# Approximation of Functions on the Real Line

• Volker Michel
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

## Abstract

As we know from functional analysis, if we have a Hilbert space $$(H,\langle \cdot, \cdot \rangle )$$ and a corresponding orthonormal basis $${\left\{{b}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$$, then every element $$f \in H$$ can be represented by a (generalized) Fourier series
$$f ={ \sum \limits_{n=0}^{\infty }}\,\langle f,{b}_{ n}\rangle \,{b}_{n}$$
(3.1)
and can, consequently, be approximated by a truncation
$$f \approx {\sum \limits_{n=0}^{N}}\,\langle f,{b}_{ n}\rangle \,{b}_{n},$$
(3.2)
provided that N is sufficiently large. More general, in the case of an orthogonal basis $${\left\{{b}_{n}\right\}}_{n\in {\mathbb{N}}_{0}}$$, the series has the form
$$f ={ \sum \limits_{n=0}^{\infty }}\,\left\langle f, \frac{1} {\left\|{b}_{n}\right\|}\,{b}_{n}\right\rangle \, \frac{1} {\left\|{b}_{n}\right\|}\,{b}_{n} ={ \sum \limits_{n=0}^{\infty }}\,\frac{\left\langle f,{b}_{n}\right\rangle } {{\left\|{b}_{n}\right\|}^{2}} \,{b}_{n}.$$
(3.3)
In practice, observations of 1D-functions usually have a bounded domain. Thus, we will restrict our attention here to functions on intervals [a, b]. The intuitive choice of a corresponding Hilbert space is $${\mathrm{L}}^{2}([a,b])$$, which is usually abbreviated as $${\mathrm{L}}^{2}[a,b]$$. We will consider a generalization of this space as follows.

## Keywords

Hilbert Space Orthogonal Polynomial Legendre Polynomial Mother Wavelet Jacobi Polynomial
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. 1.
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972)Google Scholar
2. 2.
Akhtar, N.: A multiscale harmonic spline interpolation method for the inverse spheroidal gravimetric problem. Ph.D. thesis, University of Siegen, Department of Mathematics, Geomathematics Group. Shaker, Aachen (2009)Google Scholar
3. 3.
Akhtar, N., Michel, V.: Reproducing Kernel based splines for the regularization of the inverse ellipsoidal gravimetric problem. Appl. Anal. (2011). Accepted for publication, pre-published online via doi:10.1080/00036811.2011.590479Google Scholar
4. 4.
Akram, M.: Constructive approximation on the 3-dimensional ball with focus on locally supported kernels and the Helmholtz decomposition. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen (2009)Google Scholar
5. 5.
Akram, M., Amina, I., Michel, V.: A study of differential operators for particular complete orthonormal systems on a 3D ball. Int. J. Pure Appl. Math. 73, 489–506 (2011)Google Scholar
6. 6.
Alfeld, P., Neamtu, M., Schumaker, L.L.: Fitting scattered data on sphere-like surfaces using spherical splines. J. Comput. Appl. Math. 73, 5–43 (1996)Google Scholar
7. 7.
Amann, H., Escher, J.: Analysis III, 2nd edn. Birkhäuser, Basel (2008)Google Scholar
8. 8.
Amirbekyan, A.: The application of reproducing kernel based spline approximation to seismic surface and body wave tomography: theoretical aspects and numerical results. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group (2007). www.kluedo.ub.uni-kl.de/volltexte/2007/2103/pdf/ThesisAbel.pdfGoogle Scholar
9. 9.
Amirbekyan, A., Michel, V.: Splines on the three-dimensional ball and their application to seismic body wave tomography. Inverse Probl. 24, 1–25 (2008)Google Scholar
10. 10.
Antoine, J.P., Demanet, L., Jacques, L., Vandergheynst, P.: Wavelets on the sphere: implementations and approximations. Appl. Comput. Harm. Anal. 13, 177–200 (2002)Google Scholar
11. 11.
Antoine, J.P., Vandergheynst, P.: Wavelets on the 2-sphere: A group-theoretic approach. Appl. Comput. Harm. Anal. 7, 1–30 (1999)Google Scholar
12. 12.
Ballani, L., Engels, J., Grafarend, E.W.: Global base functions for the mass density in the interior of a massive body (Earth). Manuscr. Geodaet. 18, 99–114 (1993)Google Scholar
13. 13.
Bäni, W.: Wavelets: Eine Einführung für Ingenieure. Oldenburg, München (2002)Google Scholar
14. 14.
Barron, A.R., Cohen, A., Dahmen, W., DeVore, R.A.: Approximation and learning by greedy algorithms. Ann. Stat. 36, 64–94 (2008)Google Scholar
15. 15.
Beckmann, J., Mhaskar, H.N., Prestin, J.: Quadrature formulas for integration of multivariate trigonometric polynomials on spherical triangles. Int. J. Geomath. 3, 119–138 (2012)Google Scholar
16. 16.
