Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0025-5718-99-01080-7
MathSciNet review: 1642746
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [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), 115-138. MR 96i:41009
  • [2] E. W. Cheney, Approximation using positive definite functions, Approximation Theory VIII, vol. 1: Approximation and Interpolation (C. K. Chui and L. L. Schumaker, eds.), World Scientific, Singapore, 1995, pp. 145-168. MR 98e:41036
  • [3] S. Dahlke, W. Dahmen, E. Schmitt and I. Weinreich, Multiresolution analysis and wavelets on $S^{2}$ and $S^{3}$, Numer. Funct. Anal. Optim. 16 (1995), 19-41. MR 96a:42044
  • [4] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin-Heidelberg, 1993. MR 95f:41001
  • [5] J. Duchon, Splines minimizing rotation-invariant semi-norms in Sobolev spaces, Constructive Theory of Functions of Several Variables (W. Schempp and K. Zeller, eds.), Springer-Verlag, Berlin-Heidelberg, 1979, pp. 85-100. MR 58:12146
  • [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), 551-575. MR 83m:41003
  • [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), 1-37. MR 97j:86004
  • [10] M. Golomb and H. F. Weinberger, Optimal approximation and error bounds, On Numerical Approximation (R. E. Langer, ed.), The Univ. of Wisconsin Press, Madison, 1959, pp. 117-190. MR 22:12697
  • [11] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. I, Springer-Verlag, New York, 1972. MR 50:2670
  • [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] C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, vol. 17, Springer-Verlag, Berlin-Heidelberg, 1966. MR 33:7593
  • [15] F. J. Narcowich, Generalized Hermite interpolation and positive definite kernels on a Riemannian manifold, J. Math. Anal. Appl. 190 (1995), 165-193. MR 96c:41009
  • [16] F. J. Narcowich and J. D. Ward, Nonstationary wavelets on the $m$-sphere for scattered data, Appl. Comput. Harmon. Anal. 3 (1996), 324-336. MR 97h:42020
  • [17] M. Reimer, Constructive Theory of Multivariate Functions, B. I. Wissenschaftsverlag, Mannheim, 1990. MR 92m:41003
  • [18] R. Schaback, Comparison of radial basis function interpolants, Multivariate Approximation: From Theory to Software (K. Jetter and F. I. Utreras, eds.), World Scientific, Singapore, 1995, pp. 293-305. MR 96f:65004
  • [19] I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J. 9 (1942), 96-108. MR 3:232c
  • [20] L. L. Schumaker and C. Traas, Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines, Numer. Math. 60 (1991), 133-144. MR 92j:65012
  • [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] G. Wahba, Surface fitting with scattered noisy data on Euclidian $d$-space and on the sphere, Rocky Mountain J. Math. 14 (1984), 281-299. MR 86c:65017

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 41A05, 41A25, 41A30, 41A63

Retrieve articles in all journals with MSC (1991): 41A05, 41A25, 41A30, 41A63


Additional Information

Kurt Jetter
Affiliation: Institut für Angewandte Mathematik und Statistik, Universität Hohenheim, D-70593 Stuttgart
Email: kjetter@uni-hohenheim.de

Joachim Stöckler
Affiliation: Institut für Angewandte Mathematik und Statistik, Universität Hohenheim, D-70593 Stuttgart
Email: stockler@uni-hohenheim.de

Joseph D. Ward
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843
Email: jward@math.tamu.edu

DOI: https://doi.org/10.1090/S0025-5718-99-01080-7
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

American Mathematical Society