Hypercomplex Functional Calculi and Function Theories in Clifford Analysis

The monogenic functional calculus is a means of constructing functions of a finite system of bounded or unbounded operators. For bounded noncommuting operators, a polynomial function produces a polynomial of the operators in which all possible operator orderings are equally weighted. For example, for two bounded selfadjoint operators A 1, A 2, the operator \(\frac{1} {2}(A_{1}A_{2} + A_{2}A_{1})\) is associated with polynomial \(z\mapsto z_{1}z_{2}\) in two variables by the monogenic functional calculus. The same formula applies just when the spectrum \(\sigma (\xi _{1}A_{1} +\xi _{2}A_{2})\) of a finite linear combination \(\xi _{1}A_{1} +\xi _{2}A_{2}\) of A 1 and A 2 is a subset of the real numbers for any \(\xi _{1},\xi _{2} \in \mathbb{R}\). The article begins with a discussion of Clifford algebras and Clifford analysis and points out the connection with Weyl’s functional calculus for a finite system of selfadjoint operators in Hilbert space. The construction of the Cauchy kernel in the monogenic functional calculus is achieved with the plane wave decomposition of the Cauchy kernel in Clifford analysis. The formulation applies to any finite system \((A_{1},\ldots ,A_{n})\) of closed unbounded operators such that the spectrum \(\sigma \left (\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}\right )\) of the closed operator \(\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}\) is contained in a two-sided sector in \(\mathbb{C}\) for almost all \(\xi \in \mathbb{R}^{n}\). The connection with harmonic analysis and irregular boundary value problems is emphasized. The monogenic functional calculus may be applied to the finite commuting system \(D_{\Sigma } = i(D_{1},\ldots ,D_{n})\) of differentiating operators on a Lipschitz surface \(\Sigma \) so that an H ∞-functional calculus \(f\mapsto f(D_{\Sigma })\) is obtained for functions f uniformly bounded and left monogenic in a sector containing almost all tangent planes to \(\Sigma \).KeywordsBounded Linear OperatorClifford AlgebraFunctional CalculusSelfadjoint OperatorMonogenic FunctionThese 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, using the recently discovered notion of the S-spectrum, we prove the spectral theorem for a bounded or unbounded normal operator on a Clifford module (i.e., a two-sided Hilbert module over a Clifford algebra based on units that all square to be \(-1\)). Moreover, we establish the existence of a Borel functional calculus for bounded or unbounded normal operators on a Clifford module. Towards this end, we have developed many results on functional analysis, operator theory, integration theory and measure theory in a Clifford setting which may be of an independent interest. Our spectral theory is the natural spectral theory for the Dirac operator on manifolds in the non-self adjoint case. Moreover, our results provide a new notion of spectral theory and a Borel functional calculus for a class of n-tuples of commuting or non-commuting operators on a real or complex Hilbert space. Moreover, for a special class of n-tuples of operators on a Hilbert space our results provide a complementary functional calculus to the functional calculus of J. L. Taylor.

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.

The $S$-functional calculus for slice hyperholomorphic functions generalizes the Riesz-Dunford-functional calculus for holomorphic functions to quaternionic linear operators and to $n$-tuples of noncommuting operators. For an unbounded closed operator, it is defined, as in the classical case, using an appropriate transformation and the $S$-functional calculus for bounded operators. This is however only possible if the $S$-resolvent set of the operator contains a real point. In this paper, we define the $S$-functional calculus directly via a Cauchy integral, which allows us to consider also operators whose resolvent sets do not contain real points. We show that the product rule and the spectral mapping theorem also hold true with this definition and that the $S$-functional calculus is compatible with polynomials, although polynomials are not included in the set of admissible functions if the operator is unbounded. We also prove that the $S$-functional calculus is able to create spectral projections. In order to do this, we remove another assumption usually made: we do not assume that admissible functions are defined on slice domains. Instead, we also consider functions that are defined on not necessarily connected sets. This leads to an unexpected phenomenon: the $S$-functional calculi for left and right slice hyperholomorphic functions become inconsistent and give different operators for functions that are both left and right slice hyperholomorphic. We show that any such function is the sum of a locally constant and an intrinsic function. For intrinsic functions both functional calculi agree, but for locally constant functions they must give different operators unless any spectral projection commutes with arbitrary scalars.

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.

In this article we exploit some general results on the tensor product of free resolutions to deduce important new algebraic properties of biregular functions in 2n quaternionic variables. In particular, we are able to construct a minimal resolution for the associated module and compute all the relevant graded Betti numbers. §Dedicated to Richard Delanghe on the occasion of his 65th birthday. This article is dedicated, with friendship and admiration, to Professor Richard Delanghe, who has done so much, as a mathematician and as a leader, to further the work on Clifford Analysis around the world.

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.

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 develop a plane system of first-order differential equations, describing nullsolutions of the Dirac operator in biaxially symmetric domains. The solutions to the equations are of pseudoanalytic type and may be used to construct on orthogonal basis for the spaces of so-called spherical monogenics. which are related to spherical harmonics with spin. Finally, after certain restrictions on dimensins. We give a complete characterization of these pscudoanalytic functions in terms of holomorphic ones. This paper may thus ge considered as a bridge between Clifford analysis and the theory of pscudoanalytic functions.

This survey is intended as an overview of discrete Clifford analysis and its current developments. Since in the discrete case one has to replace the partial derivative with two difference operators, backward and forward partial difference, one needs to modify the main tools for a development of a discrete function theory, such as the replacement of a real Clifford algebra by a complexified Clifford algebra or of the classic Weyl relations by so-called S-Weyl relations. The main results, like Cauchy integral formula, Fischer decomposition, CK-extension, and Taylor series, will be derived. To give a better idea of the differences between the discrete and continuous case, this chapter contains the problem of discrete Hardy spaces as well as some discrete objects which do not have an equivalent object in continuous Clifford analysis, such as the CK-extension of a discrete Delta function.

We define a new modified basis which is an association of two bases, e 1 and 2 . We give an expression of the form = 0 + 0 , where 0 is a real number and 0 is a complex number on three-dimensional real skew field. And we research the properties of regular functions with values in ternary field and reduced quaternions by Clifford analysis.

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 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.

Rotations in discrete Clifford analysis

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.