Berg, A.P., Mikhael, W.B.: A survey of mixed transform techniques for speech and image coding. In: Proceedings of the 1999 IEEE International Symposium on Circuits and Systems, vol. 4, pp. 106–109 (1999)Google Scholar
17. 17.
Berkel, P.: Multiscale methods for the combined inversion of normal mode and gravity variations. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen (2009)Google Scholar
18. 18.
Berkel, P., Fischer, D., Michel, V.: Spline multiresolution and numerical results for joint gravitation and normal mode inversion with an outlook on sparse regularisation. Int. J. Geomath. 1, 167–204 (2011)Google Scholar
19. 19.
Berkel, P., Michel, V.: On mathematical aspects of a combined inversion of gravity and normal mode variations by a spline method. Math. Geosci. 42, 795–816 (2010)Google Scholar
20. 20.
Blatter, C.: Wavelets: Eine Einführung. Vieweg, Braunschweig (1998)Google Scholar
21. 21.
Blick, C., Freeden, W.: Spherical spline application to radio occultation data. J. Geodetic Sci 1, 379–395 (2011)Google Scholar
22. 22.
Bogdanova, I., Vandergheynst, P., Antoine, J.P., Jacques, L., Morvidone, M.: Stereographic wavelet frames on the sphere. Appl. Comput. Harm. Anal. 19, 223–252 (2005)Google Scholar
23. 23.
Böhme, M., Potts, D.: A fast algorithm for filtering and wavelet decomposition on the sphere. Electron. Trans. Numer. Anal. 16, 70–93 (2003)Google Scholar
24. 24.
Chambodut, A., Panet, I., Mandea, M., Diament, M., Holschneider, M., Jamet, O.: Wavelet frames: An alternative to spherical harmonic representation of potential fields. Geophys. J. Int. 163, 875–899 (2005)Google Scholar
25. 25.
Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM Rev. 43, 129–159 (2001)Google Scholar
26. 26.
Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)Google Scholar
27. 27.
28. 28.
Cohen, A.: Numerical Analysis of Wavelet Methods. Elsevier, Amsterdam (2003)Google Scholar
29. 29.
Coifman, R., Meyer, Y., Wickerhauser, V.: Adapted wave form analysis; waveletpackets and applications. In: ICIAM 91, Proceedings of the Second International Conference on Industrial and Applied Mathematics, pp. 41–50 (1992)Google Scholar
30. 30.
Coifman, R., Wickerhauser, V.: Entropy-based algorithms for best basis selection. IEEE Trans. Inform. Theory 38, 713–718 (1992)Google Scholar
31. 31.
Conrad, M., Prestin, J.: Multiresolution on the sphere. In: Iske, A., Quak, E., Floater, M.S. (eds.) Summer School Lecture Notes on Principles of Multiresolution in Geometric Modelling, pp. 165–202, Munich (2001)Google Scholar
32. 32.
Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19, 297–301 (1965)Google Scholar
33. 33.
Cui, J., Freeden, W.: Equidistribution on the sphere. SIAM J. Sci. Comput. 18, 595–609 (1997)Google Scholar
34. 34.
Dahlen, F.A., Simons, F.J.: Spectral estimation on a sphere in geophysics and cosmology. Geophys. J. Int. 174, 774–807 (2008)Google Scholar
35. 35.
Dahlen, F.A., Tromp, J.: Theoretical Global Seismology. Princeton University Press, Princeton (1998)Google Scholar
36. 36.
Dahlke, S., Dahmen, W., Schmitt, E., Weinreich, I.: Multiresolution analysis and wavelets on S 2 and S 3. Numer. Func. Anal. Opt. 16, 19–41 (1995)Google Scholar
37. 37.
Dahlke, S., Fornasier, M., Raasch, T.: Multilevel preconditioning and adaptive sparse solution of inverse problems. Math. Comput. 81, 419–446 (2009)Google Scholar
38. 38.
Dahlke, S., Steidl, G., Teschke, G.: Coorbit spaces and Banach frames on homogeneous spaces with applications to the sphere. Adv. Comput. Math. 21, 147–180 (2004)Google Scholar
39. 39.
Dahlke, S., Steidl, G., Teschke, G.: Frames and coorbit theory on homogeneous spaces with a special guidance on the sphere. J. Fourier Anal. Appl. 13, 387–403 (2007)Google Scholar
40. 40.
41. 41.
Daubechies, I., Defrise, M., DeMol, C.: An iterative thresholding algorithm for linear inverse problems with sparsity constraint. Commun. Pur. Appl. Math. 57, 1413–1457 (2004)Google Scholar
42. 42.
Daubechies, I., Fornasier, M., Loris, I.: Accelerated projected gradient method for linear inverse problems with sparsity constraints. J. Fourier Anal. Appl. 14, 764–792 (2008)Google Scholar
43. 43.
Davis, P.J.: Interpolation and Approximation. Dover, New York (1975)Google Scholar
44. 44.
Deuflhard, P.: On algorithms for the summation of certain special functions. Computing 17, 37–48 (1975)Google Scholar
45. 45.
