Multidual Complex Numbers and the Hyperholomorphicity of Multidual Complex-Valued Functions

In recent work a radial deformation of the Fourier transform in the setting of Clifford analysis was introduced. The key idea behind this deformation is a family of new realizations of the Lie superalgebra $${\mathfrak {osp}}(1|2)$$ in terms of a so-called radially deformed Dirac operator $${\mathbf {D}}$$ depending on a deformation parameter c such that for $$c=0$$ the classical Dirac operator is reobtained. In this paper, several versions of the Paley–Wiener theorems for this radially deformed Fourier transform are investigated, which characterize the supports of functions associated to this generalized Fourier transform in Clifford analysis.

In this paper we derive for the even dimensional case a closed form of the Fourier–Borel kernel in the Clifford analysis setting. This kernel is obtained as the monogenic component in the Fischer decomposition of the exponential function \({e^{\langle \underline{x}, \underline{u} \rangle}}\) where \({\langle . , . \rangle}\) denotes the standard inner product on the m-dimensional Euclidean space. A first approach based on Clifford analysis techniques leads to a conceptual formula containing the Gamma operator and the so-called Clifford–Bessel function, two fundamental objects in the theory of Clifford analysis. To obtain an explicit expression for the Fourier–Borel kernel in terms of a finite sum of Bessel functions, this formula remains however hard to work with. To that end we have also elaborated a more direct approach based on special functions leading to recurrence formulas for a closed form of the Fourier–Borel kernel.

Rotations in discrete Clifford analysis

This paper consists of two parts. In the first part we deal with specific higher dimensional distributions within the framework of Clifford analysis. These distributions are "classical" in the sense that they were already introduced, albeit dispersed, in the literature on harmonic analysis and on Clifford analysis. Amongst these classical distributions are the fundamental solutions of the natural powers of the Laplace and the Dirac operators, and the integral kernel of the Hilbert transform. The strength of our approach is its unifying character.In the second part new higher dimensional distributions are introduced, generalizing the distributions of part I. Crucial to this generalization is the use of so-called surface spherical monogenics. The whole picture thus obtained offers structural clarity and unity.

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.

In this paper, we first define Teodorescu operator related to the -Dirac operator in Clifford analysis. Then we discuss the existence, q-times integrability, differentiability and mapping properties of Teodorescu operator in a nonempty open set , which is bounded connected.

Let Ω be a bounded, simply connected domain in the complex plane, with C ∞ smooth boundary and take u ∈ C∞(∂Ω) real-valued, then there exists a unique real-valued harmonic function for which the restriction to the boundary ∂Ω is precisely u. Let be the conjugate harmonic to U in the sense that F = U+iV is holomorphic in Ω, for which moreover V(a)=0, a ∈ Ω, then the restriction v of V to the boundary ∂Ω is called the Hilbert transform of u. In this article, we extend the principles of this construction to m-dimensional space , more specifically in a Clifford analysis setting, in order to define a Hilbert-like integral transform on the unit sphere S m−1, based upon a specific notion of conjugate harmonic functions using spherical co-ordinates. ‡Dedicated to Richard Delanghe on the occasion of his 65th birthday.

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 \({\mathbb{R}}^{2n}\) should originate from the non-tangential boundary limits of a corresponding Cauchy kernel in \({\mathbb{R}}^{2n+2}\). 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.

In this article, we mainly deal with the boundary value problem for harmonic function with values in Clifford algebra: urn:x-wiley:dummy:mana201100302:equation:mana201100302-math-0001where is a Liapunov surface in , the Dirac operator , are unknown functions with values in a universal Clifford algebra Under some assumptions, we show that the boundary value problem is solvable.

