## Fast Approximation on the 2-Sphere by Optimally Localized Approximate Identities

• We introduce a method to construct approximate identities on the 2-sphere which have an optimal localization. This approach can be used to accelerate the calculations of approximations on the 2-sphere essentially with a comparably small increase of the error. The localization measure in the optimization problem includes a weight function which can be chosen under some constraints. For each choice of weight function existence and uniqueness of the optimal kernel are proved as well as the generation of an approximate identity in the bandlimited case. Moreover, the optimally localizing approximate identity for a certain weight function is calculated and numerically tested.

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Author: Volker Michel urn:nbn:de:hbz:386-kluedo-14710 Schriften zur Funktionalanalysis und Geomathematik (29) Preprint English 2006 2006 Technische Universität Kaiserslautern 2006/12/13 Approximative Identität; Konvergenz; schnelle Approximationapproximate identity; convergence; fast approximation Approximation ; Faltung; L2-Approximation ; Lokalisation; Multivariate Approximation ; Numerische Mathematik; Polynomapproximation ; Sphäre Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 41-XX APPROXIMATIONS AND EXPANSIONS (For all approximation theory in the complex domain, see 30E05 and 30E10; for all trigonometric approximation and interpolation, see 42A10 and 42A15; for numerical approximation, see 65Dxx) / 41Axx Approximations and expansions / 41A35 Approximation by operators (in particular, by integral operators) 41-XX APPROXIMATIONS AND EXPANSIONS (For all approximation theory in the complex domain, see 30E05 and 30E10; for all trigonometric approximation and interpolation, see 42A10 and 42A15; for numerical approximation, see 65Dxx) / 41Axx Approximations and expansions / 41A55 Approximate quadratures 42-XX FOURIER ANALYSIS / 42Cxx Nontrigonometric harmonic analysis / 42C25 Uniqueness and localization for orthogonal series 65-XX NUMERICAL ANALYSIS / 65Dxx Numerical approximation and computational geometry (primarily algorithms) (For theory, see 41-XX and 68Uxx) / 65D15 Algorithms for functional approximation 86-XX GEOPHYSICS [See also 76U05, 76V05] / 86-08 Computational methods Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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