Rolf Sören Krausshar

Generalized analytic automorphic forms in hypercomplex spaces

Introduction.- 1. Function Theory in Hypercomplex Spaces.- 2. Clifford-analytic Eisenstein Series Associated to Translation Groups.- 3. Clifford-analytic Modular Forms.- Bibliography.- Index.

Szegö and Polymonogenic Bergman Kernels for Half-Space and Strip Domains, and Single-Periodic Functions in Clifford Analysis

In this paper, we consider half-space domains (semi-infinite in one of the dimensions) and strip domains (finite in one of the dimensions) in real Euclidean spaces of dimension at least 2. The Szegö reproducing kernel for the space of monogenic and square integrable functions on a strip domain is obtained in closed form as a monogenic single-periodic function, viz a monogenic cosecant. The relationship between the Szegö and Bergman kernel for monogenic functions in a strip domain is explicitated in the transversally Fourier transformed setting. This relationship is then generalised to the polymonogenic Bergman case. Finally, the half-space case is considered specifically and the simplifications are pointed out.

Hilbert spaces of solutions to polynomial Dirac equations, Fourier transforms and reproducing Kernel functions for cylindrical domains

In this paper, we consider L2 spaces of functions that satisfy polynomial Dirac equations. Fourier transformation methods and methods from harmonic analysis are then applied to treat Hilbert spaces of Clifford algebra valued functions that are either square-integrable over a cylinder or square-integrable over its boundary, and which satisfy in its interior the generalized Cauchy-Riemann system. In particular, explicit representation formulas for the Bergman and Szegö reproducing kernel of several types of cylindrical domains are developed.

Clifford and Harmonic Analysis on Cylinders and Tori

Cotangent type functions in Rn are used to construct Cauchy kernels and Green kernels on the conformally flat manifolds Rn/Zk where 1 < = k ? M. Basic properties of these kernels are discussed including introducing a Cauchy formula, Greens formula, Cauchy transform, Poisson kernel, Szego kernel and Bergman kernel for certain types of domains. Singular Cauchy integrals are also introduced as are associated Plemelj projection operators. These in turn are used to study Hardy spaces in this context. Also the analogues of Calderon-Zygmund type operators are introduced in this context, together with singular Clifford holomorphic, or monogenic, kernels defined on sector domains in the context of cylinders. Fundamental differences in the context of the n-torus arising from a double singularity for the generalized Cauchy kernel on the torus are also discussed.

Representation Formulas for the General Derivatives of the Fundamental Solution to the Cauchy-Riemann Operator in Clifford Analysis and Applications

In this paper, we discuss several essentially different formulas for the general derivatives q(n)(z) of the fundamental solution of the Cauchy-Riemann operator in Clifford Analysis, upon,which - among other important applications - the theory of monogenic Eisenstein series is based. Using Fourier and plane wave decomposition methods, we obtain a compact integral representation formula over a half-space, which also lends itself to establish upper bounds on the values parallel toq(n)(z)parallel to. A second formula that we discuss is a recurrence formula involving permutational products of hypercomplex variables by which these estimates can be obtained immediately. We further prove several formulas for q(n)(z) in terms of explicit, non-recurrent finite sums, leading themselves to further representations in terms of permutational products but using different and fewer hypercomplex variables than used in the recurrence relations. Summing up a fixed q(n). over a given discrete lattice leads to a variant of the Riemann zeta function. We apply one of the closed representation formulas for q(n)(z) to express this variant of the Riemann zeta function as a finite sum of real-valued Dirichlet series.

The Bergman Kernels for the half-ball and for fractional wedge-shaped domains in Clifford Analysis

Abstract We compute the Bergman reproducing kernel for monogenic functions for half-ball, more general orthogonal ball sectors, and for fractional wedge domains. In the results we obtain the terms to be expected from analogy with complex analysis, viz in the first case the Bergman kernels for the half-space and the entire ball, and in the second the sum of rotated half-space Bergman kernels, but in both cases there also occur supplementary, purely hypercomplex correction terms. Finally, applying a periodisation argument we obtain closed and explicit formulas for the Bergman kernels of wedge shaped domains that are rectangularly bounded.

Applications of the maximum term and the central index in the asymptotic growth analysis of entire solutions to higher dimensional polynomial Cauchy-Riemann equations

In this article we deal with entire Clifford algebra valued solutions to polynomial Cauchy–Riemann equations in higher dimensional Euclidean spaces. We introduce generalizations of the maximum term and the central index within the context of this family of elliptic partial differential equations. These notions enable us to perform a basic study of the asymptotic growth behaviour of entire solutions to these systems. In this article we set up generalizations of some classical fundamental results of Wiman and Valirons theory. Our results then enable us to get some insight on the structure of the solutions of a certain class of higher dimensional PDE.

Automorphic forms in Clifford analysis

In this paper we discuss the possibility of extending the classical theory of automorphic forms to Clifford analysis within the framework of its regularity concepts. To several weights we construct with special functions from Clifford analysis Clifford-valued automorphic forms in a hypercomplex variable that are solutions of iterated homogeneous Dirac equations in

On a new type of Eisenstein series in Clifford analysis

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Poincaré Series in Clifford Analysis

, in particular, generalizations of the classical Eisenstein series and Poincaré series on the upper half-space, on spatial octants and on the unit ball within classes of polymonogenic functions.