Fueter's theorem and its generalizations in Dunkl–Clifford analysis

In this article we establish an alternative proof of the generalized Fueter method presented in a former paper [Qian, T. and Sommen, F., 2003, Deriving harmonic functions in higher dimensional spaces. Zeitschrift fur Analysis und ihre Anwendungen, 22(2), 275–288] leading to the construction of special harmonic and monogenic functions in higher dimensions. At the same time, we also obtain a generalization of this result. §Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

In this article, we first suggest an alternative approach to extend the original Fueter's theorem in Dunkl–Clifford analysis to a version of the higher order case. We then use this result to prove a generalized version of Fueter's theorem with an extra homogeneous Dunkl-monogenic polynomial P n (x 0, x ) in instead of the classical factor P n ( x ) in ℝ d .

This paper presents results on some boundary value problems for holomorphic functions of several complex variables in polydomains. The Cauchy kernel is one of the significant tools for solving the Riemann and the Riemann-Hilbert boundary value problems for holomorphic functions as well as for establishment of the connection between them. For polydomains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in polydomains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. The general integral representation formulas for the functions, holomorphic in polydomains, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary polydomains is given and an exact, yet compact way of notation for holomorphic functions in arbitrary polydomains is introduced and applied. The Riemann jump problem and the Riemann-Hilbert problem are solved for holomorphic functions of several complex variables with the unit torus as the jump manifold. The higher-dimensional Plemelj-Sokhotzki formula for holomorphic functions in polydomains is established. The canonical functions of the Riemann problem for polydomains are represented and applied in order to construct solutions for both of the homogeneous and inhomogeneous problems. For all three boundary value problems, well-posed formulations are given which does not demand more solvability conditions than in the one variable case. The connection between the Riemann and the Riemann-Hilbert problem for polydomains is proven. Thus contrary to earlier research the results are similar to the respective ones for just one variable.

Special Bi-Axial Monogenic Functions

In Bures et al. (Elements of quaternionic analysis and Radon transform, 2009), the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on \({\mathbb {R}}^{n+1}\) using the Radon and dual Radon transforms. The main aim of this paper is to further develop this approach. In fact, the Radon transform for functions with values in the Clifford algebra \({\mathbb {R}}_n\) is mapping solutions of the generalized Cauchy–Riemann equation, i.e., monogenic functions, to a parametric family of holomorphic functions with values in \({\mathbb {R}}_n\) and, analogously, the dual Radon transform is mapping parametric families of holomorphic functions as above to monogenic functions. The parametric families of holomorphic functions considered in the paper can be viewed as a generalization of the so-called slice monogenic functions. An important part of the problem solved in the paper is to find a suitable definition of the function spaces serving as the domain and the target of both integral transforms.

We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph yields more instructive proofs of discrete analogs of several classical theorems and even new results. We provide discrete counterparts of fundamental concepts in complex analysis such as holomorphic functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss discrete versions of important basic theorems such as Green's identities and Cauchy's integral formulae. For the first time, we discretize Green's first identity and Cauchy's integral formula for the derivative of a holomorphic function. In this paper, we focus on planar quad-graphs, but we would like to mention that many notions and theorems can be adapted to discrete Riemann surfaces in a straightforward way. In the case of planar parallelogram-graphs with bounded interior angles and bounded ratio of side lengths, we construct a discrete Green's function and discrete Cauchy's kernels with asymptotics comparable to the smooth case. Further restricting to the integer lattice of a two-dimensional skew coordinate system yields appropriate discrete Cauchy's integral formulae for higher order derivatives.

Contragenic functions are defined to be reduced-quaternion-valued harmonic functions which are orthogonal to all monogenic and antimonogenic functions in the $$L^2$$ L 2 norm of a given domain. The parallelism between the spaces of contragenic functions in the interior and exterior of the unit sphere in $$\mathbb {R}^3$$ R 3 is described in detail. Bergman reproducing kernels for the spaces of contragenic functions are given, mirroring the corresponding kernels for the spaces of vector parts of monogenic functions. Numerical examples are given showing the accuracy of truncations of the integral kernels. A striking duality is observed between the basic interior contragenic functions and the vector parts of exterior monogenic functions, and vice versa.

