# Clifford Research Group

## Seminars Academic year 2005-2006

- Thursday 04-05-2006

Uwe Kähler

*Wavelet frames for the sphere*

- Thursday 20-04-2006

Nele De Schepper

*Metrodynamics with applications to wavelet analysis*

- Thursday 16-03-2006

Hendrik De Bie

*The Poincaré and the dual Poincaré lemma*

- Thursday 09-03-2006

David Eelbode

*Clifford Analysis in the Hermitian Setting Part II*

- Thursday 23-02-2006

David Eelbode

*Clifford Analysis in the Hermitian Setting*

- Thursday 16-02-2006

Dixan Peña Peña

Fueter's Theorem, revisited

- Thursday 02-02-2006

Hendrik De Bie

*Cauchy-Kowalewska extensions in superspace*

- Thursday 26-01-2006

David Eelbode

*Lie groups and Lie algebras: hopefulLie well-explained*

#### 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.