A pair of Clifford-Fourier transforms is defined in the framework of Clifford analysis, as operator exponentials with a Clifford algebra-valued kernel. It is a genuine Clifford analysis construct, which is shown to be a refinement of the classical multi-dimensional Fourier transform.An adequate operational calculus is developed.

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Abstract. Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of the classical Fourier transform. This is applied to the so-called . The integral kernel of this transform is a particular solution of a system of PDEs in a Clifford algebra, but is, contrary to the classical Fourier transform, not the unique solution. Here we determine an entire class of solutions of this system of PDEs, under certain constraints. For each solution, series expressions in terms of Gegenbauer polynomials and Bessel functions are obtained. This allows to compute explicitly the eigenvalues of the associated integral transforms. In the even-dimensional case, this also yields the inverse transform for each of the solutions. Finally, several properties of the entire class of solutions are proven.

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Hypercomplex Fourier transforms are increasingly used in signal processing for the analysis of higher-dimensional signals such as color images. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. The present paper develops and studies two conceptually new ways to define convolution products for such transforms. As a by-product, convolution theorems are obtained that will enable the development and fast implementation of new filters for quaternionic signals and systems, as well as for their higher dimensional counterparts. MSC 2010 : 30G35; 42B10; 44A35; 94A12

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. F ± := e iπm 4 e iπ 4 (∆−|x| 2 ∓2Γ) = e iπm 4 e ∓ iπ 2 Γ e iπ 4 (∆−|x| 2 )

Orthogonal Clifford analysis in flat m-dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator ∂ = m j=1 ej∂x j , where (e1, . . . , em) forms an orthogonal basis for the quadratic space R m underlying the construction of the Clifford algebra R0,m. When allowing for complex constants and taking the dimension to be even: m = 2n, the same set of generators produces the complex Clifford algebra C2n, which we equip with a Hermitean Clifford conjugation and a Hermitean inner product. Hermitean Clifford analysis then focusses on the simultaneous null solutions of two mutually conjugate Hermitean Dirac operators, naturally arising in the present context and being invariant under the action of a realization of the unitary group U (n). In this so-called Hermitean setting Clifford-Hermite polynomials are constructed, starting from a Rodrigues formula involving both Dirac operators mentioned. Due to the specific features of the Hermitean setting, four different types of polynomials are obtained, two types of even degree and two types of odd degree. We investigate their properties: recurrence relations, structure, explicit form and orthogonality w.r.t. a deliberately chosen weight function. They also give rise to the definition of the Hermitean Clifford-Hermite functions, and may be used to develop a Hermitean continuous wavelet transform, see [4].