Some properties of the spinor fourier transform

In this paper, the theory of the spinor Fourier transform introduced in [Batard T, Berthier M, Saint-Jean C, Clifford-Fourier Transform for Color Image Processing, Geometric Algebra Computing for Engineering and Computer Science (E. Bayro-Corrochano and G. Scheuermann Eds.), Springer, London, 2010, pp. 135–161] is further developed. While in the original paper, the transform was determined for vector-valued functions only, it now will be extended to functions taking values in the entire Clifford algebra. Next, two bases are determined under which this Fourier transform is diagonalizable. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. This problem will be tackled in the final section of this paper

Models for some irreducible representations of so(m,C) in discrete Clifford analysis

In this paper we work in the `split' discrete Clifford analysis setting, i.e. the m-dimensional function theory concerning null-functions of the discrete Dirac operator d, defined on the grid Zm, involving both forward and backward differences. This Dirac operator factorizes the (discrete) Star-Laplacian (Delta*= d2). We show how the space Hk of discrete k-homogeneous spherical harmonics, which is a reducible so(m;C)-representation, may explicitly be decomposed into 2^2m isomorphic copies of irreducible so(m;C)-representations with highest weight (k, 0, ..., 0). The key element is the introduction of 2^2m idempotents, dividing the discrete Clifford algebra in 2^2m subalgebras of dimension (k+m-1,k) - (k+m-3, k)

The higher spin Laplace operator in several vector variables

In this paper, an explicit expression is obtained for the conformally invariant higher spin Laplace operator $\mathcal{D}_{\lambda}$, which acts on functions taking values in an arbitrary (finite-dimensional) irreducible representation for the orthogonal group with integer valued highest weight. Once an explicit expression is obtained, a special kind of (polynomial) solutions of this operator is determined

On a special type of solutions of arbitrary higher spin Dirac operators

In this paper an explicit expression is determined for the elliptic higher spin Dirac operator, acting on functions f(x) taking values in an arbitrary irreducible finite-dimensional module for the group Spin(m) characterized by a half-integer highest weight. Also a special class of solutions of these operators is constructed, and the connection between these solutions and transvector algebras is explained

Polynomial solutions for arbitrary higher spin dirac operators

In a series of recent papers, we have introduced higher spin Dirac operators, which are far-reaching generalisations of the classical Dirac operator. Whereas the latter acts on spinor-valued functions, the former acts on functions taking values in arbitrary irreducible half-integer highest weight representations for the spin group. In this paper, we describe a general procedure to decompose the polynomial kernel spaces for these operators in irreducible summands for the regular action of the spin group. We will do this in an inductive way, making use of twisted higher spin operators

On an inductive construction of higher spin Dirac operators

In this contribution, we introduce higher spin Dirac operators, i.e. a specific class of differential operators in Clifford analysis of several vector variables, motivated by equations from theoretical physics. In particular, the higher spin Dirac operator in three vector variables will be explicitly constructed, starting from a description of the so-called twisted Rarita-Schwinger operator

Twisted higher spin Dirac operators

In this paper, we define twisted higher spin Dirac operators and explain how these invariant differential operators can be used to define more general higher spin Dirac operators acting on functions on which then take values in general half-integer representations for the spin group

Bargmann–Radon transform for axially monogenic functions

In this paper, we study the Bargmann-Radon transform and a class of monogenic functions called axially monogenic functions. First, we compute the explicit formula of the Bargmann-Radon transform for axially monogenic functions, by making use of the Funk-Hecke theorem. Then we present the explicit form of the general Cauchy-Kowalewski extension for radial function. Finally, by making use of the results we obtained, we give an application of the Bargmann-Radon transform for Cauchy-Kowalewski extension