7,137 research outputs found
New techniques for the two-sided quaternionic fourier transform
In this paper, it is shown that there exists a Hermite basis for the two-sided quaternionic Fourier transform. This basis is subsequently used to give an alternative proof for the inversion theorem and to give insight in translation and convolution for the quaternionic Fourier transform
An Alternative Definition of the Hermite Polynomials Related to the Dunkl Laplacian
We introduce the so-called Clifford-Hermite polynomials in the framework of
Dunkl operators, based on the theory of Clifford analysis. Several properties
of these polynomials are obtained, such as a Rodrigues formula, a differential
equation and an explicit relation connecting them with the generalized Laguerre
polynomials. A link is established with the generalized Hermite polynomials
related to the Dunkl operators (see [R\"osler M., Comm. Math. Phys. 192 (1998),
519-542, q-alg/9703006]) as well as with the basis of the weighted
space introduced by Dunkl.Comment: This is a contribution to the Special Issue on Dunkl Operators and
Related Topics, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Clifford algebras, Fourier transforms and quantum mechanics
In this review, an overview is given of several recent generalizations of the
Fourier transform, related to either the Lie algebra sl_2 or the Lie
superalgebra osp(1|2). In the former case, one obtains scalar generalizations
of the Fourier transform, including the fractional Fourier transform, the Dunkl
transform, the radially deformed Fourier transform and the super Fourier
transform. In the latter case, one has to use the framework of Clifford
analysis and arrives at the Clifford-Fourier transform and the radially
deformed hypercomplex Fourier transform. A detailed exposition of all these
transforms is given, with emphasis on aspects such as eigenfunctions and
spectrum of the transform, characterization of the integral kernel and
connection with various special functions.Comment: Review paper, 39 pages, to appear in Math. Methods. Appl. Sc
Fractional fourier transforms of hypercomplex signals
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 is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained. For the case of dimension 2, also an explicit expression for the kernel is given
Fourier transforms of hypercomplex signals
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 is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained
The fractional Clifford-Fourier transform
In this paper, a fractional version of the Clifford-Fourier transform is
introduced, depending on two numerical parameters. A series expansion for the
kernel of the resulting integral transform is derived. In the case of even
dimension, also an explicit expression for the kernel in terms of Bessel
functions is obtained. Finally, the analytic properties of this new integral
transform are studied in detail.Comment: 17 pages, accepted for publication in Complex Anal. Oper. T
Conformal symmetries of the super Dirac operator
In this paper, the Dirac operator, acting on super functions with values in super spinor space, is defined along the lines of the construction of generalized Cauchy-Riemann operators by Stein and Weiss. The introduction of the superalgebra of symmetries osp(m|2n) is a new and essential feature in this approach. This algebra of symmetries is extended to the algebra of conformal symmetries osp(m + 1, 1|2n). The kernel of the Dirac operator is studied as a representation of both algebras. The construction also gives an explicit realization of the Howe dual pair osp(1|2) x osp(m|2n) \subset osp(m + 4n|2m + 2n). Finally, the super Dirac operator gives insight into the open problem of classifying invariant first order differential operators in super parabolic geometries
The Dunkl kernel and intertwining operator for dihedral groups
Dunkl operators associated with finite reflection groups generate a
commutative algebra of differential-difference operators. There exists a unique
linear operator called intertwining operator which intertwines between this
algebra and the algebra of standard differential operators. There also exists a
generalization of the Fourier transform in this context called Dunkl transform.
In this paper, we determine an integral expression for the Dunkl kernel, which
is the integral kernel of the Dunkl transform, for all dihedral groups. We also
determine an integral expression for the intertwining operator in the case of
dihedral groups, based on observations valid for all reflection groups. As a
special case, we recover the result of [Xu, Intertwining operators associated
to dihedral groups. Constr. Approx. 2019]. Crucial in our approach is a
systematic use of the link between both integral kernels and the simplex in a
suitable high dimensional space.Comment: Revision version. A missing factor in formula (6) for the Bochner
expression is added. Related formulas and some typos are corrected. All
comments are welcom
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