366 research outputs found
Hilbert transforms in Clifford analysis
The Hilbert transform on the real line has applications in many fields. In particular in one–dimensional signal processing, the Hilbert operator is used to extract global as well as instantaneous characteristics, such as frequency, amplitude and phase, from real signals. The multidimensional approach to the Hilbert transform usually is a tensorial one, considering the so-called Riesz transforms in each of the cartesian variables separately. In this paper we give an overview of generalized Hilbert transforms in Euclidean space, developed within the framework of Clifford analysis. Roughly speaking, this is a function theory of higher dimensional holomorphic functions, which is particularly suited for a treatment of multidimensional phenomena since all dimensions are encompassed at once as an intrinsic feature
Fundamental solutions for the super Laplace and Dirac operators and all their natural powers
The fundamental solutions of the super Dirac and Laplace operators and their
natural powers are determined within the framework of Clifford analysis.Comment: 12 pages, accepted for publication in J. Math. Anal. App
Introductory clifford analysis
In this chapter an introduction is given to Clifford analysis and the underlying Clifford algebras. The functions under consideration are defined on Euclidean space and take values in the universal real or complex Clifford algebra, the structure and properties of which are also recalled in detail. The function theory is centered around the notion of a monogenic function, which is a null solution of a generalized Cauchy–Riemann operator, which is rotation invariant and factorizes the Laplace operator. In this way, Clifford analysis may be considered as both a generalization to higher dimension of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. A notion of monogenicity may also be associated with the vectorial part of the Cauchy–Riemann operator, which is called the Dirac operator; some attention is paid to the intimate relation between both notions. Since a product of monogenic functions is, in general, no longer monogenic, it is crucial to possess some tools for generating monogenic functions: such tools are provided by Fueter’s theorem on one hand and the Cauchy–Kovalevskaya extension theorem on the other hand. A corner stone in this function theory is the Cauchy integral formula for representation of a monogenic function in the interior of its domain of monogenicity. Starting from this representation formula and related integral formulae, it is possible to consider integral transforms such as Cauchy, Hilbert, and Radon transforms, which are important both within the theoretical framework and in view of possible applications
Rarita-Schwinger Type Operators on Spheres and Real Projective Space
In this paper we deal with Rarita-Schwinger type operators on spheres and
real projective space. First we define the spherical Rarita-Schwinger type
operators and construct their fundamental solutions. Then we establish that the
projection operators appearing in the spherical Rarita-Schwinger type operators
and the spherical Rarita-Schwinger type equations are conformally invariant
under the Cayley transformation. Further, we obtain some basic integral
formulas related to the spherical Rarita-Schwinger type operators. Second, we
define the Rarita-Schwinger type operators on the real projective space and
construct their kernels and Cauchy integral formulas.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1106.358
q-deformed harmonic and Clifford analysis and the q-Hermite and Laguerre polynomials
We define a q-deformation of the Dirac operator, inspired by the one
dimensional q-derivative. This implies a q-deformation of the partial
derivatives. By taking the square of this Dirac operator we find a
q-deformation of the Laplace operator. This allows to construct q-deformed
Schroedinger equations in higher dimensions. The equivalence of these
Schroedinger equations with those defined on q-Euclidean space in quantum
variables is shown. We also define the m-dimensional q-Clifford-Hermite
polynomials and show their connection with the q-Laguerre polynomials. These
polynomials are orthogonal with respect to an m-dimensional q-integration,
which is related to integration on q-Euclidean space. The q-Laguerre
polynomials are the eigenvectors of an su_q(1|1)-representation
Ontwikkeling van een milieuvriendelijke visserijmethode voor de garnalenvisserij, gebaseerd op stimulering door elektrische pulsen. Nr. EOGFL 30-A41
Segal-Bargmann-Fock modules of monogenic functions
In this paper we introduce the classical Segal-Bargmann transform starting
from the basis of Hermite polynomials and extend it to Clifford algebra-valued
functions. Then we apply the results to monogenic functions and prove that the
Segal-Bargmann kernel corresponds to the kernel of the Fourier-Borel transform
for monogenic functionals. This kernel is also the reproducing kernel for the
monogenic Bargmann module.Comment: 11 page
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