Generalized Hermitean Clifford–Hermite polynomials and the associated wavelet transform

Clifford analysis is a higher‐dimensional function theory offering a refinement of classical harmonic analysis, which has proven to be an appropriate framework for developing higher‐dimensional continuous wavelet transforms, the construction of the wavelets being based on generalizations to a higher dimension of classical orthogonal polynomials on the real line. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the standard Euclidean case; it focusses on so‐called Hermitean monogenic functions, i.e. simultaneous null solutions of two Hermitean Dirac operators. In this Hermitean setting, Clifford–Hermite polynomials and their associated families of wavelet kernels have been constructed starting from a Rodrigues formula involving both Hermitean Dirac operators mentioned. Unfortunately, the property of the so‐called vanishing moments of the corresponding mother wavelets, ensuring that polynomial behaviour in the analyzed signal is filtered out, is only partially satisfied and has to be interpreted with care, the underlying mathematical reason being the fact that the Hermitean Clifford–Hermite polynomials show a too restrictive structure. In this paper, we will remediate this drawback by considering generalized Hermitean Clifford–Hermite polynomials, involving in their definition homogeneous Hermitean monogenic polynomials. The ultimate goal being the construction of new continuous wavelet transforms by means of these polynomials, we first deeply investigate their properties, amongst which are their connection with the traditional Laguerre polynomials, their structure and recurrence relations. Copyright © 2008 John Wiley & Sons, Ltd.     

A matrix Hilbert transform in Hermitean Clifford analysis

Orthogonal Clifford analysis is a higher dimensional function theory offering both a generalization of complex analysis in the plane and a refinement of classical harmonic analysis. During the last years, Hermitean Clifford analysis has emerged as a new and successful branch of it, offering yet a refinement of the orthogonal case. Recently in [F. Brackx, B. De Knock, H. De Schepper, D. Peña Peña, F. Sommen, submitted for publication], a Hermitean Cauchy integral was constructed in the framework of circulant (2×2) matrix functions. In the present paper, a new Hermitean Hilbert transform is introduced, arising naturally as part of the non-tangential boundary limits of that Hermitean Cauchy integral. The resulting matrix operator is shown to satisfy properly adapted analogues of the characteristic properties of the Hilbert transform in classical analysis and orthogonal Clifford analysis.     

Cauchy Integral Formulae in Quaternionic Hermitean Clifford Analysis

The theory of complex Hermitean Clifford analysis was developed recently as a refinement of Euclidean Clifford analysis; it focusses on the simultaneous null solutions, called Hermitean monogenic functions, of two Hermitean Dirac operators constituting a splitting of the traditional Dirac operator. In this function theory, the fundamental integral representation formulae, such as the Borel?CPompeiu and the Clifford?CCauchy formula have been obtained by using a (2 ×?2) circulant matrix formulation. In the meantime, the basic setting has been established for so-called quaternionic Hermitean Clifford analysis, a theory centred around the simultaneous null solutions, called q-Hermitean monogenic functions, of four Hermitean Dirac operators in a quaternionic Clifford algebra setting. In this paper we address the problem of establishing a quaternionic Hermitean Clifford?CCauchy integral formula, by following a (4?× 4) circulant matrix approach.     

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.     

The Hermitean Hilbert–Dirac Connection

Hermitean Clifford analysis is a recent branch of Clifford analysis, refining the Euclidean case; it focusses on the simultaneous null solutions, called Hermitean monogenic functions, of two complex Dirac operators which are invariant under the action of the unitary group. The specificity of the framework, introduced by means of a complex structure creating a Hermitean space, forces the underlying vector space to be even dimensional. Thus, any Hilbert convolution kernel in should originate from the non-tangential boundary limits of a corresponding Cauchy kernel in . In this paper we show that the difficulties posed by this inevitable dimensional jump can be overcome by following a matrix approach. The resulting matrix Hermitean Hilbert transform also gives rise, through composition with the matrix Dirac operator, to a Hermitean Hilbert–Dirac convolution operator “factorizing” the Laplacian and being closely related to Riesz potentials. Received: October, 2007. Accepted: February, 2008.     

