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Error estimates for scattered data interpolation on spheres

Authors: Kurt Jetter, Joachim Stöckler and Joseph D. Ward
Journal: Math. Comp. 68 (1999), 733-747
MSC (1991): Primary 41A05, 41A25; Secondary 41A30, 41A63
MathSciNet review: 1642746
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Abstract: We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the $n$-sphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.

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  • [1] L. Bos, N. Levenberg, P. Milman, and B. A. Taylor, Tangential Markov inequalities characterize algebraic submanifolds of 𝑅^{𝑁}, Indiana Univ. Math. J. 44 (1995), no. 1, 115–138. MR 1336434, 10.1512/iumj.1995.44.1980
  • [2] E. W. Cheney, Approximation using positive definite functions, Approximation theory VIII, Vol. 1 (College Station, TX, 1995) Ser. Approx. Decompos., vol. 6, World Sci. Publ., River Edge, NJ, 1995, pp. 145–168. MR 1471725
  • [3] Stephan Dahlke, Wolfgang Dahmen, Ilona Weinreich, and Eberhard Schmitt, Multiresolution analysis and wavelets on 𝑆² and 𝑆³, Numer. Funct. Anal. Optim. 16 (1995), no. 1-2, 19–41. MR 1322896, 10.1080/01630569508816605
  • [4] Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635
  • [5] Jean Duchon, Splines minimizing rotation-invariant semi-norms in Sobolev spaces, Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976) Springer, Berlin, 1977, pp. 85–100. Lecture Notes in Math., Vol. 571. MR 0493110
  • [6] N. Dyn, F. J. Narcowich, and J. D. Ward, Variational principles and Sobolev-type estimates for generalized interpolation on a Riemannian manifold, Constr. Approx. (to appear).
  • [7] W. Freeden, On spherical spline interpolation and approximation, Math. Methods Appl. Sci. 3 (1981), no. 4, 551–575. MR 657073, 10.1002/mma.1670030139
  • [8] W. Freeden and E. W. Grafarend, Mathematische Methoden der Geodäsie, Oberwolfach Conference Report 41/1995.
  • [9] W. Freeden and U. Windheuser, Combined spherical harmonic and wavelet expansion—a future concept in Earth’s gravitational determination, Appl. Comput. Harmon. Anal. 4 (1997), no. 1, 1–37. MR 1429676, 10.1006/acha.1996.0192
  • [10] Michael Golomb and Hans F. Weinberger, Optimal approximation and error bounds, On numerical approximation. Proceedings of a Symposium, Madison, April 21–23, 1958, Edited by R. E. Langer. Publication No. 1 of the Mathematics Research Center, U.S. Army, the University of Wisconsin, The University of Wisconsin Press, Madison, Wis., 1959, pp. 117–190. MR 0121970
  • [11] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177
  • [12] W. R. Madych and S. A. Nelson, Multivariate interpolation: a variational theory, manuscript, 1983.
  • [13] U. Maier and J. Fliege, Charge distribution of points on the sphere and corresponding cubature formulae, Multivariate Approximation: Recent Trends and Results (W. Haußmann, K. Jetter and M. Reimer, eds.), Mathematical Research, vol. 101, Akademie-Verlag, Berlin, 1997, pp. 147-159. CMP 98:12
  • [14] Claus Müller, Spherical harmonics, Lecture Notes in Mathematics, vol. 17, Springer-Verlag, Berlin-New York, 1966. MR 0199449
  • [15] Francis J. Narcowich, Generalized Hermite interpolation and positive definite kernels on a Riemannian manifold, J. Math. Anal. Appl. 190 (1995), no. 1, 165–193. MR 1314111, 10.1006/jmaa.1995.1069
  • [16] Francis J. Narcowich and Joseph D. Ward, Nonstationary wavelets on the 𝑚-sphere for scattered data, Appl. Comput. Harmon. Anal. 3 (1996), no. 4, 324–336. MR 1420501, 10.1006/acha.1996.0025
  • [17] Manfred Reimer, Constructive theory of multivariate functions, Bibliographisches Institut, Mannheim, 1990. With an application to tomography. MR 1115901
  • [18] Kurt Jetter and Florencio I. Utreras (eds.), Multivariate approximation: from CAGD to wavelets, Series in Approximations and Decompositions, vol. 3, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1359541
  • [19] I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J. 9 (1942), 96–108. MR 0005922
  • [20] Larry L. Schumaker and Cornelis Traas, Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines, Numer. Math. 60 (1991), no. 1, 133–144. MR 1131503, 10.1007/BF01385718
  • [21] G. Wahba, Spline interpolation and smoothing on the sphere, SIAM J. Sci. Statist. Comput. 2 (1981), 5-16; errata, ibid. 3 (1982), 385-386. MR 84j:65016a,b
  • [22] Grace Wahba, Surface fitting with scattered noisy data on Euclidean 𝐷-space and on the sphere, Rocky Mountain J. Math. 14 (1984), no. 1, 281–299. Surfaces (Stanford, Calif., 1982). MR 736179, 10.1216/RMJ-1984-14-1-281

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Additional Information

Kurt Jetter
Affiliation: Institut für Angewandte Mathematik und Statistik, Universität Hohenheim, D-70593 Stuttgart

Joachim Stöckler
Affiliation: Institut für Angewandte Mathematik und Statistik, Universität Hohenheim, D-70593 Stuttgart

Joseph D. Ward
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843

Keywords: Scattered data interpolation, spherical harmonics, Markov inequality, norming set, best approximation
Received by editor(s): August 25, 1997
Additional Notes: Research supported by NSF Grant DMS-9303705 and Air Force AFOSR Grant F49620-95-1-0194.
Article copyright: © Copyright 1999 American Mathematical Society