Nonorthogonal Expansions on the Sphere

  • Discrete families of functions with the property that every function in a certain space can be represented by its formal Fourier series expansion are developed on the sphere. A Fourier series type expansion is obviously true if the family is an orthonormal basis of a Hilbert space, but it also can hold in situations where the family is not orthogonal and is overcomplete. Furthermore, all functions in our approach are axisymmetric (depending only on the spherical distance) so that they can be used adequately in (rotation) invariant pseudodifferential equations on the frames (ii) Gauss- Weierstrass frames, and (iii) frames consisting of locally supported kernel functions. Abel-Poisson frames form families of harmonic functions and provide us with powerful approximation tools in potential theory. Gauss-Weierstrass frames are intimately related to the diffusion equation on the sphere and play an important role in multiscale descriptions of image processing on the sphere. The third class enables us to discuss spherical Fourier expansions by means of axisymmetric finite elements.

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Metadaten
Author:Willi Freeden, Michael Schreiner
URN (permanent link):urn:nbn:de:hbz:386-kluedo-7048
Serie (Series number):Berichte der Arbeitsgruppe Technomathematik (AGTM Report) (97)
Document Type:Preprint
Language of publication:English
Year of Completion:1993
Year of Publication:1993
Publishing Institute:Technische Universität Kaiserslautern
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:510 Mathematik

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