Hermitean Clifford-Hermite Polynomials

Orthogonal Clifford analysis in flat m–dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator , where ( ) forms an orthogonal basis for the quadratic space underlying the construction of the Clifford algebra . When allowing for complex constants and taking the dimension to be even: m = 2n, the same set of generators produces the complex Clifford algebra , which we equip with a Hermitean Clifford conjugation and a Hermitean inner product. Hermitean Clifford analysis then focusses on the simultaneous null solutions of two mutually conjugate Hermitean Dirac operators, naturally arising in the present context and being invariant under the action of a realization of the unitary group U (n). In this so–called Hermitean setting Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Dirac operators mentioned. Due to the specific features of the Hermitean setting, four different types of polynomials are obtained, two types of even degree and two types of odd degree. We investigate their properties: recurrence relations, structure, explicit form and orthogonality w.r.t. a deliberately chosen weight function. They also give rise to the definition of the Hermitean Clifford–Hermite functions, and may be used to develop a Hermitean continuous wavelet transform, see [4].     

On Fundamental Solutions in Clifford Analysis

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions, which constitute the kernel of a first order vector valued, rotation invariant, differential operator ?{\underline{\partial}} called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on a subclass of monogenic functions, i.e. the simultaneous null solutions, called Hermitean (or h−) monogenic functions, of two Hermitean Dirac operators ?_z{\partial_{\underline{z}}} and ?_z^f{\partial_{\underline{z}^\dagger}} which are invariant under the action of the unitary group, and constitute a splitting of the original Euclidean Dirac operator. In Euclidean Clifford analysis, the Clifford–Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Also a Hermitean Clifford–Cauchy integral formula has been established by means of a matrix approach. Naturally Cauchy integral formulae rely upon the existence of fundamental solutions of the Dirac operators under consideration. The aim of this paper is twofold. We want to reveal the underlying structure of these fundamental solutions and to show the particular results hidden behind a formula such as, e.g. ?E = d{\underline{\partial}E = \delta}. Moreover we will refine these relations by constructing fundamental solutions for the differential operators issuing from the Euclidean and Hermitean Dirac operators by splitting the Clifford algebra product into its dot and wedge parts.     

On Cauchy and Martinelli-Bochner integral formulae in Hermitean Clifford analysis

Euclidean Clifford analysis is a higher dimensional function theory, refining harmonic analysis, centred around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential operator, called the Dirac operator. More recently, Hermitean Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on the simultaneous null solutions of two Hermitean Dirac operators, invariant under the action of the unitary group. In this paper, a Cauchy integral formula is established by means of a matrix approach, allowing the recovering of the traditional Martinelli-Bochner formula for holomorphic functions of several complex variables as a special case.     

Hermitean Matrix Ensembles and Orthogonal Polynomials

In this article, we investigate orthogonal polynomials associated with complex Hermitean matrix ensembles using the combination of the methods of Coulomb fluid (or potential theory), chain sequences, and Birkhoff–Trjitzinsky theory. We give a general formula for the largest eigenvalue of the N×N Jacobi matrices (which is equivalent to estimating the largest zero of a sequence of orthogonal polynomials) and the two-level correlation function for the α ensembles (α>0) introduced previously for α>1. In the case of 0<α<1, we give a natural representation for the weight function that is a special case of the general Nevanlinna parametrization. We also discuss Hermitean matrix ensembles associated with general indeterminate moment problems.     

Holder Norm Estimate for a Hilbert Transform in Hermitean Clifford Analysis

A Hilbert transform for H?lder continuous circulant (2 × 2) matrix functions, on the d-summable (or fractal) boundary Γ of a Jordan domain Ω in ?²ⁿ , has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the H?lder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the H?lder exponents, the diameter of Γ and a specific d-sum (d > d) of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary.     

