Generalized Taylor series in hermitian Clifford analysis

Clifford analysis is a higher-dimensional function theory offering a refinement of classical harmonic analysis. It is centred around monogenic functions, i.e. null solutions of the rotation invariant vector-valued Dirac operator. In this context, generalizations of classical orthogonal polynomials on the real line have been introduced in order to use them as building blocks for wavelets, as has been the case, e.g. for the generalized Clifford–Hermite polynomials. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the standard Euclidean case; it focuses on simultaneous null solutions of two Hermitean Dirac operators which are invariant under the action of the unitary group. In this Hermitean setting, generalized Hermitean Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Hermitean Dirac operators. In this paper, we establish their mutual orthogonality relations w.r.t. a well-chosen weight function.

Half Dirichlet problem for matrix functions on the unit ball in Hermitian Clifford analysis

Disjoint mixing linear fractional composition operators in the unit ball

This monograph provides an introduction and a survey of recent results in potential theory with respect to the Laplace–Beltrami operator D in several complex variables, with special emphasis on the unit ball in Cn. Topics covered include Poisson–Szego integrals on the ball, the Green's function for D and the Riesz decomposition theorem for invariant subharmonic functions. The extension to the ball of the classical Fatou theorem on non-tangible limits of Poisson integrals, and Littlewood's theorem on the existence of radial limits of subharmonic functions are covered in detail. The monograph also contains recent results on admissible and tangential boundary limits of Green potentials, and Lp inequalities for the invariant gradient of Green potentials. Applications of some of the results to Hp spaces, and weighted Bergman and Dirichlet spaces of invariant harmonic functions are included. The notes are self-contained, and should be accessible to anyone with some basic knowledge of several complex variables.

Spin actions in Euclidean and Hermitian Clifford analysis in superspace

In this note, we describe the Gel’fand‐Tsetlin procedure for the construction of an orthogonal basis in spaces of Hermitean monogenic polynomials of a fixed bidegree. The algorithm is based on the Cauchy‐Kovalevskaya extension theorem and the Fischer decomposition in Hermitean Clifford analysis.

We examine translation-invariant measures on Banach spaces lp, where p ∈ [1,∞]. We construct analogs of the Lebesgue measure on Borel σ-algebras generated by the topology of pointwise convergence (σ-additive, invariant under shifts by arbitrary vectors, regular measures). We show that these measures are not σ-finite. We also study spaces of functions integrable with respect to measures constructed and prove that these spaces are not separable. We consider various dense subspaces in spaces of functions that are integrable with respect to a translation-invariant measure. We specify spaces of continuous functions, which are dense in the functional spaces considered. We discuss Borel σ-algebras corresponding to various topologies in the spaces lp, where p ∈ [1,∞]. For p ∈ [1,∞), we prove the coincidence of Borel σ-algebras corresponding to certain natural topologies in the given spaces of sequences and the Borel σ-algebra corresponding to the topology of pointwise convergence. We also verify that the space l∞ does not possess similar properties.

Carleson measures, riemann-stieltjes operators and multipliers on F(p, q, s) spaces in the unit ball of ℂn

We introduce two new linear differential operators which are invariant with respect to the unitary group SU(n). They constitute analogues of the twistor and the Rarita-Schwinger operator in the orthogonal case. The natural setting for doing this is Hermitian Clifford Analysis. Such operators are constructed by twisting the two versions of the Hermitian Dirac operator \( \partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} } \) and \( \partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} }^\dag \) and then projecting on irreducible modules for the unitary group. We then study some properties of their spaces of nullsolutions and we find a formulation of the Hermitian Rarita-Schwinger operators in terms of Hermitian monogenic polynomials.

In this paper, we discuss the so‐called Witt basis in a Clifford algebra and we axiomatically define an algebra of abstract Hermitian vector variables similar to the ‘radial algebra’. In this setting, we introduce some linear partial differential operators and we study their resolutions. Copyright © 2002 John Wiley & Sons, Ltd.

We first review some facts on parabolic geometries and Weyl structures. Using this framework, we introduce CR geometry, basic differential operators on CR manifolds and we study their relation to basic differential operators in Hermitian Clifford analysis.

Explicit differentiation matrices in various polynomial bases are presented in this work. The idea is to avoid any change of basis in the process of polynomial differentiation. This article concerns both degree-graded polynomial bases such as orthogonal bases, and non-degree-graded polynomial bases including the Lagrange and Bernstein bases.

Classical Segal–Bargmann theory studies three Hilbert space unitary isomorphisms that describe the wave-particle duality and the configuration space-phase space. In this work, we generalized these concepts to Clifford algebra-valued functions. We establish the unitary isomorphisms among the space of Clifford algebra-valued square-integrable functions on $$\mathbb {R}^n$$ with respect to a Gaussian measure, the space of monogenic square-integrable functions on $$\mathbb {R}^{n+1}$$ with respect to another Gaussian measure and the space of Clifford algebra-valued linear functionals on symmetric tensor elements of $$\mathbb {R}^n$$ .

Matrix methods for quadrature formulas on the unit circle. A survey

We consider the approximation by spectral and pseudo‐spectral methods of the solution of the Cauchy problem for a scalar linear hyperbolic equation in one space dimension posed in the whole real line. To deal with the fact that the domain of the equation is unbounded, we use Hermite functions as orthogonal basis. Under certain conditions on the coefficients of the equation, we prove the spectral convergence rate of the approximate solutions for regular initial data in a weighted space related to the Hermite functions. Numerical evidence of this convergence is also presented. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012