DeVore, R.A.: Nonlinear approximation. Acta Numerica 7, 51–150 (1998)Google Scholar
46. 46.
Driscoll, J.R., Healy, R.M.: Computing Fourier transforms and convolutions on the 2-sphere. Adv. Appl. Math. 15, 202–250 (1994)Google Scholar
47. 47.
Dufour, H.M.: Fonctions orthogonales dans la sphère. résolution théorique du problème du potentiel terrestre. B. Geod. 51, 227–237 (1977)Google Scholar
48. 48.
Dunkl, C.F., Xu, Y.: Orthogonal polynomials of several variables. In: Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2001)Google Scholar
49. 49.
Engl, H.W., Grever, W.: Using the L-curve for determining optimal regularization parameters. Numer. Math. 69, 25–31 (1994)Google Scholar
50. 50.
Fasshauer, G.E., Schumaker, L.L.: Scattered data fitting on the sphere. In: Dæhlen, M., Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces II, pp. 117–166. Vanderbilt University Press, Nashville, TN (1998)Google Scholar
51. 51.
Feinerman, R.P., Newman, D.J.: Polynomial Approximation. The Williams and Wilkins Company, Baltimore (1974)Google Scholar
52. 52.
Fengler, M.J., Freeden, W., Kohlhaas, A., Michel, V., Peters, T.: Wavelet modelling of regional and temporal variations of the Earth’s gravitational potential observed by GRACE. J. Geodesy 81, 5–15 (2007)Google Scholar
53. 53.
Fengler, M.J., Michel, D., Michel, V.: Harmonic spline-wavelets on the 3-dimensional ball and their application to the reconstruction of the Earth’s density distribution from gravitational data at arbitrarily shaped satellite orbits. Z. Angew. Math. Mech. 86, 856–873 (2006)Google Scholar
54. 54.
Fischer, D.: Sparse regularization of a joint inversion of gravitational data and normal mode anomalies. Ph.D. thesis, University of Siegen, Department of Mathematics, Geomathematics Group (2011). Verlag Dr. Hut, MünchenGoogle Scholar
55. 55.
Fischer, D., Michel, V.: How to combine spherical harmonics and localized bases for regional gravity modelling and inversion. In: Siegen Preprints on Geomathematics, vol. 8. University of Siegen, Germany (2012, Preprint)Google Scholar
56. 56.
Fischer, D., Michel, V.: Inverting GRACE gravity data for local climate effects. In: Siegen Preprints on Geomathematics, vol. 9. University of Siegen, Germany (2012, Preprint)Google Scholar
57. 57.
Fischer, D., Michel, V.: Sparse regularization of inverse gravimetry — case study: spatial and temporal mass variations in South America. Inverse Probl. 28 (2012). 065012Google Scholar
58. 58.
Fletcher, N.H., Rossing, T.D.: The Physics of Musical Instruments, 2nd edn. Springer, New York (1998)Google Scholar
59. 59.
Fokas, A.S., Hauk, O., Michel, V.: Electro-magneto-encephalography for the three-shell model: numerical implementation via splines for distributed current in spherical geometry. Inverse Probl. 28 (2012). 035009 (28 pp.)Google Scholar
60. 60.
Fornasier, M., Pitolli, F.: Adaptive iterative thresholding algorithms for magnetoencephalography (MEG). J. Comput. Appl. Math. 221, 386–395 (2008)Google Scholar
61. 61.
Freeden, W.: On approximation by harmonic splines. Manuscr. Geodaet. 6, 193–244 (1981)Google Scholar
62. 62.
Freeden, W.: On spherical spline interpolation and approximation. Math. Methods Appl. Sci. 3, 551–575 (1981)Google Scholar
63. 63.
Freeden, W.: Multiscale Modelling of Spaceborne Geodata. B G Teubner. Stuttgart, Leipzig (1999)Google Scholar
64. 64.
Freeden, W., Gerhards, C.: Poloidal and toroidal field modeling in terms of locally supported vector wavelets. Math. Geosci. 42, 817–838 (2010)Google Scholar
65. 65.
Freeden, W., Gervens, T., Schreiner, M.: Tensor spherical harmonics and tensor spherical splines. Manuscr. Geodaet. 19, 70–100 (1994)Google Scholar
66. 66.
Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere with Applications to Geomathematics. Oxford University Press, Oxford (1998)Google Scholar
67. 67.
Freeden, W., Mayer, C.: Wavelets generated by layer potentials. Appl. Comput. Harm. Anal. 14, 195–237 (2003)Google Scholar
68. 68.
Freeden, W., Michel, V.: Constructive approximation and numerical methods in geodetic research today—an attempt at a categorization based on an uncertainty principle. J. Geodesy 73, 452–465 (1999)Google Scholar
69. 69.
Freeden, W., Michel, V.: Multiscale Potential Theory (with Applications to Geoscience). Birkhäuser, Boston (2004)Google Scholar
70. 70.
Freeden, W., Michel, V.: Orthogonal zonal, tesseral and sectorial wavelets on the sphere for the analysis of satellite data. Adv. Comput. Math. 21, 181–217 (2004)Google Scholar
71. 71.
