arXiv subjects

Rolf Soeren Krausshar

Publications and source records attributed to Rolf Soeren Krausshar.

14 recordsLinked to original sources

arXiv2023

Slice monogenic theta series

In this paper we introduce a generalization of theta series in the context of the slice monogenic function theory in $\mathbb{R}^{n+1}$ where me make use of the so-called $*$-exponential function in a hypercomplex variable. Together with the Eisenstein and Poincaré series that we introduced in a previous paper, the theta series construction in this paper completes the fundament of the basic theory of modular forms in the slice monogenic setting. We introduce a suitable generalized Poisson summation formula in this framework and we apply a properly adapted Fourier transform. As a direct application we prove a transformation formula for slice monogenic theta series. Then we introduce a family of conjugated theta functions. These are used to construct a slice monogenic generalization of the third power of the Dedekind eta function and of the modular discriminant. We also investigate their transformation behavior. Finally, we show that these theta series are special solutions to a generalization of the heat equation associated with the slice derivative. We round off by discussing the monogenic case.

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arXiv2023

Octonionic monogenic and slice monogenic Hardy and Bergman spaces

In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm $1$ in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity.

math.CV↗

arXiv2023

Variational principles in quaternionic analysis with applications to the stationary MHD equations

In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder's fixed point theorem. The advantage of the approach using Schauder's fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These consideration will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness.

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arXiv2023

On the generating functions and special functions associated with superoscillations

The aim of this paper is to study generating functions for the coefficients of the classical superoscillatory function associated with weak measurements. We also establish some new relations between the superoscillatory coefficients and many well-known families of special polynomials, numbers, and functions such as Bernstein basis functions, the Hermite polynomials, the Stirling numbers of second kind, and also the confluent hypergeometric functions. Moreover, by using generating functions, we are able to develop a recurrence relation and a derivative formula for the superoscillatory coefficients.

math.CA↗

arXiv2019

Conformal mappings revisited in the octonions and Clifford algebras of arbitrary dimension

In this paper we revisit the concept of conformality in the sense of Gauss in the context of octonions and Clifford algebras. We extend a characterization of conformality in terms of a system of partial differential equations and differential forms using special orthonormal sets of continuous functions that have been used before in the particular quaternionic setting. The aim is to describe to which higher dimensional algebras this characterization can exactly be extended and under which circumstances. It turns out to be crucial that this characterization requires a domain of definition that lies in a subalgebra that has the norm composition property and that is either associative (Clifford algebra case) or at least alternative (octonionic case). The orthonormal frames are elements of the spin group Spin(n+1). We round off by relating the nature of the orthonormal frames to the associated Möbius transformation which are related to SO(9,1) in the octonionic case and to the Ahlfors-Vahlen group in the case of a Clifford algebra.

math.CV↗

arXiv2019

Function Theories in Cayley-Dickson algebras and Number Theory

In the recent years a lot of effort has been made to extend the theory of hyperholomorphic functions from the setting of associative Clifford algebras to non-associative Cayley-Dickson algebras, starting with the octonions. An important question is whether there appear really essentially different features in the treatment with Cayley-Dickson algebras that cannot be handled in the Clifford analysis setting. Here we give one concrete example. Cayley-Dickson algebras namely admit the construction of direct analogues of CM-lattices, in particular lattices that are closed under multiplication. Canonical examples are lattices with components from the algebraic number fields $\mathbb{Q}[\sqrt{m_1},\ldots\sqrt{m_k}]$. Note that the multiplication of two non-integer lattice paravectors does not give anymore a lattice paravector in the Clifford algebra. In this paper we exploit the tools of octonionic function theory to set up an algebraic relation between different octonionic generalized elliptic functions which give rise to octonionic elliptic curves. We present formulas for the trace of the octonionic CM-division values.

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arXiv2013

Applications of hypercomplex automorphic forms in Yang-Mills gauge theories

In this paper we show how hypercomplex function theoretical objects can be used to construct explicitly self-dual SU(2)-Yang-Mills instanton solutions on certain classes of conformally flat 4-manifolds. We use a hypercomplex argument principle to establish a natural link between the fundamental solutions of $D Δf = 0$ and the second Chern class of the SU(2) principal bundles over these manifolds. The considered base manifolds of the bundles are not simply-connected, in general. Actually, this paper summarizes an extension of the corresponding results of Gürsey and Tze on a hyper-complex analytical description of SU(2) instantons. Furthermore, it provides an application of the recently introduced new classes of hypercomplex-analytic automorphic forms.

