Rotations in discrete Clifford analysis

We develop a discrete version of Clifford analysis, i.e., a higher-dimensional discrete function theory in a Clifford algebra context. On the simplest of all graphs, the rectangular ℤm grid, the concept of a discrete monogenic function is introduced. To this end new Clifford bases are considered, involving so-called forward and backward basis vectors, controlling the support of the involved operators. Following a proper definition of a discrete Dirac operator and of some topological concepts, function theoretic results amongst which Stokes' theorem, Cauchy's theorem and a Cauchy integral formula are established.

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).

Discrete Clifford analysis is a higher dimensional discrete function theory based on skew Weyl relations. It is centered around the study of Clifford algebra valued null solutions, called discrete monogenic functions, of a discrete Dirac operator, i.e. a first order, Clifford vector valued difference operator. In this contribution, we establish a Cauchy‐Kovalevskaya extension theorem for discrete monogenic functions defined on the grid Zhm of m‐tuples of integer multiples of a variable mesh width h. Convergence to the continuous case is investigated. As illustrative examples we explicitly construct the Cauchy‐Kovalevskaya extensions of the discrete delta function and of a discretized exponential.

We establish dynamical localization for random Dirac operators on the d-dimensional lattice, with \(d\in \left\{ 1, 2, 3\right\} \), in the three usual regimes: large disorder, band edge and 1D. These operators are discrete versions of the continuous Dirac operators and consist in the sum of a discrete free Dirac operator with a random potential. The potential is a diagonal matrix formed by different scalar potentials, which are sequences of independent and identically distributed random variables according to an absolutely continuous probability measure with bounded density and of compact support. We prove the exponential decay of fractional moments of the Green function for such models in each of the above regimes, i.e., (j) throughout the spectrum at larger disorder, (jj) for energies near the band edges at arbitrary disorder and (jjj) in dimension one, for all energies in the spectrum and arbitrary disorder. Dynamical localization in theses regimes follows from the fractional moments method. The result in the one-dimensional regime contrast with one that was previously obtained for 1D Dirac model with Bernoulli potential.

Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on R N . The Dunkl operators are denoted by T j for 1 ≤ j ≤ N , and the Laplacian Δ κ = ∑ j = 1 N T j 2 . This paper finds the homogeneous harmonic polynomials annihilated by all T j for j > 2 . The structure constants with respect to the Gaussian and sphere inner products are computed. These harmonic polynomials are used to produce monogenic polynomials, those annihilated by a Dirac-type operator.

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.

We develop a rigorous algebraic–analytic framework for multidual complex numbers DCn within the setting of Clifford analysis and establish a comprehensive theory of hyperholomorphic multidual complex-valued functions. Our main contributions are (i) a fully coupled multidual Cauchy–Riemann system derived from the Dirac operator, yielding precise differentiability criteria; (ii) generalized conjugation laws and the associated norms that clarify metric and geometric structure; and (iii) explicit operator and kernel constructions—including generalized Cauchy kernels and Borel–Pompeiu-type formulas—that produce new representation theorems and regularity results. We further provide matrix–exponential and functional calculus representations tailored to DCn, which unify algebraic and analytic viewpoints and facilitate computation. The theory is illustrated through a portfolio of examples (polynomials, rational maps on invertible sets, exponentials, and compositions) and a solvable multidual boundary value problem. Connections to applications are made explicit via higher-order automatic differentiation (using nilpotent infinitesimals) and links to kinematics and screw theory, highlighting how multidual analysis expands classical holomorphic paradigms to richer, nilpotent-augmented coordinate systems. Our results refine and extend prior work on dual/multidual numbers and situate multidual hyperholomorphicity within modern Clifford analysis. We close with a concise summary of notation and a set of concrete open problems to guide further development.

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.

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 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.

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.

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.

In differential geometry of surfaces the Dirac operator appears intrinsically as a tool to address the immersion problem as well as in an extrinsic flavor (that comes with spin transformations to comformally transfrom immersions) and the two are naturally related. In this paper we consider a corresponding pair of discrete Dirac operators, the latter on discrete surfaces with polygonal faces and normals defined on each face, and show that many key properties of the smooth theory are preserved. In particular, the corresponding spin transformations, conformal invariants for them, and the relation between this operator and its intrinsic counterpart are discussed.