Freeden, W., Michel, V., Nutz, H.: Satellite-to-satellite tracking and satellite gravity gradiometry (advanced techniques for high-resolution geopotential field determination). J. Eng. Math. 43, 19–56 (2002)Google Scholar
72. 72.
Freeden, W., Nutz, H.: Satellite gravity gradiometry as tensorial inverse problem. Int. J. Geomath. 2, 177–218 (2011)Google Scholar
73. 73.
Freeden, W., Schneider, F.: Regularization wavelets and multiresolution. Inverse Probl. 14, 225–243 (1998)Google Scholar
74. 74.
Freeden, W., Schreiner, M.: Orthogonal and non-orthogonal multiresolution analysis, scale discrete and exact fully discrete wavelet transform on the sphere. Constr. Appr. 14, 493–515 (1998)Google Scholar
75. 75.
Freeden, W., Schreiner, M.: Spherical Functions of Mathematical Geosciences, a Scalar, Vectorial, and Tensorial Setup. Springer, Berlin (2009)Google Scholar
76. 76.
Freeden, W., Windheuser, U.: Earth’s gravitational potential and its MRA approximation by harmonic singular integrals. Z. Angew. Math. Mech. 75, 633–634 (1995)Google Scholar
77. 77.
Freeden, W., Windheuser, U.: Spherical wavelet transform and its discretization. Adv. Comput. Math. 5, 51–94 (1996)Google Scholar
78. 78.
Freeden, W., Windheuser, U.: Combined spherical harmonic and wavelet expansion—a future concept in Earth’s gravitational determination. Appl. Comput. Harm. Anal. 4, 1–37 (1997)Google Scholar
79. 79.
Gerhards, C.: Spherical decompositions in a global and local framework: theory and application to geomagnetic modeling. Int. J. Geomath. 1, 205–256 (2011)Google Scholar
80. 80.
Gerhards, C.: Spherical multiscale methods in terms of locally supported wavelets: theory and application to geomagnetic modeling. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group (2011). Verlag Dr. Hut, MünchenGoogle Scholar
81. 81.
Gledhill, J.A.: Aeronomic effects of the South Atlantic anomaly. Rev. Geophys. 14, 173–187 (1976)Google Scholar
82. 82.
Göttelmann, J.: Locally supported wavelets on the sphere. Z. Angew. Math. Mech. 78, 919–920 (1998)Google Scholar
83. 83.
Goupillaud, P., Grossmann, A., Morlet, J.: Cycle-octave and related transforms in seismic signal analysis. Geoexploration 23, 85–102 (1984/85)Google Scholar
84. 84.
Greville, T.N.E.: Introduction to spline functions. In: Greville, T.N.E. (ed.) Theory and Applications of Spline Functions, pp. 1–35. Academic, New York (1969)Google Scholar
85. 85.
Gronwall, T.: On the degree of convergence of Laplace series. Trans. Am. Math. Soc. 15, 1–30 (1914)Google Scholar
86. 86.
Haar, A.: Zur Theorie der orthogonalen Funktionen-Systeme. Math. Ann. 69, 331–371 (1910)Google Scholar
87. 87.
Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34, 561–580 (1992)Google Scholar
88. 88.
Hansen, P.C.: The L-curve and its use in the numerical treatment of inverse problems. In: Johnston, P. (ed.) Computational Inverse Problems in Electrocardiology, pp. 119–142. WIT Press, Southampton (2000)Google Scholar
89. 89.
Hebinger, G., Michel, V., Richter, M., Simon, A.: Speech Recognition Support of Assisted Living. Schriften zur Funktionalanalysis und Geomathematik 40 (2008)Google Scholar
90. 90.
Heirtzler, J.R.: The future of the South Atlantic anomaly and implications for radiation damage in space. J. Atmos. Sol.-Terr. Phy. 64, 1701–1708 (2002)Google Scholar
91. 91.
Heiskanen, W.A., Moritz, H.: Physical Geodesy, Reprint. Institute of Physical Geodesy, Technical University Graz/Austria (1981)Google Scholar
92. 92.
Hesse, K., Sloan, I.H., Womersly, R.S.: Numerical integration on the sphere. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, pp. 1187–1219. Springer, Heidelberg (2010)Google Scholar
93. 93.
Heuser, H.: Funktionalanalysis, 3rd edn. B G Teubner, Stuttgart (1992)Google Scholar
94. 94.
Hobson, E.W.: The Theory of Spherical and Ellipsoidal Harmonics. Chelsea, New York (1965)Google Scholar
95. 95.
Holschneider, M.: Continuous wavelet transforms on the sphere. J. Math. Phys. 37, 4156–4165 (1996)Google Scholar
96. 96.
Holschneider, M., Chambodut, A., Mandea, M.: From global to regional analysis of the magnetic field on the sphere using wavelet frames, Phys. Earth Planet. In. 135, 107–124 (2003)Google Scholar
97. 97.
