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In the framework of Clifford analysis a chain of harmonic and monogenic potentials is constructed in the upper half of Euclidean space Rm+1 Their distributional limits at the boundary are computed, obtaining in this way well-known distributions in Rm such as the Dirac distribution, the Hilbert kernel, 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. [ABSTRACT FROM AUTHOR]

The fundamental solution is constructed of a particular higher spin operator in Rm, acting on functions taking values in an irreducible representation space for Spin(m) with highest weight (k12+,l+12,12,...,12), with k, l ∈ N and k≥l. Some basic integral formulae are stated. [ABSTRACT FROM AUTHOR]

Considering the foundation of Quaternionic Analysis by R. Fueter and his collaborators in the beginning of the 1930s as starting point of Clifford Analysis, we can look back to 80 years of work in this field. However the interest in multivariate analysis using Clifford algebras only started to grow significantly in the 70s. Since then a great amount of papers on Clifford Analysis referring different classes of Special Functions have appeared. This situation may have been triggered by a more systematic treatment of monogenic functions by their multiple series development derived from Gegenbauer or associated Legendre polynomials (and not only by their integral representation). Also approaches to Special Functions by means of algebraic methods, either Lie algebras or through Lie groups and symmetric spaces gained by that time importance and influenced their treatment in Clifford Analysis. In our talk we will rely on the generalization of the classical approach to Special Functions through differential equations with respect to the hypercomplex derivative, which is a more recently developed tool in Clifford Analysis. In this context special attention will be payed to the role of Special Functions as intermediator between continuous and discrete mathematics. This corresponds to a more recent trend in combinatorics, since it has been revealed that many algebraic structures have hidden combinatorial underpinnings. [ABSTRACT FROM AUTHOR]

In this note, we describe the Gel’fand-Tsetlin procedure for the construction of an orthogonal basis in spaces of Hermitean monogenic polynomials of a fixed bidegree. The algorithm is based on the Cauchy-Kovalevskaya extension theorem and the Fischer decomposition in Hermitean Clifford analysis. [ABSTRACT FROM AUTHOR]

Classical results on the expansion of complex functions in a series of special polynomials (namely inverse similar sets of polynomials) are extended to the Clifford setting. This expansion is shown to be valid in closed balls. [ABSTRACT FROM AUTHOR]

Spherical monogenics were studied from the very beginning of Clifford analysis (see [2]) and a lot of their properties are well understood. In particular, the monogenic projection πM (i.e., the projection from the space of homogeneous polynomials of order k to the space of spherical monogenics of order k) plays a key role in many different investigations. There is a standard integral formula for the projection (see e.g. [4]). Recently, an explicit differential formula was given ([7, 3]) for the projection, analogous to the classical formula for spherical harmonics. The aim of the article is to describe some other differential formulae useful in further applications. [ABSTRACT FROM AUTHOR]

Finely monogenic functions have been recently introduced in [14, 15, 17] as a higher dimensional generalization of Fuglede’s finely holomorphic ones. It seems to remain open whether such functions are infinitely fine differentiable and satisfy the unique continuation property. In this note, some examples related to this question are given. [ABSTRACT FROM AUTHOR]

In this talk we show how the Fourier-Bros-Iagolnitzer transform may be related to the theory of monogenic functions in Clifford analysis. This leads then to a constructive proof, using the Radon transform of monogenic functions, for the non-linear Radon decomposition of the delta-distribution, which corresponds to its micro-localization. [ABSTRACT FROM AUTHOR]