TY - CHAP A1 - Ebert, Svend A1 - Bernstein, Swanhild A1 - Cerejeiras, Paula A1 - Kähler, Uwe ED - Gürlebeck, Klaus ED - Könke, Carsten T1 - NONZONAL WAVELETS ON S^N N2 - In the present article we will construct wavelets on an arbitrary dimensional sphere S^n due the approach of approximate Identities. There are two equivalently approaches to wavelets. The group theoretical approach formulates a square integrability condition for a group acting via unitary, irreducible representation on the sphere. The connection to the group theoretical approach will be sketched. The concept of approximate identities uses the same constructions in the background, here we select an appropriate section of dilations and translations in the group acting on the sphere in two steps. At First we will formulate dilations in terms of approximate identities and than we call in translations on the sphere as rotations. This leads to the construction of an orthogonal polynomial system in L²(SO(n+1)). That approach is convenient to construct concrete wavelets, since the appropriate kernels can be constructed form the heat kernel leading to the approximate Identity of Gauss-Weierstra\ss. We will work out conditions to functions forming a family of wavelets, subsequently we formulate how we can construct zonal wavelets from a approximate Identity and the relation to admissibility of nonzonal wavelets. Eventually we will give an example of a nonzonal Wavelet on $S^n$, which we obtain from the approximate identity of Gauss-Weierstraß. KW - Angewandte Informatik KW - Angewandte Mathematik KW - Architektur KW - Computerunterstütztes Verfahren KW - Computer Science Models in Engineering; Multiscale and Multiphysical Models; Scientific Computing Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20170314-28406 UR - http://euklid.bauing.uni-weimar.de/ikm2009/paper.html SN - 1611-4086 ER - TY - JOUR A1 - Cerejeiras, Paula A1 - Kähler, Uwe A1 - Legatiuk, Anastasiia A1 - Legatiuk, Dmitrii T1 - Discrete Hardy Spaces for Bounded Domains in Rn JF - Complex Analysis and Operator Theory N2 - Discrete function theory in higher-dimensional setting has been in active development since many years. However, available results focus on studying discrete setting for such canonical domains as half-space, while the case of bounded domains generally remained unconsidered. Therefore, this paper presents the extension of the higher-dimensional function theory to the case of arbitrary bounded domains in Rn. On this way, discrete Stokes’ formula, discrete Borel–Pompeiu formula, as well as discrete Hardy spaces for general bounded domains are constructed. Finally, several discrete Hilbert problems are considered. KW - Dirac-Operator KW - Randwertproblem KW - Funktionentheorie KW - discrete Dirac operator KW - discrete monogenic functions KW - discrete boundary value problems Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20210804-44746 UR - https://link.springer.com/article/10.1007/s11785-020-01047-6 VL - 2021 IS - Volume 15, article 4 SP - 1 EP - 32 PB - Springer CY - Heidelberg ER - TY - CHAP A1 - Cerejeiras, Paula T1 - Hyperbolic Qp-scales N2 - The Qp-scales were first introduced in [1] as interpolation spaces between the Bloch and Dirichlet spaces in the complex space. ... However, such treatment presents the disadvantage of only considering the Euclidean case. In order to obtain an approach to homogeneous hyperbolic manifolds, the projective model of Gel'fand was retaken in [2]. With the help of a convenient fundamental solution for the hyperbolic (homogeneous of degree ®) D® (see [5]) it was introduced in [7] and [3] equivalent Qp scales for homogeneous hyperbolic spaces. In this talk we shall present and study some properties of this hyperbolic scale. KW - Quaternion Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20111215-2853 ER -