Holschneider, M., Iglewska-Nowak, I.: Poisson wavelets on the sphere. J. Fourier Anal. Appl. 13, 405–419 (2007)Google Scholar
98. 98.
Johnston, I.: Measured Tones. The Interplay of Physics and Music. Institute of Physics Publishing, Bristol (1989)Google Scholar
99. 99.
Jones, F.: Lebesgue Integration on Euclidean Spaces. Jones and Bartlett Publishers, Boston (1993)Google Scholar
100. 100.
Keiner, J., Prestin, J.: A Fast Algorithm for Spherical Basis Approximation. In: Govil, N.K., Mhaskar, H.N., Mohapatra, R.N., Nashed, Z., Szabados, J. (eds.) Frontiers in Interpolation and Approximation, pp. 259–286. Chapman & Hall/CRC, Boca Raton (2006)Google Scholar
101. 101.
Kellogg, O.D.: Foundations of Potential Theory. Springer, Berlin (1967)Google Scholar
102. 102.
Kress, R.: Numerical Analysis. Springer, New York (1998)Google Scholar
103. 103.
Kufner, A., John, O., Fučík, S.: Function Spaces. Noordhoff International Publishing, Leyden (1977)Google Scholar
104. 104.
Kunis, S., Potts, D.: Fast spherical Fourier algorithms. J. Comput. Appl. Math. 161, 75–98 (2003)Google Scholar
105. 105.
Lai, M.J., Shum, C.K., Baramidze, V., Wenston, P.: Triangulated spherical splines for geopotential reconstruction. J. Geodesy 83, 695–708 (2009)Google Scholar
106. 106.
Laín Fernández, N.: Optimally space-localized band-limited wavelets on $${\mathbb{S}}^{q-1}$$. J. Comput. Appl. Math. 199, 68–79 (2007)Google Scholar
107. 107.
Landau, H.J., Pollak, H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty—II. Bell Syst. Tech. J. 40, 65–84 (1961)Google Scholar
108. 108.
109. 109.
Le Gia, Q.T., Mhaskar, H.N.: Localized linear polynomial operators and quadrature formulas on the sphere. SIAM J. Numer. Anal. 47, 440–466 (2008)Google Scholar
110. 110.
Lemoine, F.G., Smith, D.E., Kunz, L., Smith, R., Pavlis, E.C., Pavlis, N.K., Klosko, S.M., Chinn, D.S., Torrence, M.H., Williamson, R.G., Cox, C.M., Rachlin, K.E., Wang, Y.M., Kenyon, S.C., Salman, R., Trimmer, R., Rapp, R.H., Nerem, R.S.: The development of the NASA GSFC and NIMA joint geopotential model. In: Proceedings of the International Symposium on Gravity, Geoid, and Marine Geodesy (GRAGEOMAR 1996), The University of Tokyo. Springer (1996)Google Scholar
111. 111.
Li, T.H.: Multiscale representation and analysis of spherical data by spherical wavelets. SIAM J. Sci. Comput. 21, 924–953 (1999)Google Scholar
112. 112.
Louis, A.K., Maaß, P., Rieder, A.: Wavelets: Theory and Applications. Wiley, Chichester (1997)Google Scholar
113. 113.
Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, Berlin (1966)Google Scholar
114. 114.
Mallat, S.: A Wavelet Tour of Signal Processing, 3rd edn. Academic, Burlington (2009)Google Scholar
115. 115.
Mallat, S.G., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41, 3397–3415 (1993)Google Scholar
116. 116.
Masters, G., Richards-Dinger, K.: On the efficient calculation of ordinary and generalized spherical harmonics. Geophys. J. Int. 135, 307–309 (1998)Google Scholar
117. 117.
Maus, S., Rother, M., Hemant, K., Stolle, C., Lühr, H., Kuvshinov, A., Olsen, N.: Earth’s lithospheric magnetic field determined to spherical harmonic degree 90 from CHAMP satellite measurements. Geophys. J. Int. 164, 319–330 (2006)Google Scholar
118. 118.
Maus, S., Rother, M., Holme, R., Lühr, H., Olsen, N., Haak, V.: First scalar magnetic anomaly map from CHAMP satellite data indicates weak lithospheric field. Geophys. Res. Lett. 29, 47–1 to 47–4 (2002)Google Scholar
119. 119.
McShane, E.J.: Integration. Princeton University Press, Princeton (1974)Google Scholar
120. 120.
121. 121.
Mhaskar, H.N.: Local quadrature formulas on the sphere, II. In: Neamtu M., Saff, E.B. (eds.) Advances in Constructive Approximation, pp. 333–344. Nashboro Press, Brentwood (2004)Google Scholar
122. 122.
Mhaskar, H.N., Narcowich, F.J., Prestin, J., Ward, J.D.: Polynomial frames on the sphere. Adv. Comput. Math. 13, 387–403 (2000)Google Scholar
123. 123.
Mhaskar, H.N., Narcowich, F.J., Ward, J.D.: Spherical Marcinkiewicz–Zygmund inequalities and positive quadrature. Math. Comput. 70, 1113–1130 (2000)Google Scholar
124. 124.
