169 research outputs found
Two powerful theorems in Clifford analysis
Two useful theorems in Euclidean and Hermitean Clifford analysis are discussed: the Fischer decomposition and the Cauchy-Kovalevskaya extension
The Clifford-Fourier integral kernel in even dimensional Euclidean space
AbstractRecently, we devised a promising new multi-dimensional integral transform within the Clifford analysis setting, the so-called Fourier–Bessel transform. In the specific case of dimension two, it coincides with the Clifford–Fourier transform introduced earlier as an operator exponential. Moreover, the L2-basis elements, consisting of generalized Clifford–Hermite functions, appear to be simultaneous eigenfunctions of both integral transforms. In the even dimensional case, this allows us to express the Clifford–Fourier transform in terms of the Fourier–Bessel transform, leading to a closed form of the Clifford–Fourier integral kernel
Hardy spaces of solutions of generalized Riesz and Moisil-Teodorescu systems
Hardy spaces of solutions of generalized Riesz and generalized Moisil-Teodorescu systems in half space Rm+1,+ , and of their non-tangential L2-boundary values in Rm are characterized
On the radial derivative of the delta distribution
Possibilities for defining the radial derivative of the delta distribution delta((x) under bar) in the setting of spherical coordinates are explored. This leads to the introduction of a new class of continuous linear functionals similar to but different from the standard distributions. The radial derivative of delta((x) under bar) then belongs to that new class of so-called signumdistributions. It is shown that these signumdistributions obey easy-to-handle calculus rules which are in accordance with those for the standard distributions in R-m
On the structure of complex Clifford algebra
The structure of a complex Clifford algebra is studied by direct sum decompositions into eigenspaces of specific linear operators
On a chain of harmonic and monogenic potentials in Euclidean half-space
In the framework of Clifford analysis, a chain of harmonic and monogenic potentials is constructed in the upper half of Euclidean space R^(m+1), including a higher dimensional generalization of the complex logarithmic function. Their distributional limits at the boundary R^(m) turn out to be well-known distributions such as the Dirac distribution, the Hilbert kernel, the fundamental solution of the Laplace and Dirac operators, the square root of the negative Laplace operator, and the like. It is shown how each of those potentials may be recovered from an adjacent kernel in the chain by an appropriate convolution with such a distributional limit
Representation of Distributions by Harmonic and Monogenic Potentials in Euclidean Space
In the framework of Clifford analysis, a chain of harmonic and monogenic
potentials in the upper half of (m+1)-dimensional Euclidean space was recently
constructed, including a higher dimensional analogue of the logarithmic
function in the complex plane, and their distributional boundary values were
computed. In this paper we determine these potentials in lower half-space, and
investigate whether they can be extended through the boundary R^m. This is a
stepping stone to the representation of a doubly infinite sequence of
distributions in R^m, consisting of positive and negative integer powers of the
Dirac and the Hilbert-Dirac operators, as the jump across R^m of monogenic
functions in the upper and lower half-spaces, in this way providing a sequence
of interesting examples of Clifford hyperfunctions.Comment: arXiv admin note: substantial text overlap with arXiv:1210.238
Higher spin Dirac operators between spaces of simplicial monogenics in two vector variables
The higher spin Dirac operator Q_{k,l} acting on functions taking values in an irreducible representation space for Spin(m) with highest weight (k+1/2,l+1/2,1/2,...,1/2), with k, l in N and k>= l, is constructed. The structure of the kernel space containing homogeneous polynomial solutions is then also studied
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