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arXiv2011

The Szegö metric associated to Hardy spaces of Clifford algebra valued functions and some geometric properties

In analogy to complex function theory we introduce a Szegö metric in the context of hypercomplex function theory dealing with functions that take values in a Clifford algebra. In particular, we are dealing with Clifford algebra valued functions that are annihilated by the Euclidean Dirac operator in $\mathbb{R}^{m+1}$. These are often called monogenic functions. As a consequence of the isometry between two Hardy spaces of monogenic functions on domains that are related to each other by a conformal map, the generalized Szegö metric turns out to have a pseudo-invariance under Möbius transformations. This property is crucially applied to show that the curvature of this metric is always negative on bounded domains. Furthermore, it allows us to establish that this metric is complete on bounded domains.

math.CV↗

arXiv2011

The Klein-Gordon operator on Möbius strip domains and the Klein bottle in $\mathbb{R}^n$

In this paper we present explicit formulas for the fundamental solution to the Klein-Gordon operator on some higher dimensional generalizations of the Möbius strip and the Klein bottle with values in distinct pinor bundles. The fundamental solution is described in terms of generalizations of the Weierstraß $\wp$-function that are adapted to the context of these geometries. The explicit formulas for the kernel then allow us to express all solutions to the homogeneous and inhomogeneous Klein-Gordon problem with given boundary data in the context of these manifolds. In the case of the Klein bottle we are able to describe all null solutions of the Klein-Gordon equation in terms of finite linear combinations of the fundamental solution and its partial derivatives.

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arXiv2011

A new class of hypercomplex analytic cusp forms

In this paper we deal with a new class of Clifford algebra valued automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. The forms that we consider are in the kernel of the operator $D Δ^{k/2}$ for some even $k \in {\mathbb{Z}}$. They will be called $k$-holomorphic Cliffordian automorphic forms. $k$-holomorphic Cliffordian functions are well equipped with many function theoretical tools. Furthermore, the real component functions have also the property that they are solutions to the homogeneous and inhomogeneous Weinstein equation. This function class includes the set of $k$-hypermonogenic functions as a special subset. While we have not been able so far to propose a construction for non-vanishing $k$-hypermonogenic cusp forms for $k \neq 0$, we are able to do so within this larger set of functions. After having explained their general relation to hyperbolic harmonic automorphic forms we turn to the construction of Poincaré series. These provide us with non-trivial examples of cusp forms within this function class. Then we establish a decomposition theorem of the spaces of $k$-holomorphic Cliffordian automorphic forms in terms of a direct orthogonal sum of the spaces of $k$-hypermonogenic Eisenstein series and of $k$-holomorphic Cliffordian cusp forms.

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arXiv2006

Dirac Type Operators for Arithmetic Subgroups of Generalized Modular Groups

Fundamental solutions of Dirac type operators are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of $\mathbb{R}^n$ by arithmetic subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented together with associated Eisenstein and Poincaré type series.

math.AP↗

arXiv2006

k-hypermonogenic automorphic forms

In this paper we deal with monogenic and $k$-hypermonogenic automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. Monogenic automorphic forms are exactly the 0-hypermonogenic automorphic forms. In the first part we establish an explicit relation between $k$-hypermonogenic automorphic forms and Maaß wave forms. In particular, we show how one can construct from any arbitrary non-vanishing monogenic automorphic form a Clifford algebra valued Maaß wave form. In the second part of the paper we compute the Fourier expansion of the $k$-hypermonogenic Eisenstein series which provide us with the simplest non-vanishing examples of $k$-hypermonogenic automorphic forms.

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arXiv2004

Generalized Analytic Automorphic Forms for some Arithmetic Congruence subgroups of the Vahlen group on the n-Dimensional Hyperbolic Space

This paper deals with a new analytic type of vector- and Clifford algebra valued automorphic forms in one and two vector variables. For hypercomplex generalizations of the classical modular group and their arithmetic congruence subgroups Eisenstein- and Poincaré type series that are annihilated by Dirac operators, and more generally, by iterated Dirac operators on the upper half-space of $R^n$ are discussed. In particular we introduce (poly-)monogenic modular forms on hypercomplex generalizations of the classical theta group.

math.NT↗

arXiv2002

Some Conformally Flat Spin Manifolds, Dirac Operators and Automorphic Forms

In this paper we study Clifford and harmonic analysis on some conformal flat spin manifolds. In particular we treat manifolds that can be parametrized by $U / Γ$ where $U$ is a simply connected subdomain of either $S^{n}$ or $R^{n}$ and $Γ$ is a Kleinian group acting discontinuously on $U$. Examples of such manifolds treated here include for example $RP^{n}$ and $S^{1}\times S^{n-1}$. Special kinds of Clifford-analytic automorphic forms associated to the different choices of $Γ$ are used to construct Cauchy kernels, Cauchy Integral formulas, Green's kernels and formulas together with Hardy spaces, Plemelj projection operators and Szegö kernels for $L^{p}$ spaces of hypersurfaces lying in these manifolds.

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