Mhaskar, H.N., Prestin, J.: Polynomial frames: a fast tour. In: Chui, C.K., Neamtu, M., Schumaker, L.L. (eds.) Approximation Theory XI: Gatlinburg 2004, pp. 101–132. Nashboro Press, Brentwood (2004)Google Scholar
125. 125.
Michel, D.: Framelet based multiscale operator decomposition. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen (2006)Google Scholar
126. 126.
Michel, V.: A wavelet based method for the gravimetry problem. In: Freeden, W. (ed.) Progress in Geodetic Science, Proceedings of the Geodetic Week, pp. 283–298. Shaker, Aachen (1998)Google Scholar
127. 127.
Michel, V.: A multiscale method for the gravimetry problem: theoretical and numerical aspects of harmonic and anharmonic modelling. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen (1999)Google Scholar
128. 128.
Michel, V.: A multiscale approximation for operator equations in separable Hilbert spaces—case study: reconstruction and description of the Earth’s interior, Habilitation thesis. Shaker, Aachen (2002)Google Scholar
129. 129.
Michel, V.: Scale continuous, scale discretized and scale discrete harmonic wavelets for the outer and the inner space of a sphere and their application to an inverse problem in geomathematics. Appl. Comput. Harm. Anal. 12, 77–99 (2002)Google Scholar
130. 130.
Michel, V.: Regularized wavelet-based multiresolution recovery of the harmonic mass density distribution from data of the Earth’s gravitational field at satellite height. Inverse Probl. 21, 997–1025 (2005)Google Scholar
131. 131.
Michel, V.: Wavelets on the 3-dimensional ball. Proc. Appl. Math. Mech. 5, 775–776 (2005)Google Scholar
132. 132.
Michel, V.: Tomography—problems and multiscale solutions. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, pp. 949–972. Springer, Heidelberg (2010)Google Scholar
133. 133.
Michel, V.: Optimally localized approximate identities on the 2-sphere. Numer. Func. Anal. Opt. 32, 877–903 (2011)Google Scholar
134. 134.
Michel, V., Fokas, A.S.: A unified approach to various techniques for the non-uniqueness of the inverse gravimetric problem and wavelet-based methods. Inverse Probl. 24 (2008). 045019 (25 pp.)Google Scholar
135. 135.
Michel, V., Wolf, K.: Numerical aspects of a spline-based multiresolution recovery of the harmonic mass density out of gravity functionals. Geophys. J. Int. 173, 1–16 (2008)Google Scholar
136. 136.
Mikhlin, S.G.: Mathematical Physics, an Advanced Course. North-Holland Publishing Company, Amsterdam (1970)Google Scholar
137. 137.
Mohlenkamp, M.J.: A fast transform for spherical harmonics. J. Fourier Anal. Appl. 5, 159–184 (1999)Google Scholar
138. 138.
Müller, C.: Über die ganzen Lösungen der Wellengleichung. Math. Ann. 124, 235–264 (1952)Google Scholar
139. 139.
Müller, C.: Spherical Harmonics. Springer, Berlin (1966)Google Scholar
140. 140.
Müller, C.: Foundations of the Mathematical Theory of Electromagnetic Waves. Springer, Berlin (1969)Google Scholar
141. 141.
Narcowich, F.J., Petrushev, P., Ward, J.D.: Localized tight frames on spheres. SIAM J. Math. Anal. 38, 574–594 (2006)Google Scholar
142. 142.
Narcowich, F.J., Ward, J.D.: Nonstationary wavelets on the m-sphere for scattered data. Appl. Comput. Harm. Anal. 3, 324–336 (1996)Google Scholar
143. 143.
144. 144.
Nikiforov, A.F., Uvarov, V.B.: Special Functions of Mathematical Physics—A Unified Introduction with Applications. Birkhäuser, Basel (1988). Translated from the Russian by R. P. BossGoogle Scholar
145. 145.
Olson, H.F.: Music, Physics and Engineering, 2nd edn. Dover, New York (1967)Google Scholar
146. 146.
Pavlis, N.K., Holmes, S.A., Kenyon, S.C., Factor, J.K.: An Earth gravitational model to degree 2160: EGM2008. Presentation given at the 2008 European Geosciences Union General Assembly held in Vienna, Austria, 13–18 Apr 2008. http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008/NPavlis&al_EGU2008.ppt
147. 147.
Plato, R.: Numerische Mathematik kompakt, 4th edn. Vieweg + Teubner, Wiesbaden (2010)Google Scholar
148. 148.
Potts, D., Steidl, G., Tasche, M.: Kernels of spherical harmonics and spherical frames. In: Fontanella, F., Jetter, K., Laurent, P.J. (eds.) Advanced Topics in Multivariate Approximation, pp. 287–301. World Scientific, Singapore (1996)Google Scholar
149. 149.
Prestin, J., Rosca, D.: On some cubature formulas on the sphere. J. Approx. Theory 142, 1–19 (2006)Google Scholar
150. 150.