We lay the foundations of Fatou theory in one and several complex variables. We describe the main contributions contained in E. M. Stein’s book Boundary Behavior of Holomorphic Functions, published in 1972 and still a source of inspiration. We also give an account of his contributions to the study of the boundary behavior of harmonic functions. The point of this paper is not simply to exposit well-known ideas. Rather, we completely reorganize the subject in order to bring out the profound contributions of E. M. Stein to the study of the boundary behavior both of holomorphic and harmonic functions in one and several variables. In an appendix, we provide a self-contained proof of a new result which is relevant to the differentiation of integrals, a topic which, as witnessed in Stein’s work, and especially by the aforementioned book, has deep connections with the boundary behavior of harmonic and holomorphic functions.

We describe several different representations of nilpotent step two Lie groups in spaces of monogenic Clifford-valued functions. We are inspired by the classic representation of the Heisenberg group in the Segal-Bargmann space of holomorphic functions. Connections with quantum mechanics are described.

In this paper we show a version of the Fueter mapping theorem that can be stated in integral form based on the Cauchy formulas for slice monogenic (or slice regular) functions. More precisely, given a holomorphic function f of a paravector variable, we generate a monogenic function by an integral transform whose kernel is particularly simple. This procedure allows us to define a functional calculus for n-tuples of commuting operators (called ℱ-functional calculus) based on a new notion of spectrum, called ℱ-spectrum, for the n-tuples of operators. Analogous results are shown for the quaternionic version of the theory and for the related ℱ-functional calculus. Copyright © 2010 John Wiley & Sons, Ltd.

We discuss the Fourier–Borel transform for the dual of spaces of monogenic functions. This transform may be seen as a restriction of the classical Fourier–Borel transform for holomorphic functionals, and it transforms spaces of monogenic functionals into quotients of spaces of entire holomorphic functions of exponential type. We prove that, for the Lie ball, these quotient spaces are isomorphic to spaces of monogenic functions of exponential type.

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.

A finite group W generated by reflections is called a finite reflection group and have been studied deeply in different important contexts such as geometry and group theory, matrix algebra and Lie Theory. In this paper we study a class of fuzzy subgroups of finite reflection groups which are called parabolic fuzzy subgroups using preferential equality. In the classical case parabolic subgroups are associated with a set of generators, each a simple reflection, indexed by elements of subsets of a fixed root system Δ . We establish a one-to-one correspondence between the class of fuzzy parabolic subgroups of W and the class of fuzzy subsets of S where S is a set of simple reflections associated with Δ .

On elliptic Calogero–Moser systems for complex crystallographic reflection groups

In recent literature, moonshine has been explored for some groups beyond the Monster, for example the sporadic O’Nan and Thompson groups. This collection of examples may suggest that moonshine is a rare phenomenon, but a fundamental and largely unexplored question is how general the correspondence is between modular forms and finite groups. For every finite group G, we give constructions of infinitely many graded infinite-dimensional $$\mathbb {C}[G]$$ -modules where the McKay–Thompson series for a conjugacy class [g] is a weakly holomorphic modular function properly on $$\varGamma _0({{\mathrm{ord}}}(g))$$ . As there are only finitely many normalized Hauptmoduln, groups whose McKay–Thompson series are normalized Hauptmoduln are rare, but not as rare as one might naively expect. We give bounds on the powers of primes dividing the order of groups which have normalized Hauptmoduln of level $${{\mathrm{ord}}}(g)$$ as the graded trace functions for any conjugacy class [g], and completely classify the finite abelian groups with this property. In particular, these include $$(\mathbb {Z}/ 5 \mathbb {Z})^5$$ and $$(\mathbb {Z}/ 7 \mathbb {Z})^4$$ , which are not subgroups of the Monster.