Taylor Series in Hermitean Clifford Analysis

In this paper, we consider the Taylor decomposition for h-monogenic functions in Hermitean Clifford analysis. The latter is to be considered as a refinement of the classical orthogonal function theory, in which the structure group underlying the equations is reduced from \mathfrakso(2m){\mathfrak{so}(2m)}to the unitary Lie algebra u(m).     

On the eigenproblem for normal matrices

Summary The Householder-Givens method for the solution of the Hermitean eigenproblem is used for the solution of the skew-Hermitean and the normal eigenproblem.This research was performed at the (Mathematical Institute of the) State University in Utrecht.     

Hermitian modular forms and the Burkhardt quartic

In a recent paper [3] Dern and Krieg investigate the ring of Hermitean modular forms of degree two with respect to the Eisenstein number field. There is a relation to our paper [1] on the Burkhardt group, which we make explicit.     

Concomitants and syzygies of spinors in 3-dimensional euclidean space

Summary In 3-dimensional Euclidean space, explicit formulae for the Hermitean concomitants of two irreducible spinsors of type[n+1/2] are obtained, where n is a positive integer. Further, a method, by means of which the syzygies of degree2 in these concomitants can be obtained, is described and is illustrated by considering two irreducible spinors of type[3/2]. Entrata in Redazione il 16 maggio 1972.     

Solvability of elliptic systems with square integrable boundary data

We consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L ₂ Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when A is either Hermitean, block or constant. Our methods apply to more general systems of partial differential equations and as an example we prove perturbation results for boundary value problems for differential forms.     

First Colonization of a Hard-Edge in Random Matrix Theory

We describe the spectral statistics of the first finite number of eigenvalues in a newly-forming band on the hard-edge of the spectrum of a random Hermitean matrix model, a phenomenon also known as the “birth of a cut” near a hard-edge. It is found that in a suitable scaling regime, they are described by the same spectral statistics of a finite-size Laguerre-type matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials.     

Hermitean (semi) lattices and Rolf��s lattice

We investigate the indexed Hermitean lattice of type 0 generated by a single element a subject to the relation ${a = a^\amalg \leq b = b^\amalg}$ . Although the lattice is infinite, we are able to recursively describe its subdirectly irreducible factors. We also give necessary and sufficient conditions about indices for this lattice to be finite.     

Matrix Models and Matrix Integrals

We discuss the relations between arbitrary solutions of the loop equations describing the Hermitean one-matrix model and particular (multicut) solutions corresponding to concrete matrix integrals. These latter have a series of specific properties and, in particular, are described in terms of the Seiberg-Witten-Whitham theory. We consider the simplest example of an ordinary integral in detail. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 77–89, January, 2006.     

Duality for Hermitean Systems in {\mathbb R^{2n}}

In this paper, using the ring structure of the space of circulant (2 × 2)-matrix, we characterize the dual of the (Fréchet) space of germs of left Hermitean monogenic matrix functions in a compact set _boxclose_boxclose_boxclose_boxclose_boxclose_boxclose_boxclose_boxclose R^2n{{\bf E}\subset\mathbb R^{2n}}. As an application we describe the dual space of the so-called h-monogenic functions satisfying simultaneously two Dirac type equations.     

The Structure of Nonthin Irreducible T-modules of Endpoint 1: Ladder Bases and Classical Parameters

Building on the work of Terwilliger, we find the structure of nonthin irreducible T-modules of endpoint 1 for P- and Q-polynomial association schemes with classical parameters. The isomorphism class of such a given module is determined by the intersection numbers of the scheme and one additional parameter which must be an eigenvalue for the first subconstituent graph. We show that these modules always have what we call a ladder basis, and find the structure explicitly for the bilinear, Hermitean, and alternating forms schemes.