Protter, M.H., Morrey, C.B.: A First Course in Real Analysis, 2nd edn. Springer, New York (1977)Google Scholar
151. 151.
Purucker, M.E., Dyment, J.: Satellite magnetic anomalies related to seafloor spreading in the South Atlantic ocean. Geophys. Res. Lett. 27, 2765–2768 (2000)Google Scholar
152. 152.
Qian, S., Chen, D.: Signal representation using adaptive normalized Gaussian functions. Signal Process. 36, 1–11 (1994)Google Scholar
153. 153.
Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, 2nd edn. Springer, Berlin (2007)Google Scholar
154. 154.
Reigber, C., Balmino, G., Schwintzer, P., Biancale, R., Bode, A., Lemoine, J-M., König, R., Loyer, S., Neumayer, H., Marty, J-C., Barthelmes, F., Perosanz, F., Zhu, S.Y.: A high-quality global gravity field model from CHAMP GPS tracking data and accelerometry (EIGEN-1S). Geophys. Res. Lett. 29, 37–1 to 37–4 (2002)Google Scholar
155. 155.
Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations. Springer, New York (1996)Google Scholar
156. 156.
Renka, R.J.: Interpolation of data on the surface of a sphere. ACM T. Math. Software 10, 417–436 (1984)Google Scholar
157. 157.
Reuter, R.: Über Integralformeln der Einheitssphäre und harmonische Splinefunktionen. Ph.D. thesis, Veröff. Geod. Inst. RWTH Aachen, RWTH Aachen, vol. 33 (1982)Google Scholar
158. 158.
Riley, K.F., Hobson, M.P., Bence, S.J.: Mathematical Methods for Physics and Engineering, 4th edn. Cambridge University Press, Cambridge (2008)Google Scholar
159. 159.
Rivlin, T.J.: An Introduction to the Approximation of Functions. Blaisdell Publishing Company, Waltham (1969)Google Scholar
160. 160.
Robin, L.: Fonctions Sphérique de Legendre et Fonctions Sphéroidale, vol. 1. Gauthier-Villars, Paris (1957)Google Scholar
161. 161.
Robin, L.: Fonctions Sphérique de Legendre et Fonctions Sphéroidale, vol. 2. Gauthier-Villars, Paris (1958)Google Scholar
162. 162.
Robin, L.: Fonctions Sphérique de Legendre et Fonctions Sphéroidale, vol. 3. Gauthier-Villars, Paris (1959)Google Scholar
163. 163.
Sard, A.: Linear Approximation. American Mathematical Society, Providence (1963)Google Scholar
164. 164.
Schaeben, H., Bernstein, S., Hielscher, R., Beckmann, J., Keiner, J., Prestin, J.: High resolution texture analysis with spherical wavelets. Mater. Sci. Forum 495–497, 245–254 (2005)Google Scholar
165. 165.
Schmidt, M., Fengler, M., Mayer-Gürr, T., Eicker, A., Kusche, J., Sánchez, L., Han, S-C.: Regional gravity modeling in terms of spherical base functions. J. Geodesy 81, 17–38 (2007)Google Scholar
166. 166.
Schneider, F.: Inverse problems in satellite geodesy and their approximate solution by splines and wavelets. Ph.D. thesis, University of Kaiserslautern, Geomathematics Group. Shaker, Aachen (1997)Google Scholar
167. 167.
Schoenberg, I.J.: On best approximations of linear operators. Nederl. Akad. Wetensch. Proc. Ser. A 67, 155–163 (1964)Google Scholar
168. 168.
Schreiner, M.: On a new condition for strictly positive definite functions on spheres. Proc. Am. Math. Soc. 125, 531–539 (1997)Google Scholar
169. 169.
Schröder, P., Sweldens, W.: Spherical wavelets: efficiently representing functions on the sphere. In: SIGGRAPH’95 Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques pp. 161–172. ACM, New York (1995)Google Scholar
170. 170.
Schwarz, H.R.: Numerical Analysis: A Comprehensive Introduction. Wiley, Chichester (1989)Google Scholar
171. 171.
Sethares, W.A.: Tuning, Timbre, Spectrum, Scale. Springer, London (2005)Google Scholar
172. 172.
Simons, F.J.: Slepian functions and their use in signal estimation and spectral analysis. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, pp. 891–923. Springer, Heidelberg (2010)Google Scholar
173. 173.
Simons, F.J., Dahlen, F.A.: Spherical Slepian functions and the polar gap in geodesy. Geophys. J. Int. 166, 1039–1061 (2006)Google Scholar
174. 174.
Simons, F.J., Dahlen, F.A., Wieczorek, M.A.: Spatiospectral concentration on a sphere. SIAM Rev. 48, 504–536 (2006)Google Scholar
175. 175.
Simons, F.J., Loris, I., Brevdo, E., Daubechies, I.C.: Wavelets and wavelet-like transforms on the sphere and their application to geophysical data inversion. Proc. SPIE 8138 (2011). 81380XGoogle Scholar
176. 176.
