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- Hendrik De Bie
- 2008

In this paper extensions of the classical Fourier, fractional Fourier and Radon transforms to superspace are studied. Previously, a Fourier transform in superspace was already studied, but with a different kernel. In this work, the fermionic part of the Fourier kernel has a natural symplectic structure, derived using a Clifford analysis approach. Several… (More)

- Hendrik De Bie, F. Sommen
- 2008

The Clifford-Hermite and the Clifford-Gegenbauer polynomials of standard Clifford analysis are generalized to the new framework of Clifford analysis in superspace in a merely symbolic way. This means that one does not a priori need an integration theory in superspace. Furthermore a lot of basic properties, such as orthogonality relations, differential… (More)

The study of spherical harmonics in superspace, introduced in [J. Phys. A: Math. Theor. 40 (2007) 7193-7212], is further elaborated. A detailed description of spherical harmonics of degree k is given in terms of bosonic and fermionic pieces, which also determines the irreducible pieces under the action of SO(m) × Sp(2n). In the second part of the paper,… (More)

- Hendrik De Bie, F. Sommen
- 2008

A new framework for studying superspace is given, based on methods from Clifford analysis. This leads to the introduction of both orthogonal and symplectic Clifford algebra generators, allowing for an easy and canonical introduction of a super-Dirac operator, a super-Laplace operator and the like. This framework is then used to define a super-Hodge… (More)

- Roxana Bujack, Hendrik De Bie, Nele De Schepper, Gerik Scheuermann
- Journal of Mathematical Imaging and Vision
- 2013

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… (More)

- Hendrik De Bie, Nele De Schepper
- Signal, Image and Video Processing
- 2012

An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention… (More)

- Hendrik De Bie, F. Sommen
- 2009

In previous work the framework for a hypercomplex function theory in superspace was established and amply investigated. In this paper a Cauchy integral formula is obtained in this new framework by exploiting techniques from orthogonal Clifford analysis. After introducing Clifford algebra valued surfaceand volume-elements first a purely fermionic Cauchy… (More)

- K. Coulembier, Hendrik De Bie, F. Sommen
- 2009

In this paper, a new class of Cauchy integral formulae in superspace is obtained, using formal expansions of distributions. This allows to solve five open problems in the study of harmonic and Clifford analysis in superspace. MSC 2000 : 58C50, 30G35, 26B20

- Hendrik De Bie, Dixan Peña Peña, Frank Sommen
- Journal of Approximation Theory
- 2014

In this paper two important classes of orthogonal polynomials in higher dimensions using the framework of Clifford analysis are considered, namely the Clifford–Hermite and the Clifford–Gegenbauer polynomials. For both classes an explicit generating function is obtained. c ⃝ 2013 Elsevier Inc. All rights reserved.

- Hendrik De Bie, Nele De Schepper, Todd A. Ell, K. Rubrecht, Stephen J. Sangwine
- Applied Mathematics and Computation
- 2015

The quaternion Fourier transform (qFT) is an important tool in multi-dimensional data analysis, in particular for the study of color images. An important problem when applying the qFT is the mismatch between the spatial and frequency domains: the convolution of two quaternion signals does not map to the pointwise product of their qFT images. The recently… (More)