A note on stability results for scattered data interpolation on Euclidean spheres

Advances in Computational Mathematics

Volumn 30, Issue 4, 2009, Pages 303-314

  Kunis, Stefan ^a  

^a CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Interpolation; Localized polynomials; Separation distance; Zonal functions

Indexed keywords

EID: 63049101481     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10444-008-9069-4     Document Type: Article

References (27)

1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C. (1972)
2. B.K. Alpert V. Rokhlin 1991 A fast algorithm for the evaluation of Legendre expansions SIAM J. Sci. Stat. Comput. 12 158 179
3. K. Ball 1992 Eigenvalues of Euclidean distance matrices J. Approx. Theory 68 74 82
4. R.F. Bass K. Gröchenig 2004 Random sampling of multivariate trigonometric polynomials SIAM J. Math. Anal. 36 773 795
5. J.R. Driscoll D. Healy D. Rockmore 1996 Fast discrete polynomial transforms with applications to data analysis for distance transitive graphs SIAM J. Comput. 26 1066 1099
6. Fasshauer, G.E., Schumaker, L.L.: Scattered data fitting on the sphere. In: Dahlen, M., Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces II, pp. 117-166. Vanderbilt University Press, Nashville (1998)
7. Filbir, F., Themistoclakis, W.: Polynomial approximation on the sphere using scattered data. Math. Nachr. (2008, in press)
8. Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere. Oxford University Press, Oxford (1998)
9. J. Keiner S. Kunis D. Potts 2007 Efficient reconstruction of functions on the sphere from scattered data J. Fourier Anal. Appl. 13 435 458
10. A.B.J. Kuijlaars E.B. Saff X. Sun 2007 On separation of minimal Riesz energy points on spheres in euclidean spaces J. Comput. Appl. Math. 199 172 180
11. S. Kunis D. Potts 2007 Stability results for scattered data interpolation by trigonometric polynomials SIAM J. Sci. Comput. 29 1403 1419
12. J. Levesley Z. Luo X. Sun 1999 Norm estimates of interpolation matrices and their inverses associated with strictly positive definite functions Proc. Amer. Math. Soc. 127 2127 2134
13. H.N. Mhaskar 2004 Polynomial operators and local smoothness classes on the unit interval J. Approx. Theory 131 243 267
14. H.N. Mhaskar F.J. Narcowich J.D. Ward 2001 Spherical Marcinkiewicz- Zygmund inequalities and positive quadrature Math. Comp. 71 1113 1130 (Corrigendum on the positivity of the quadrature weights in 71, 453-454 (2002))
15. H.N. Mhaskar J. Prestin 2000 On the detection of singularities of a periodic function Adv. Comput. Math. 12 95 131
16. Mhaskar, H.N., Prestin, J.: Polynomial frames: a fast tour. In: Approximation theory XI: Gatlinburg 2004, Mod. Methods Math., pp. 287-318. Nashboro, Brentwood, TN (2005)
17. Müller, C.: Spherical Harmonics. Springer, Aachen (1966)
18. F.J. Narcowich P. Petrushev J.D. Ward 2006 Localized tight frames on spheres SIAM J. Math. Anal. 38 574 594
19. F.J. Narcowich N. Sivakumar J.D. Ward 1998 Stability results for scattered-data interpolation on euclidean spheres Adv. Comput. Math. 8 137 163
20. F.J. Narcowich J.D. Ward 1991 Norms of inverses and condition numbers for matrices associated with scattered data J. Approx. Theory 64 69 94
21. F.J. Narcowich J.D. Ward 2002 Scattered data interpolation on spheres: error estimates and locally supported basis functions SIAM J. Math. Anal. 33 1393 1410
22. P. Petrushev Y. Xu 2008 Localized polynomial frames on the ball Constr. Approx. 27 121 148
23. D. Potts G. Steidl M. Tasche 1998 Fast algorithms for discrete polynomial transforms Math. Comp. 67 1577 1590
24. Reimer, M.: Multivariate Polynomial Approximation. Birkhäuser Verlag, Basel (2003)
25. E.B. Saff A.B.J. Kuijlaars 1997 Distributing many points on a sphere Math. Intelligencer 19 5 11
26. Szego, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence (1975)
27. Wendland, H.: Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2005)