Hermitian Clifford Analysis Research Articles

Discrete weierstrass transform in discrete hermitian clifford analysis

The classical Weierstrass transform is an isometric operator mapping elements of the weighted L2−space L2(R,exp(−x2/2)) to the Fock space. It has numereous applications in physics and applied mathematics. In this paper, we define an analogue version of this transform in discrete Hermitian Clifford analysis, where functions are defined on a grid rather than the continuous space. This new transform is based on the classical definition, in combination with a discrete version of the Gaussian function and discrete counterparts of the classical Hermite polynomials. Furthermore, a discrete Weierstrass space with appropriate inner product is constructed, for which the discrete Hermite polynomials form a basis. In this setting, we also investigate the behaviour of the discrete delta functions and check if they are elements of this newly defined Weierstrass space.

Bochner‐Martinelli formula in superspace

In a series of recent papers, a harmonic and hypercomplex function theory in superspace has been established and amply developed. In this paper, we address the problem of establishing Cauchy integral formulae in the framework of Hermitian Clifford analysis in superspace. This allows us to obtain a successful extension of the classical Bochner‐Martinelli formula to superspace by means of the corresponding projections on the space of spinor‐valued superfunctions.

Hermitian Clifford Analysis on Superspace

In this paper we first recall the proper algebraic framework, i.e. the radial algebra, needed to extend Hermitian Clifford analysis to the superspace setting. The fundamental objects for this extension then are introduced by means of an abstract complex structure on the Hermitian radial algebra. This leads to a natural representation of this Hermitian radial algebra on superspace.

Spin actions in Euclidean and Hermitian Clifford analysis in superspace

In [4] we studied the group invariance of the inner product of supervectors as introduced in the framework of Clifford analysis in superspace. The fundamental group SO0 leaving invariant such an inner product turns out to be an extension of SO(m)×Sp(2n) and gives rise to the definition of the spin group in superspace through the exponential of the so-called extended superbivectors, where the spin group can be seen as a double covering of SO0 by means of the representation h(s)[x]=sxs‾. In the present paper, we study the invariance of the Dirac operator in superspace under the classical H and L actions of the spin group on superfunctions. In addition, we consider the Hermitian Clifford setting in superspace, where we study the group invariance of the Hermitian inner product of supervectors introduced in [3]. The group of complex supermatrices leaving this inner product invariant constitutes an extension of U(m)×U(n) and is isomorphic to the subset SO0J of SO0 of elements that commute with the complex structure J. The realization of SO0J within the spin group is studied together with the invariance under its actions of the super Hermitian Dirac system. It is interesting to note that the spin element leading to the complex structure can be expressed in terms of the n-dimensional Fourier transform.

Hermitian Clifford analysis and its connections with representation theory

This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We argue the generalized gradient construction of Stein and Weiss based on representation theory of Lie groups is the natural way to construct such a Dirac-type operator because applied to a Riemannian spin manifold, it provides the Atiyah–Singer Dirac operator. This method, however, does not apply to these Hermitian Dirac operators because the representations of the unitary group used are not irreducible, causing problems in considering invariance under a group larger than U(n). This motivates either the development of Clifford analysis over a complex vector space with respect to a Hermitian inner product or the development of Dirac-type operators on Cauchy–Riemann structures.

The Radial Algebra as an Abstract Framework for Orthogonal and Hermitian Clifford Analysis

We recall the algebra of endomorphisms on the so-called radial algebra of abstract vector variables which generalizes both polynomial and Clifford algebras, and the defining building blocks of a function theory in this abstract framework. These building blocks are given by the abstract versions of the Dirac operator and the directional derivatives. Together with other fundamental endomorphisms, they lead to an algebraic structure which can be considered as the abstract equivalent of orthogonal Clifford analysis. Following a similar approach, we present the axiomatic definitions of the Hermitian radial algebra, leading to the abstract equivalent of Hermitian Clifford analysis.

Szegö kernel for hardy space of matrix functions

By the characterization of the matrix Hilbert transform in the Hermitian Clifford analysis, we introduce the matrix Szegö projection operator for the Hardy space of Hermitean monogenic functions defined on a bounded sub-domain of even dimensional Euclidean space, establish the Kerzman-Stein formula which closely connects the matrix Szegö projection operator with the Hardy projection operator onto the Hardy space, and get the matrix Szegö projection operator in terms of the Hardy projection operator and its adjoint. Furthermore, we construct the explicit matrix Szegö kernel function for the Hardy space on the sphere as an example, and get the solution to a boundary value problem for matrix functions.

