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Special Functions in Mathematical Geosciences: An Attempt at a Categorization

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Handbook of Geomathematics

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Abstract

This chapter reports on the current activities and recent progress in the field of special functions of mathematical geosciences. The chapter focuses on two major topics of interest, namely, trial systems of polynomial (i.e., spherical harmonics) and polynomially based (i.e., zonal kernel) type. A fundamental tool is an uncertainty principle, which gives appropriate bounds for both the qualification and quantification of space and frequency (momentum) localization of the special (kernel) function under consideration. The essential outcome is a better understanding of constructive approximation in terms of zonal kernel functions such as splines and wavelets.

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References

  • Antoine JP, Vandergheynst P (1999) Wavelets on the 2-sphere: a group-theoretic approach. Appl Comput Harmon Anal 7:1–30

    Article  MathSciNet  Google Scholar 

  • Antoine JP, Demanet L, Jaques L, Vandergheynst P (2002) Wavelets on the sphere: implementations and approximations. Appl Comput Harm Anal 13:177–200

    Article  MATH  Google Scholar 

  • Backus GE (1966) Potentials for tangent tensor fields on spheroids. Arch Ration Mech Anal 22:210–252

    Article  MathSciNet  MATH  Google Scholar 

  • Backus GE (1967) Converting vector and tensor equations to scalar equations in spherical coordinates. Geophys J R Astron Soc 13:61–101

    Article  Google Scholar 

  • Backus GE (1986) Poloidal and toroidal fields in geomagnetic field modelling. Rev Geophys 24:75–109

    Article  MathSciNet  Google Scholar 

  • Clebsch RFA (1861) Ãœber eine Eigenschaft der Kugelfunktionen. Crelles J 60:343

    MathSciNet  Google Scholar 

  • Dahlke S, Maass P (1996) Continuous wavelet transforms with applications to analyzing functions on spheres. J Fourier Anal Appl 2(4):379–396

    MathSciNet  MATH  Google Scholar 

  • Dahlke S, Dahmen W, Schmitt W, Weinreich I (1995) Multiresolution analysis and wavelets on S 2. Numer Funct Anal Optim 16(1–2):19–41

    Article  MathSciNet  MATH  Google Scholar 

  • de Laplace PS (1785) Theorie des attractions des sphéroides et de la figure des planètes. Mèm de l’Acad, Paris

    Google Scholar 

  • Freeden W (1981) On spherical spline interpolation and approximation. Math Methods Appl Sci 3:551–575

    Article  MathSciNet  MATH  Google Scholar 

  • Freeden W (1999) Multiscale modelling of spaceborne geodata. B.G. Teubner, Leipzig

    MATH  Google Scholar 

  • Freeden W, Gutting M (2013) Special functions of mathematical (geo-)physics. Birkhäuser, Basel

    Book  MATH  Google Scholar 

  • Freeden W, Michel V (2004) Multiscale potential theory (with applications to geoscience). Birkhäuser Verlag, Boston/Basel/Berlin

    Book  MATH  Google Scholar 

  • Freeden W, Schreiner M (1995) Non-orthogonal expansions on the sphere. Math Methods Appl Sci 18:83–120

    Article  MathSciNet  MATH  Google Scholar 

  • Freeden W, Schreiner M (2007) Biorthogonal locally supported wavelets on the sphere based on zonal kernel functions. J Fourier Anal Appl 13:693–709

    Article  MathSciNet  MATH  Google Scholar 

  • Freeden W, Schreiner M (2009) Spherical functions of mathematical geosciences-a scalar, vectorial, and tensorial setup. Springer, Berlin/Heidelberg

    MATH  Google Scholar 

  • Freeden W, Windheuser U (1996) Spherical wavelet transform and its discretization. Adv Comput Math 5:51–94

    Article  MathSciNet  MATH  Google Scholar 

  • Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere (with applications to geomathematics). Oxford/Clarendon, Oxford

