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In an open subset Ω of the complex plane ℂ the Morera theorem gives a simple looking necessary and sufficient condition for a continuous function f to be holomorphic in Ω. Namely the vanishing of all the integrals ∫γf(z) dz, where γ is an arbitrary Jordan curve in Ω whose interior also lies in Ω. The Morera problem consists in finding relatively small families Γ of Jordan curves such that...
There is a deep connection between the theory of several complex variables and complex Clifford analysis. We will use a Borel—Pompeiu formula in ℂn and the representation of holomorphic functions obtained in the context of Clifford analysis to study the inverse scattering problem for an n-dimensional Schròdinger-type equation. Equations are found for reconstructing the potential...
The main goal of the paper is to apply the theory of discrete analytic functions to the solution of Dirichlet problems for the Stokes and Navier—Stokes equations, respectively. The Cauchy—Riemann operator will be approximated by certain finite difference operators. Approximations of the classical T-operator as well as for the Bergman projections are constructed in such a way that the algebraic properties...
For a system A = (Ai,…, An) of linear operators whose real linear combinations have spectra contained in a fixed sector in ℂ and satisfy resolvent bounds there, functions f(A) of the system A of operators can be formed for monogenic functions f having decay at zero and infinity in a corresponding sector in ℝn+1....
In this paper we deal with Clifford-valued generalizations of several families of classical complex-analytic Eisenstein series and Poincaré series for discrete subgroups of Vahlen’s group in the framework of Clifford analysis.
We develop a function theory associated with non-elliptic, variable co-efficient operators of Dirac type on Lipschitz domains. Boundary behavior, global regularity, integral representation formulas, are studied by means of tools originating in PDE and harmonic analysis.
This paper is concerned with the classical Paley—Wiener theorems in one and several complex variables, the generalization to Euclidean spaces in the Clifford analysis setting and their proofs. We prove a new Shannon sampling theorem in the Clifford analysis setting.
We study weighted Bergman projections in the monogenic Bergman spaces of the real unit ball \mathbb{B} in ℝn. We extend results of Forelli—Rudin, Coifman—Rochberg, and Djrbashian to Clifford analysis. The main result is as follows: Let Pα be the orthogonal projection from the Hilbert space L2( \mathbb{B} , Cl0,n, dVα)...
We study Galpern—Sobolev equations with the help of a quaternionic operator calculus. Previous work is extended to the case of a variable dispersive term. We approximate the time derivative by forward finite differences. Solving the resulting stationary problems by means of a quaternionic calculus, we obtain representation formulae.
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