Bicomplex Hermitian Clifford analysis

Complex Hermitian Clifford analysis emerged recently as a refinement of the theory of several complex variables, while at the same time, the theory of bicomplex numbers motivated by the bicomplex version of quantum mechanics is also under full development. This stimulates us to combine the Hermitian Clifford analysis with the theory of bicomplex number so as to set up the theory of bicomplex Hermitian Clifford analysis. In parallel with the Euclidean Clifford analysis, the bicomplex Hermitian Clifford analysis is centered around the bicomplex Hermitian Dirac operator Open image in new window: \(C^\infty (\mathbb{R}^{4n} ,\mathbb{W}_{4n} ) \to C^\infty (\mathbb{R}^{4n} ,\mathbb{W}_{4n} )\), where \(\mathbb{W}_{4n}\) is the tensor product of three algebras, i.e., the hyperbolic quaternion Open image in new window, the bicomplex number \(\mathbb{B}\), and the Clifford algebra ℝ0,4n. The operator Open image in new window is a square root of the Laplacian in ℝ4n, introduced by the formula Open image in new window with Kj being the basis of Open image in new window, and \(\partial _{Z_j }\) denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra \(\mathbb{B} \otimes \mathbb{R}_{0,4n}\) whose definition involves a delicate construction of the bicomplex Witt basis. The introduction of the operator Open image in new window can also overturn the prevailing opinion in the Hermitian Clifford analysis in the complex or quaternionic setting that the complex or quaternionic Hermitiean monogenic functions are described by a system of equations instead of by a single equation like classical monogenic functions which are null solutions of Dirac operator. In contrast to the Hermitian Clifford analysis in quaternionic setting, the Poisson brackets of the twisted real Clifford vectors do not vanish in general in the bicomplex setting. For the operator Open image in new window, we establish the Cauchy integral formula, which generalizes the Martinelli-Bochner formula in the theory of several complex variables.

Split-quaternionic Hermitian Clifford analysis

The theory of split-quaternion analysis and split-quaternion geometry is nowadays under full development. In this article, we extend quaternionic Hermitian Clifford analysis to the case of split-quaternions. The split-quaternionic Hermitian Dirac operator is introduced as a self-map of smooth functions defined in domains of with values in the tensor product of the hyperbolic quaternions , the split-quaternions and the Clifford algebra . It is defined by with being the basis of and denoting the twisted Hermitian Dirac operators in the split-quaternionic Clifford algebra whose definition is based on a delicate construction of the split-quaternionic Witt basis. We establish the counterpart of the Cauchy integral formula.

Generalized Taylor series in hermitian Clifford analysis

An orthogonal Appell basis of homogeneous polynomials is constructed for the Bergman space of square-integrable hermitian monogenic functions in the unit ball of Cn, with values in a homogeneous subspace of spinor space. This construction then leads to a generalized Taylor series expansion for spherical monogenics.

Fischer decomposition in symplectic harmonic analysis

In the framework of quaternionic Clifford analysis in Euclidean space \(\mathbb {R}^{4p}\), which constitutes a refinement of Euclidean and Hermitian Clifford analysis, the Fischer decomposition of the space of complex valued polynomials is obtained in terms of spaces of so-called (adjoint) symplectic spherical harmonics, which are irreducible modules for the symplectic group Sp\((p)\). Its Howe dual partner is determined to be \(\mathfrak {sl}(2,\mathbb {C}) \oplus \mathfrak {sl}(2,\mathbb {C}) = \mathfrak {so}(4,\mathbb {C})\).

Riemann boundary value problem for H-2-monogenic function in Hermitian Clifford analysis

Hermitian Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions of two Hermitian Dirac operators. Using a circulant matrix approach, we will study the R 0 Riemann type problems in Hermitian Clifford analysis. We prove a mean value formula for the Hermitian monogenic function. We obtain a Liouville-type theorem and a maximum module for the function above. Applying the Plemelj formula, integral representation formulas, and a Liouville-type theorem, we prove that the R 0 Riemann type problems for Hermitian monogenic and Hermitian-2-monogenic functions are solvable. Explicit representation formulas of the solutions are also given.

Dirichlet type problems in Hermitian Clifford analysis

Solvability conditions for some Dirichlet type boundary value problems in the framework of Hermitian Clifford analysis are established.

