Special Functions in Mathematical Geosciences: An Attemptat a Categorization

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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.

Keywords

Spherical Harmonic Uncertainty Principle Pseudodifferential Operator Frequency Localization Space Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Antoine JP, Vandergheynst P (1999) Wavelets on the 2-sphere: a group-theoretic approach. Appl Comput Harmon Anal 7:1–30MathSciNetCrossRefGoogle Scholar
  2. Antoine JP, Demanet L, Jaques L, Vandergheynst P (2002) Wavelets on the sphere: implementations and approximations. Appl Comput Harm Anal 13:177–200CrossRefMATHGoogle Scholar
  3. Backus GE (1966) Potentials for tangent tensor fields on spheroids. Arch Ration Mech Anal 22:210–252MathSciNetCrossRefMATHGoogle Scholar
  4. Backus GE (1967) Converting vector and tensor equations to scalar equations in spherical coordinates. Geophys J R Astron Soc 13:61–101CrossRefGoogle Scholar
  5. Backus GE (1986) Poloidal and toroidal fields in geomagnetic field modelling. Rev Geophys 24:75–109MathSciNetCrossRefGoogle Scholar
  6. Clebsch RFA (1861) Über eine Eigenschaft der Kugelfunktionen. Crelles J 60:343MathSciNetGoogle Scholar
  7. Dahlke S, Maass P (1996) Continuous wavelet transforms with applications to analyzing functions on spheres. J Fourier Anal Appl 2(4):379–396MathSciNetMATHGoogle Scholar
  8. Dahlke S, Dahmen W, Schmitt W, Weinreich I (1995) Multiresolution analysis and wavelets on S 2. Numer Funct Anal Optim 16(1–2):19–41MathSciNetCrossRefMATHGoogle Scholar
  9. de Laplace PS (1785) Theorie des attractions des sphéroides et de la figure des planètes. Mèm de l’Acad, ParisGoogle Scholar
  10. Freeden W (1981) On spherical spline interpolation and approximation. Math Methods Appl Sci 3:551–575MathSciNetCrossRefMATHGoogle Scholar
  11. Freeden W (1999) Multiscale modelling of spaceborne geodata. B.G. Teubner, LeipzigMATHGoogle Scholar
  12. Freeden W, Gutting M (2013) Special functions of mathematical (geo-)physics. Birkhäuser, BaselCrossRefMATHGoogle Scholar
  13. Freeden W, Michel V (2004) Multiscale potential theory (with applications to geoscience). Birkhäuser Verlag, Boston/Basel/BerlinCrossRefMATHGoogle Scholar
  14. Freeden W, Schreiner M (1995) Non-orthogonal expansions on the sphere. Math Methods Appl Sci 18:83–120MathSciNetCrossRefMATHGoogle Scholar
  15. Freeden W, Schreiner M (2007) Biorthogonal locally supported wavelets on the sphere based on zonal kernel functions. J Fourier Anal Appl 13:693–709MathSciNetCrossRefMATHGoogle Scholar
  16. Freeden W, Schreiner M (2009) Spherical functions of mathematical geosciences-a scalar, vectorial, and tensorial setup. Springer, Berlin/HeidelbergMATHGoogle Scholar
  17. Freeden W, Windheuser U (1996) Spherical wavelet transform and its discretization. Adv Comput Math 5:51–94MathSciNetCrossRefMATHGoogle Scholar
  18. Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere (with applications to geomathematics). Oxford/Clarendon, OxfordMATHGoogle Scholar
  19. Gauß CF(1838) Allgemeine Theorie des Erdmagnetismus, Resultate aus den Beobachtungen des magnetischen Vereins. Göttinger Magnetischer Verein, LeipzigGoogle Scholar
  20. Heine E (1878) Handbuch der Kugelfunktionen. Verlag G. Reimer, BerlinGoogle Scholar
  21. Holschneider M (1996) Continuous wavelet transforms on the sphere. J Math Phys 37:4156–4165MathSciNetCrossRefMATHGoogle Scholar
  22. Laín Fernández N (2003) Polynomial bases on the sphere. PhD thesis, University of LübeckGoogle Scholar
  23. Legendre AM (1785) Recherches sur l’attraction des sphèroides homogènes. Mèm math phys près à l’Acad Aci par divers savantes 10:411–434Google Scholar
  24. Lyche T, Schumaker L (2000) A multiresolution tensor spline method for fitting functions on the sphere. SIAM J Sci Comput 22:724–746MathSciNetCrossRefMATHGoogle Scholar
  25. 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)
  26. Morse PM, Feshbach H (1953) Methods of theoretical physics. McGraw-Hill, New YorkMATHGoogle Scholar
  27. Narcowich FJ, Ward JD (1996) Nonstationary wavelets on the m-sphere for scattered data. Appl Comput Harmon Anal 3:324–336MathSciNetCrossRefMATHGoogle Scholar
  28. Neumann F (1887) Vorlesungen über die Theorie des Potentials und der Kugelfunktionen. Teubner, Leipzig, pp 135–154Google Scholar
  29. Potts D, Tasche M (1995) Interpolatory wavelets on the sphere. In: Chui CK, Schumaker LL (eds) Approximation theory VIII. World Scientific, Singapore, pp 335–342Google Scholar
  30. Schröder P, Sweldens W (1995) Spherical wavelets: efficiently representing functions on the sphere. In: Computer graphics proceedings (SIGGRAPH95). ACM, New York, pp 161–175Google Scholar
  31. Svensson SL (1983) Pseudodifferential operators-a new approach to the boundary value problems of physical geodesy. Manus Geod 8:1–40MathSciNetMATHGoogle Scholar
  32. Sylvester T (1876) Note on spherical harmonics. Phil Mag II 291, 400Google Scholar
  33. 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
  34. Wahba G (1990) Spline models for observational data. In: CBMS-NSF regional conference series in applied mathematics, vol 59. SIAM, PhiladelphiaGoogle Scholar
  35. Weinreich I (2001) A construction of C(1)-wavelets on the two-dimensional sphere. Appl Comput Harmon Anal 10:1–26MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Geomathematics GroupUniversity of KaiserslauternRhineland-PalatinateGermany
  2. 2.Institute for Computational EngineeringUniversity of BuchsBuchsSwitzerland

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