203 research outputs found
3D-mappings by means of monogenic functions and their approximation
We consider quasi-conformal 3D-mappings realized by hypercomplex
di erentiable (monogenic) functions and their polynomial
approximation. Main tools are the series development
of monogenic functions in terms of hypercomplex variables
and the generalization of L. V. Kantorovich's approach for
approximating conformal mappings by powers of a small parameter.
.This work was partially supported by the R&D Unit Matematica e Aplicacoes (UIMA) of the University of Aveiro, through Portuguese Foundation for Science and Technology (FCT
Regular vs. classical M\"obius transformations of the quaternionic unit ball
The regular fractional transformations of the extended quaternionic space
have been recently introduced as variants of the classical linear fractional
transformations. These variants have the advantage of being included in the
class of slice regular functions, introduced by Gentili and Struppa in 2006, so
that they can be studied with the useful tools available in this theory. We
first consider their general properties, then focus on the regular M\"obius
transformations of the quaternionic unit ball B, comparing the latter with
their classical analogs. In particular we study the relation between the
regular M\"obius transformations and the Poincar\'e metric of B, which is
preserved by the classical M\"obius transformations. Furthermore, we announce a
result that is a quaternionic analog of the Schwarz-Pick lemma.Comment: 14 page
Harmonic analysis on the Möbius gyrogroup
In this paper we propose to develop harmonic analysis on the Poincaré ball , a model of the n-dimensional real hyperbolic space. The Poincaré ball is the open ball of the Euclidean n-space with radius , centered at the origin of and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in . For any and an arbitrary parameter we study the -translation, the -convolution, the eigenfunctions of the -Laplace-Beltrami operator, the -Helgason Fourier transform, its inverse transform and the associated Plancherel's Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when the resulting hyperbolic harmonic analysis on tends to the standard Euclidean harmonic analysis on , thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on
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