Embedding Factors for Branching in Hermitian Clifford Analysis

A step 2 branching decomposition of spaces of homogeneous Hermitian monogenic polynomials in \({\mathbb {C}}^n\) is established with explicit embedding factors in terms of the generalized Jacobi polynomials, which allows for an inductive construction of an orthogonal basis for those spaces. The embedding factors and the orthogonal bases are fully worked out in the complex dimension 2 case, with special interest for the Appell property.

Paracomplex Hermitean Clifford Analysis

Substituting the complex structure by the paracomplex structure plays an important role in para-geometry and para-analysis. In this article we shall introduce the paracomplex structure into the realm of Clifford analysis and establish paracomplex Hermitean Clifford analysis by constructing a paracomplex Hermitean Dirac operator \({\mathcal {D}}\) and establishing the corresponding Cauchy integral formula. The theory of paracomplex Hermitean Clifford analysis turns out to be similar to that of complex Hermitean Clifford analysis which recently emerged as a refinement of the theory of several complex variables. It deserves to be pointed out that the introduction of a single operator \({\mathcal {D}}\) in the paracomplex setting has an advantage over the complex setting where complex Hermitean monogenic functions are described by a system of equations instead of being given as null-solution of a single Dirac operator as in the case of classic monogenic functions.

Special functions and systems in Hermitian Clifford analysis: Higher codimension

AbstractIn this paper, we investigate a Cauchy–Kowalevski (CK) extension problem that arises naturally within the framework of Hermitian Clifford analysis, which concerns the study of Dirac‐like operators in several complex variables. The work presented here includes CK extensions of higher codimension and in particular the CK extension of the Gauss distribution in several complex variables. Copyright © 2013 John Wiley & Sons, Ltd.

Matrix Cauchy and Hilbert transforms in Hermitian quaternionic Clifford analysis

Recently the basic setting has been established for the development of quaternionic Hermitian Clifford analysis, a theory centred around the simultaneous null solutions, called q-Hermitian monogenic functions, of four Hermitian Dirac operators in a quaternionic Clifford algebra setting. Borel–Pompeiu and Cauchy integral formulae have been established in this framework by means of a (4 × 4) circulant matrix approach. By means of the matricial quaternionic Hermitian Cauchy kernel involved in these formulae, a quaternionic Hermitian Cauchy integral may be defined. The subsequent study of the boundary limits of this Cauchy integral then leads to the definition of a quaternionic Hermitian Hilbert transform. These integral transforms are studied in this article.

Half Dirichlet problem for the Hölder continuous matrix functions in Hermitian Clifford analysis

The simultaneous null solutions of the two complex Hermitian Dirac operators are focused on Hermitian Clifford analysis, where the Hermitian Cauchy integral was constructed in the framework of circulant (2 × 2) matrix functions. Under this setting, we will present the half Dirichlet problem with boundary spaces of Hölder continuous circulant (2 × 2) matrix functions on the sphere of even-dimensional Euclidean space. We will give the unique solution to it merely by using the Hermitian Cauchy transformation, get the solutions to the Dirichlet problem on the unit ball for Hölder continuous circulant (2 × 2) matrix functions as the boundaries and the solutions to the classical Dirichlet problem as the special case, and derive a decomposition of the Poisson kernel for matrix Laplace operator.

On Primitives and Conjugate Harmonic Pairs in Hermitian Clifford Analysis

The notion of a conjugate harmonic pair in the context of Hermitian Clifford analysis is introduced as a pair of specific harmonic functions summing up to a Hermitian monogenic function in an open region $$\Omega $$ of $$\mathbb C ^n$$ . Hermitian monogenic functions are special monogenic functions, which are at the core of so-called Clifford analyis, a straightforward generalization to higher dimension of the holomorphic functions in the complex plane. Under certain geometric conditions on $$\Omega $$ the conjugate harmonic to a given specific harmonic is explicitly constructed and the potential or primitive of a Hermitian monogenic function is determined.

Special functions and systems in Hermitian Clifford analysis

In this paper, we study some new special functions that arise naturally within the framework of Hermitian Clifford analysis, which concerns the study of Dirac‐like systems in several complex variables. In particular, we focus on Hermite polynomials, Bessel functions, and generalized powers. We also derive a Vekua system for solutions of Hermitian systems in axially symmetric domains. Copyright © 2012 John Wiley & Sons, Ltd.