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Fri, 02 Nov 2007 15:03:29 +0100Fri, 02 Nov 2007 15:03:29 +0100Locally Supported Approximate Identities on the Unit Ball
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1837
We present a constructive theory for locally supported approximate identities on the unit ball in \(\mathbb{R}^3\). The uniform convergence of the convolutions of the derived kernels with an arbitrary continuous function \(f\) to \(f\), i.e. the defining property of an approximate identity, is proved. Moreover, an explicit representation for a class of such kernels is given. The original publication is available at www.springerlink.comMuhammad Akram; Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1837Sun, 11 Feb 2007 15:03:29 +0100Fast Approximation on the 2-Sphere by Optimally Localized Approximate Identities
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1822
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.Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1822Wed, 13 Dec 2006 13:46:52 +0100On the Multiscale Solution of Satellite Problems by Use of Locally Supported Kernel Functions Corresponding to Equidistributed Data on Spherical Orbits
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1117
Being interested in (rotation-)invariant pseudodi erential equations of satellite problems corresponding to spherical orbits, we are reasonably led to generating kernels that depend only on the spherical distance, i. e. in the language of modern constructive approximation form spherical radial basis functions. In this paper approximate identities generated by such (rotation-invariant) kernels which are additionally locally supported are investigated in detail from theoretical as well as numerical point of view. So-called spherical di erence wavelets are introduced. The wavelet transforms are evaluated by the use of a numerical integration rule, that is based on Weyl's law of equidistribution. This approximate formula is constructed such that it can cope with millions of (satellite) data. The approximation error is estimated on the orbital sphere. Finally, we apply the developed theory to the problems of satellite-to-satellite tracking (SST) and satellite gravity gradiometry (SGG).Willi Freeden; Kerstin Hessepreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1117Mon, 21 Aug 2000 00:00:00 +0200