Spherical Harmonics, Splines, and Wavelets
This contribution substantially represents a geodetically relevant collection of particularly valuable material in the diverse approximation areas involving spherical harmonics, splines, and wavelets, thereby establishing a consistent and unified setup. T
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Spherical Harmonics, Splines, and Wavelets Definitoric Constituents, Strategic Perspectives, Specific Applicability and Applications Willi Freeden and Michael Schreiner
Contents 1 Motivation and Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Signals, Spherical Harmonics, and Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . 3 Zonal Kernel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Spline Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Wavelet Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Goal of Spherical Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Sampling Methods and Recovery Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Selective Publication List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract This contribution substantially represents a geodetically relevant collection of particularly valuable material in the diverse approximation areas involving spherical harmonics, splines, and wavelets, thereby establishing a consistent and
This chapter is part of the series Handbuch der Geodäsie, volume “Mathematische Geodäsie/Mathematical Geodesy”, edited by Willi Freeden, Kaiserslautern. W. Freeden () Geomathematics Group, Mathematics Department, University of Kaiserslautern, Kaiserslautern, Germany E-Mail: [email protected]; [email protected] M. Schreiner Institute for Computational Engineering, University of Applied Sciences of Technology NTB, Buchs, Switzerland E-Mail: [email protected] © Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020 W. Freeden (Hrsg.), Mathematische Geodäsie/Mathematical Geodesy, Springer Reference Naturwissenschaften, https://doi.org/10.1007/978-3-662-55854-6_101
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unified setup. The goal of the work is to preferably convince members from geodesy that spherically oriented approximation provides a rich mathematical cornucopia that has much to offer to a large palette of applications. Geomathematically it reflects both the approximate shape of the Earth’s surface and the typical satellite geometry of a low Earth orbiter (LEO). Our essential interest is in reconstruction and decomposition characteristics corresponding to different types of data on spheres and various observables naturally occurring in geodetic context, when efficient and economic
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