Scattered data interpolation on spheres: Error estimates and locally supported basis functions

SIAM Journal on Mathematical Analysis

Volumn 33, Issue 6, 2002, Pages 1393-1410

  Narcowich, Francis J ^a   Ward, Joseph D ^a  

^a TEXAS A AND M UNIVERSITY   (United States)

Author keywords

Error estimates; Interpolation on spheres; Locally supported basis functions; Rates of convergence; Scattered data

Indexed keywords

EID: 0036433867     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141001395054     Document Type: Article

References (27)

1. W. Cheney, Approximation using positive definite functions, in Approximation Theory VIII, Vol. 1: Approximation and Interpolation, C.K. Chui and L.L. Schumaker, eds., World Scientific Publishing, Singapore, 1995, pp. 145-168.
2. R.A. Devore and G.G. Lorentz, Constructive Approximation, Grundlehren Math. Wiss. 303, Springer-Verlag, Berlin, 1993.
3. N. Dyn, F.J. Narcowich, and J.D. Ward, A framework for interpolation and approximation on Riemannian manifolds, in Approximation Theory and Optimization, M.D. Buhmann and A. Iserles, eds., Cambridge University Press, Cambridge, UK, 1997, pp. 133-144.
4. N. Dyn, F.J. Narcowich, and J.D. Ward, Variational principles and Sobolev-type estimates for generalized interpolation on a Riemannian manifold, Constr. Approx., 15 (1999), pp. 175-208.
5. G.E. Fasshauer and L.L. Schumaker, Scattered data fitting on the sphere, in Mathematical Methods for Curves and Surfaces II, M. Dæhlen, T. Lyche, and L.L. Schumaker, eds., Vanderbilt University Press, Nashville, TN, 1998, pp. 117-166.
6. P.B. Gilkey, The Index Theorem and the Heat Equation, Publish or Perish, Boston, MA, 1974.
7. M. Von Golitschek and W.A. Light, Interpolation by polynomials and radial basis functions, Constr. Approx., 17 (2001), pp. 1-18.
8. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, San Diego: CA, 1980.
9. K. Jetter, J. Stöckler, and J.D. Ward, Error estimates for scattered data interpolation, Math. Comp., 68 (1999), pp. 743-747.
10. J.L. Lions and B. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York, 1972.
11. J. Löfström and J. Peetre, Approximation theorems connected with generalized translation, Math. Ann., 181 (1969), pp. 255-268.
12. H.N. Mhaskar, F.J. Naroowich, and J.D. Ward, Representing and analyzing scattered data on spheres, in Multivariate Approximation and Applications, N. Dyn, D. Leviaton, D. Levin, and A. Pinkus, eds., Cambridge University Press, Cambridge, UK, 2001, pp. 44-72.
13. H.N. Mhaskar, F.J. Narcowich, N. Sivakumar, and J.D. Ward, Approximation with interpolatory constraints, Proc. Amer. Math. Soc., 130 (2002), pp. 1355-1364.
14. T.M. Morton and M. Neamtu, Error bounds for solving pseudodifferential equations on spheres by collocation with zonal kernels, J. Approx. Theory, 114 (2002), pp. 242-268.
15. C. Müller, Spherical Harmonics, Lecture Notes in Math. 17, Springer-Verlag, Berlin, 1966.
16. F.J. Narcowich, Generalized Hermite interpolation and positive definite kernels on a Riemannian manifold, J. Math. Anal. Appl., 190 (1995), pp. 165-193.
17. F.J. Narcowlcr, Recent developments in approximation via positive definite functions, in Approximation IX, Vol. II: Computational Aspects, C.K. Chui and L. Schumaker, eds., Vanderbilt University Press, Nashville, TN, 1998, pp. 221-242.
18. F.J. Narcowich, R. Schaback, and J.D. Ward, Approximation in Sobolev spaces by kernel expansions, J. Approx. Theory, 114 (2002), pp. 70-83.
19. S. Pawelke, Ein Satz vom Jacksonschen Typ für algebraischen Polynome, Acta Sci. Math. (Szeged), 33 (1972), pp. 323-336.
20. S. Pawelke: Über die Approximationsordnung bei Kugelfunktionen und algebraischen Polynomen, Tôhoku Math. J. (2), 24 (1972), pp. 473-486.
21. A. Ron and X. Sun, Strictly positive definite functions on spheres on spheres in Euclidean spaces, Math. Comp., 65 (1996), pp. 1513-1530.
22. I.J. Scroenberg, Positive definite functions on spheres, Duke Math. J., 9 (1942), pp. 96-108.
23. G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1966.
24. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math., 4 (1995), pp. 389-396.
25. H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory, 93 (1998), pp. 258-272.
26. Y. Xu and E.W. Cheney, Strictly positive definite functions on spheres, Proc. Amer. Math. Soc., 116 (1992), pp. 97-981.
27. p-error estimates for 'shifted' surface spline interpolation on Sobolev space, Math. Comp., to appear.