis and the underlying Clifford algebras. The functions under consideration are defined on Euclidean space and take values in the universal real or complex Clifford algebra, the structure and properties of which are also recalled in detail. The function theory is centered around the notion of a monogenic function, which is a null solution of a generalized Cauchy–Riemann operator, which is rotation invariant and factorizes the Laplace operator. In this way, Clifford analysis may be considered as both a generalization to higher dimension of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. A notion of monogenicity may also be associated with the vectorial part of the Cauchy–Riemann operator, which is called the Dirac operator; some attention is paid to the intimate relation between both notions. Since a product of monogenic functions is, in general, no longer monogenic, it is crucial to possess some tools for generating monogenic functions: such tools are provided by Fueter’s theorem on one hand and the Cauchy–Kovalevskaya extension theorem on the other hand. A corner stone in this function theory is the Cauchy integral formula for representation of a monogenic function in the interior of its domain of monogenicity. Starting from this representation formula and related integral formulae, it is possible to consider integral transforms such as Cauchy, Hilbert, and Radon transforms, which are important both within the theoretical framework and in view of possible applications

Distributional boundary values of harmonic potentials in Euclidean half-space as fundamental solutions of convolution operators in Clifford analysis

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space R^(m+1) was recently constructed, including a higher dimensional analogue of the logarithmic function in the complex plane. In this construction the distributional limits of these potentials at the boundary R^m are crucial. The remarkable relationship between these distributional boundary values and four basic pseudodifferential operators linked with the Dirac and Laplace operators is studied

From hermitean clifford analysis to subelliptic dirac operators on odd dimensional spheres and other CR manifolds

We show that the two Dirac operators arising in Hermitian Clifford analysis are identical to standard differential operators arising in several complex variables. We also show that the maximal subgroup that preserves these operators are generated by translations, dilations and actions of the unitary n-group. So the operators are not invariant under Kelvin inversion. We also show that the Dirac operators constructed via two by two matrices in Hermitian Clifford analysis correspond to standard Dirac operators in euclidean space. In order to develop Hermitian Clifford analysis in a different direction we introduce a sub elliptic Dirac operator acting on sections of a bundle over odd dimensional spheres. The particular case of the three sphere is examined in detail. We conclude by indicating how this construction could extend to other CR manifolds.publishe

On an hypercomplex generalization of Gould-Hopper and related Chebyshev polynomials

An operational approach introduced by Gould and Hopper to the construction of generalized Hermite polynomials is followed in the hypercomplex context to build multidimensional generalized Hermite polynomials by the consideration of an appropriate basic set of monogenic polynomials. Directly related functions, like Chebyshev polynomials of first and second kind are constructed