Clifford Research Group

Seminars Academic year 2005-2006



Wavelet frames for the sphere
Uwe Kähler (University of Aveiro, Portugal)

Abstract:
In recent years frame theory and frame representations (in particular sparse frame representations) in Banach spaces received much attention. Especially, the technical necessities from wireless communications, such as the OFDM-technique, are impossible to conquer without frames. In this talk we will construct wavelet frames in the case of the sphere based on representations of the conformal group and give some applications based on sparse frame representations.

Metrodynamics with applications to wavelet analysis
Nele De Schepper (UGent)

Abstract:
Clifford analysis offers a direct, elegant and powerful generalization to higher dimension of the theory of holomorphic functions in the complex plane. It focusses on so-called monogenic functions, which are in the simple but useful setting of flat m-dimensional Euclidean space, null solutions of the so called Dirac operator, a first order vector differential operator, similar to the Cauchy-Riemann operator in the complex plane.
A highly important intrinsic feature of Clifford analysis is that it encompasses all dimensions at once, in other words all concepts in this multidimensional theory are not merely tensor products of one dimensional phenomena but are directly defined and studied in multi-dimensional space and cannot be recursively reduced to lower dimension. This true multidimensional nature has allowed for among others a very specific and original approach to multi-dimensional wavelet theory.
In earlier research multi-dimensional wavelets have been constructed in the framework of Clifford analysis. These wavelets are based on Clifford generalizations of the Hermite polynomials, the Gegenbauer polynomials, the Laguerre polynomials and the Jacobi polynomials on the real line. Moreover, they arise as specific applications of a general theory for constructing multi-dimensional Clifford-wavelets. This Clifford wavelet theory might be characterized as isotropic, since the metric in the underlying space is the standard Euclidean one.
In this talk we present the idea of a metric dependent Clifford analysis leading to a so-called anisotropic Clifford wavelet theory featuring wavelet functions which are adaptable to preferential, not necessarily orthogonal, directions in the signals or textures to be analyzed.
Apart from the detailed development of Clifford analysis in a global metric dependent setting, which includes among others the concepts of Fischer inner product, monogenicity, harmonicity, spherical monogenics and spherical harmonics, we also construct new Clifford-Hermite polynomials and study the corresponding Continuous Wavelet Transform.

The Poincaré and the dual Poincaré lemma
Hendrik De Bie (UGent)

Abstract:
We start with the definition of a differential calculus on superspace. We introduce an exterior derivative which gives rise to a de Rham complex. We prove that this complex is exact when we restrict ourselves to polynomial differential forms (Poincaré lemma).
Next we introduce a Hodge coderivative in superspace. It is possible to define such an operator even though there doesn't exist a Hodge star operator. Moreover this operator has the same properties as the classical coderivative (i.e. factorization of the laplacian). We also prove the dual Poincaré lemma for this operator, which is a bit more complicated than the classical case.

Clifford Analysis in the Hermitian Setting Part II
David Eelbode (UGent)

Abstract:
In this seminar, which is the continuation of the previous seminar, we will deal with the problem of finding a suitable Hermitian generalization for the classical zonal functions (after having revised some of the concepts introduced last time). For that purpose, we will first recall the orthogonal Fischer decomposition with respect to the classical Dirac operator on R^m, which leads to the Gegenbauer differential equation. We will then show how a generalization of this procedure to the Hermitian setting leads to the hypergeometric differential equation. We also investigate the question of uniqueness.

Clifford Analysis in the Hermitian Setting
David Eelbode (UGent)

Abstract:
The aim of this talk is to give an elementary introduction to what is called the Hermitian setting for Clifford analysis. We will introduce the algebraic setting, and define the so-called Hermitian Dirac operators. These are two complex Dirac operators, in some sense factorizing the Laplacian on an m-dimensional complex vector space, which remain invariant under the unitary group (as opposed to the classical Dirac operator, invariant under the orthogonal group). We will introduce the so-called spin-Euler polynomials and show how they are used in the Hermitian version of the Fischer decomposition.

Fueter's Theorem, revisited
Dixan Peña Peña (UGent)

Abstract:
Fueter's Theorem on the construction of monogenic quaternionic functions starting with a holomorphic function in the upper half of the complex plane, is further generalized in a Clifford analysis setting. In this lecture, we will discuss a new result which contains previous generalizations as special cases.

Cauchy-Kowalewska extensions in superspace
Hendrik De Bie (UGent)

Abstract:
We start this talk with a brief review of the appropriate definitions of the diracoperator and the cliffordalgebra suited to extend the notion of clifford analysis to spaces of commuting and anti-commuting variables.
Secondly we develop a theory of spherical monogenics in superspace. More precisely we want to determine bases of these spaces. In order to do this, we discuss two different CK-extensions in superspace. First of all, the classical extension by adding a new commuting variable is still well defined, and allows us to calculate the dimension of (almost all) spaces of spherical monogenics.
Next we develop a CK-extension with respect to two anti-commuting variables. This poses more difficulties: not all superfunctions can be extended in this scheme and moreover we don't obtain a surjective mapping between spaces of polynomials and spaces of spherical monogenics. This forces us to a more thorough study of the purely fermionic case, since the classical CK-extension is not applicable there.

Lie groups and Lie algebras: hopefulLie well-explained
David Eelbode (UGent)

Abstract:
In this lecture, I will try to explain why Lie algebras are so vital for the study of representations of Lie groups. The main objective is to introduce some basic concepts, which we probably know already but never used before, in such a way that it becomes clear why people introduced them in the first place. If time doesn't run out too fast, I will give a beautiful example, leading to the Gegenbauer functions from a pure representation-theoretical point of view.

Department of Mathematical AnalysisDepartment of Mathematical Analysis