    MATH  Google Scholar 

  • Gauß CF(1838) Allgemeine Theorie des Erdmagnetismus, Resultate aus den Beobachtungen des magnetischen Vereins. Göttinger Magnetischer Verein, Leipzig

    Google Scholar 

  • Heine E (1878) Handbuch der Kugelfunktionen. Verlag G. Reimer, Berlin

    Google Scholar 

  • Holschneider M (1996) Continuous wavelet transforms on the sphere. J Math Phys 37:4156–4165

    Article  MathSciNet  MATH  Google Scholar 

  • Laín Fernández N (2003) Polynomial bases on the sphere. PhD thesis, University of Lübeck

    Google Scholar 

  • Legendre AM (1785) Recherches sur l’attraction des sphèroides homogènes. Mèm math phys près à l’Acad Aci par divers savantes 10:411–434

    Google Scholar 

  • Lyche T, Schumaker L (2000) A multiresolution tensor spline method for fitting functions on the sphere. SIAM J Sci Comput 22:724–746

    Article  MathSciNet  MATH  Google Scholar 

  • Maxwell JC (1891) A treatise on electricity and magnetism (1873, 1881, 1891) Bde 1 u. 2 Ungekürzter Nachdruck der letzten Auflage 1891, Carnegie Mellon University, Dover, 1954. (Vol 2. Available at http://posner.library.cmu/Posner/books/book.cgi?call=537_M46T_1873_VOL_2)

  • Morse PM, Feshbach H (1953) Methods of theoretical physics. McGraw-Hill, New York

    MATH  Google Scholar 

  • Narcowich FJ, Ward JD (1996) Nonstationary wavelets on the m-sphere for scattered data. Appl Comput Harmon Anal 3:324–336

    Article  MathSciNet  MATH  Google Scholar 

  • Neumann F (1887) Vorlesungen über die Theorie des Potentials und der Kugelfunktionen. Teubner, Leipzig, pp 135–154

    Google Scholar 

  • Potts D, Tasche M (1995) Interpolatory wavelets on the sphere. In: Chui CK, Schumaker LL (eds) Approximation theory VIII. World Scientific, Singapore, pp 335–342

    Google Scholar 

  • Schröder P, Sweldens W (1995) Spherical wavelets: efficiently representing functions on the sphere. In: Computer graphics proceedings (SIGGRAPH95). ACM, New York, pp 161–175

    Google Scholar 

  • Svensson SL (1983) Pseudodifferential operators-a new approach to the boundary value problems of physical geodesy. Manus Geod 8:1–40

    MathSciNet  MATH  Google Scholar 

  • Sylvester T (1876) Note on spherical harmonics. Phil Mag II 291, 400

    Google Scholar 

  • Wahba G (1981) Spline interpolation and smoothing on the sphere. SIAM J Sci Stat Comput 2: 5–16 (also errata: SIAM J Sci Stat Comput 3:385–386)

    Google Scholar 

  • Wahba G (1990) Spline models for observational data. In: CBMS-NSF regional conference series in applied mathematics, vol 59. SIAM, Philadelphia

    Google Scholar 

  • Weinreich I (2001) A construction of C(1)-wavelets on the two-dimensional sphere. Appl Comput Harmon Anal 10:1–26

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

  2. Institute for Computational Engineering, University of Buchs, Buchs, Switzerland

    Michael Schreiner

Authors
  1. Willi Freeden
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  2. Michael Schreiner
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Corresponding author

Correspondence to Willi Freeden .

Editor information

Editors and Affiliations

  1. Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    M. Zuhair Nashed

  3. Computational Mathematics, Technische Universität Braunschweig, Braunschweig, Germany

    Thomas Sonar

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Freeden, W., Schreiner, M. (2015). Special Functions in Mathematical Geosciences: An Attempt at a Categorization. In: Freeden, W., Nashed, M., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54551-1_31

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