Simons, F.J., Loris, I., Nolet, G., Daubechies, I.C., Voronin, S., Judd, J.S., Vetter, P.A., Charléty, J., Vonesch, C.: Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity. Geophys. J. Int. 187, 969–988 (2011)Google Scholar
177. 177.
Slepian, D.: Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: extensions to many dimensions; generalized prolate spheroidal functions. Bell Syst. Tech. J. 43, 3009–3057 (1964)Google Scholar
178. 178.
Slepian, D., Pollak, H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty—I. Bell Syst. Tech. J. 40, 43–63 (1961)Google Scholar
179. 179.
Sloan, I.H., Womersley, R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21, 107–125 (2004)Google Scholar
180. 180.
Szegö, G.: Orthogonal Polynomials, vol. XXIII, 14th edn. AMS Colloquium Publications, Providence (1975)Google Scholar
181. 181.
Temlyakov, V.N.: Greedy algorithms and m-term approximation. J. Approx. Theor. 98, 117–145 (1999)Google Scholar
182. 182.
Temlyakov, V.N.: Greedy algorithms with regard to multivariate systems with special structure. Constr. Approx. 16, 399–425 (1999)Google Scholar
183. 183.
Temlyakov, V.N.: Nonlinear methods of approximation. Found. Comput. Math. 3, 33–107 (2003)Google Scholar
184. 184.
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Johann Ambrosius Barth Verlag, Heidelberg (1995)Google Scholar
185. 185.
Trim, D.: Calculus. Prentice Hall, Scarborough (1993)Google Scholar
186. 186.
Tscherning, C.C.: Isotropic reproducing kernels for the inner of a sphere or spherical shell and their use as density covariance functions. Math. Geol. 28, 161–168 (1996)Google Scholar
187. 187.
Tygert, M.: Fast algorithms for spherical harmonic expansions II. J. Comput. Phys. 227, 4260–4279 (2008)Google Scholar
188. 188.
Voigt, A., Wloka, J.: Hilberträume und elliptische Differentialoperatoren. Bibliographisches Institut, Mannheim (1975)Google Scholar
189. 189.
Walnut, D.F.: An Introduction to Wavelet Analysis. Birkhäuser, Boston (2002)Google Scholar
190. 190.
Walter, W.: Einführung in die Potentialtheorie. Bibliographisches Institut, Mannheim (1971)Google Scholar
191. 191.
Walter, W.: Analysis 2, 3rd edn. Springer, Berlin (1992)Google Scholar
192. 192.
Wang, Z., Dahlen, F.A.: Spherical-spline parameterization of three-dimensional Earth models. Geophys. Res. Lett. 22, 3099–3102 (1995)Google Scholar
193. 193.
Wang, Z.X., Guo, D.R.: Special Functions. World Scientific, Singapore (1989)Google Scholar
194. 194.
Weinreich, I.: A construction of C1-wavelets on the two-dimensional sphere. Appl. Comput. Harm. Anal. 10, 1–26 (2001)Google Scholar
195. 195.
Werner, J.: Numerische Mathematik I: Lineare und nichtlineare Gleichungssysteme, Interpolation, numerische Integration. Vieweg, Braunschweig, Wiesbaden (1992)Google Scholar
196. 196.
Wesfried, E., Wickerhauser, M.V.: Adapted local trigonometric transforms and speech processing. IEEE Trans. Signal Process. 41, 3596–3600 (1993)Google Scholar
197. 197.
Wickerhauser, M.V.: INRIA lectures on wavelet packet algorithms. In: Minicourse lecture notes. INRIA, Rocquencourt (1991)Google Scholar
198. 198.
Wieczorek, M.A., Simons, F.J.: Localized spectral analysis on the sphere. Geophys. J. Int. 162, 655–675 (2005)Google Scholar
199. 199.
Wieczorek, M.A., Simons, F.J.: Minimum-variance spectral analysis on the sphere. J. Fourier Anal. Appl. 13, 665–692 (2007)Google Scholar
200. 200.
Windheuser, U.: Sphärische Wavelets: Theorie und Anwendung in der Physikalischen Geodäsie. Ph.D. thesis, University of Kaiserslautern, Geomathematics Group (1995)Google Scholar
201. 201.
Wojtaszczyk, P.: A Mathematical Introduction to Wavelets. Cambridge University Press, Cambridge (1997)Google Scholar
202. 202.
Wood, A.: The Physics of Music. University Paperbacks, London (1962)Google Scholar
203. 203.
204. 204.
205. 205.
206. 206.
Xu, Y., Cheney, E.W.: Strictly positive definite functions on spheres. Proc. Am. Math. Soc. 116, 977–981 (1992)Google Scholar
207. 207.
Yosida, K.: Functional Analysis, 6th edn. Classics in Mathematics. Springer, Berlin (1995)Google Scholar
208. 208.
Zeidler, E. (ed.): Teubner-Taschenbuch der Mathematik, originally from I.N. Bronstein and K.A. Semendjajew. Teubner, Leipzig (1996)Google Scholar