The classical Fischer decomposition of polynomials on Euclidean space makes it possible to express any polynomial as a sum of harmonic polynomials multiplied by powers of |x|2. A deformation of the Laplace operator was recently introduced by Ch.F. Dunkl. It has the property that the symmetry with respect to the orthogonal group is broken to a finite subgroup generated by reflections (a Coxeter group). It was shown by B. Orsted and S. Ben Said that there is a deformation of the Fischer decomposition for polynomials with respect to the Dunkl harmonic functions. In Clifford analysis, a Dunkl version of the Dirac operator was introduced and studied by P. Cerejeiras, U. Kahler and G. Ren. The aim of the article is to describe an analogue of the Fischer decomposition for solutions of the Dunkl Dirac operator. The main methods used are coming from representation theory, in particular, from ideas connected with Howe dual pairs.

We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree $(m,n,1)$ RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular level set behavior, and non-tangential boundary values. Our results, coupled with constructions of non-trivial RIF examples, demonstrate that much of the nice behavior seen in the two-variable case is lost in higher dimensions.

In this paper we work in the `split' discrete Cli fford analysis setting, i.e. the m-dimensional function theory concerning null-functions of the discrete Dirac operator d, defi ned on the grid Zm, involving both forward and backward di fferences. 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 Cli fford algebra in 2^2m subalgebras of dimension (k+m-1,k) - (k+m-3, k).

Let Cℓ n be the (universal) Clifford algebra generated by e1,…,e n The Cauchy-Riemann (or Dirac) operator in Cℓ n is defined by $$ D = \sum\nolimits_{{i = 0}}^{n} {{{e}_{i}}} \frac{\partial }{{\partial {{x}_{i}}}}. $$ The functions $$ f{\text{ }}:{\text{ }}\Omega \to C{{\ell }_{{n,}}}{\text{ }}\Omega {\text{ }} \subset {\text{ }}{{\mathbb{R}}^{{n + 1}}} $$ open, satisfying Df = 0, called left monogenic functions, are widely studied in Clifford analysis. The power function xm is not monogenic. The second author noticed that the power function is the solution of the modified Cauchy-Riemann system $$ xnDf + \left( {n - 1} \right)fn = 0 $$ which has strong connections to the hyperbolic metric. We study solutions of the equation $$ xnDf + \left( {n - 1} \right)Q_{{n - 1}}^{'}\left( f \right) = 0, $$ where Qf is the function given by the decomposition $$ f\left( x \right) = Pf\left( x \right) + Qf\left( x \right){{e}_{n}} $$ for $$ Pf\left( x \right),Qf\left( x \right) \notin C{{\ell }_{{n - 1}}}. $$ If $$ f = \sum\nolimits_{{i = 0}}^{n} {{{f}_{i}}} {{e}_{i}} $$ for some real functions f i , then f is the solution of the modified Cauchy-Riemann system stated above.

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space ℝm+1 was recently constructed, including a higher dimensional analogue of the logarithmic function in the complex plane. In this construction the distributional limits of these potentials at the boundary ℝm are crucial. The remarkable relationship between these distributional boundary values and four basic pseudodifferential operators linked with the Dirac and Laplace operators is studied.KeywordsFundamental SolutionLaplace OperatorDirac OperatorPseudodifferential OperatorClifford AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present results concerning analytic machines, a model of real computation introduced by Hotz which extends the well-known Blum, Shub and Smale machines (BSS machines) by infinite converging computations. The well-known representation theorem for BSS machines elucidates the structure of the functions computable in the BSS model: the domain of such a function partitions into countably many semi-algebraic sets, and on each of those sets the function is a polynomial resp. rational function. In this paper, we study whether the representation theorem can, in the univariate case, be extended to analytic machines, i.e. whether functions computable by analytic machines can be represented by power series in some part of their domain. We show that this question can be answered in the negative over the real numbers but positive under certain restrictions for functions over the complex numbers. We then use the machine model to define computability of univariate complex analytic (i.e. holomorphic) functions and examine in particular the class of analytic functions which have analytically computable power series expansions. We show that this class is closed under the basic analytic operations composition, local inversion and analytic continuation.