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The one-dimensional continuous wavelet transform is a successful tool for signal and image analysis, with applications in physics and engineering. Clifford analysis offers an appropriate framework for taking wavelets to higher dimension. In the usual orthogonal case Clifford analysis focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator ∂, defined in terms of an orthogonal basis for the quadratic space Rm underlying the construction of the Clifford algebra R0,m. An intrinsic feature of this function theory is that it encompasses all dimensions at once, as opposed to a tensorial approach with products of one-dimensional phenomena. This has allowed for a very specific construction of higher dimensional wavelets and the development of the corresponding theory, based on generalizations of classical orthogonal polynomials on the real line, such as the radial Clifford-Hermite polynomials introduced by Sommen. In this paper, we pass to the Hermitian Clifford setting, i.e. we let the same set of generators produce the complex Clifford algebra C2n (with even dimension), which we equip with a Hermitian conjugation and a Hermitian inner product. Hermitian Clifford analysis then focusses on the null solutions of two mutually conjugate Hermitian Dirac operators which are invariant under the action of the unitary group. In this setting we construct new Clifford-Hermite polynomials, starting in a natural way from a Rodrigues formula which now involves both Dirac operators mentioned. Due to the specific features of the Hermitian setting, four different types of polynomials are obtained, two types of even degree and two types of odd degree. These polynomials are used to introduce a new continuous wavelet transform, after thorough investigation of all necessary properties of the involved polynomials, the mother wavelet and the associated family of wavelet kernels.

In earlier research, generalized multidimensional Hilbert transforms have been constructed in m-dimensional Euclidean space, in the framework of Clifford analysis. Clifford analysis, centred around the notion of monogenic functions, may be regarded as a direct and elegant generalization to higher dimension of the theory of the holomorphic functions in the complex plane. The considered Hilbert transforms, usually obtained as a part of the boundary value of an associated Cauchy transform in m+1 dimensions, might be characterized as isotropic, since the metric in the underlying space is the standard Euclidean one. In this paper we adopt the idea of a so-called anisotropic Clifford setting, which leads to the introduction of a metric dependent m-dimensional Hilbert transform, showing, at least formally, the same properties as the isotropic one. The Hilbert transform being an important tool in signal analysis, this metric dependent setting has the advantage of allowing the adjustment of the co-ordinate system to possible preferential directions in the signals to be analyzed. A striking result to be mentioned is that the associated anisotropic (m+1)-dimensional Cauchy transform is no longer uniquely determined, but may stem from a diversity of (m+1)-dimensional "mother" metrics.

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centered around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential operator called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on the simultaneous null solutions, called Hermitean (or h-) monogenic functions, of two Hermitean Dirac operators which are invariant under the action of the unitary group. In Euclidean Clifford analysis, the Clifford-Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Previously, a Hermitean Clifford-Cauchy integral formula has been established by means of a matrix approach. This formula reduces to the traditional Martinelli-Bochner formula for holomorphic functions of several complex variables when taking functions with values in an appropriate part of complex spinor space. This means that the theory of Hermitean monogenic functions should encompass also other results of several variable complex analysis as special cases. At present we will elaborate further on the obtained results and refine them, considering fundamental solutions, Borel-Pompeiu representations and the Teoderescu inversion, each of them being developed at different levels, including the global level, handling vector variables, vector differential operators and the Clifford geometric product as well as the blade level were variables and differential operators act by means of the dot and wedge products. A rich world of results reveals itself, indeed including well-known formulae from the theory of several complex variables.

On the basis of the little material available (an architecture plan and some photographs) a computer model is developed for a bullet shaped dome, part of the Belgian Congo pavilion, created by the architect Henry Lacoste for the International Colonial Exhibition of 1931 in Paris. The ingenious and elegant wooden skeleton of the dome is approximated in two stages. The first approximation focusses on the curves traced on the dome by the wooden laminae, which appear to be loxodromes, cutting the meridians by a constant angle. In a second approximation the very specific joints of the laminae are taken into consideration. The resulting computer image shows an astonishing resemblance with the photographs. Finally, the shapes and dimensions of all laminae are calculated, enabling a possible reconstruction of the dome.