About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 206265, 5 pages
http://dx.doi.org/10.1155/2013/206265
Research Article

Interpolation and Best Approximation for Spherical Radial Basis Function Networks

Institute for Information and System Sciences, School of Mathematics and statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

Received 1 May 2013; Revised 17 September 2013; Accepted 18 September 2013

Academic Editor: Mieczysław Mastyło

Copyright © 2013 Shaobo Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. Freeden, T. Gervens, and M. Schreiner, Constructive Approximation on the Sphere, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, NY, USA, 1998. View at MathSciNet
  2. W. Freeden and V. Michel, “Constructive approximation and numerical methods in geodetic research today—an attempt at a categorization based on an uncertainty principle,” Journal of Geodesy, vol. 73, no. 9, pp. 452–465, 1999.
  3. Y. T. Tsai and Z. C. Shih, “All-frequency precomputed radiance transfer using spherical radial basis functions and clustered tensor approximation,” ACM Transactions on Graphics, vol. 25, pp. 967–976, 2006.
  4. Y. T. Tsai, C. C. Chang, Q. Z. Jiang, and S. C. Weng, “Importance sampling of products from illumination and BRDF using spherical radial basis functions,” The Visual Computer, vol. 24, pp. 817–826, 2008.
  5. H. Q. Minh, “Some properties of Gaussian reproducing kernel Hilbert spaces and their implications for function approximation and learning theory,” Constructive Approximation, vol. 32, no. 2, pp. 307–338, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. K. Jetter, J. Stöckler, and J. D. Ward, “Error estimates for scattered data interpolation on spheres,” Mathematics of Computation, vol. 68, no. 226, pp. 733–747, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Levesley and X. Sun, “Approximation in rough native spaces by shifts of smooth kernels on spheres,” Journal of Approximation Theory, vol. 133, no. 2, pp. 269–283, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, “Approximation properties of zonal function networks using scattered data on the sphere,” Advances in Computational Mathematics, vol. 11, no. 2-3, pp. 121–137, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. H. N. Mhaskar, F. J. Narcowich, J. Prestin, and J. D. Ward, “Lp Bernstein estimates and approximation by spherical basis functions,” Mathematics of Computation, vol. 79, no. 271, pp. 1647–1679, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. T. M. Morton and M. Neamtu, “Error bounds for solving pseudodifferential equations on spheres by collocation with zonal kernels,” Journal of Approximation Theory, vol. 114, no. 2, pp. 242–268, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. F. J. Narcowich and J. D. Ward, “Scattered data interpolation on spheres: error estimates and locally supported basis functions,” SIAM Journal on Mathematical Analysis, vol. 33, no. 6, pp. 1393–1410, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. F. J. Narcowich, X. Sun, J. D. Ward, and H. Wendland, “Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions,” Foundations of Computational Mathematics, vol. 7, no. 3, pp. 369–390, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. X. Sun and E. W. Cheney, “Fundamental sets of continuous functions on spheres,” Constructive Approximation, vol. 13, no. 2, pp. 245–250, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. X. Sun and Z. Chen, “Spherical basis functions and uniform distribution of points on spheres,” Journal of Approximation Theory, vol. 151, no. 2, pp. 186–207, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. F. J. Narcowich, R. Schaback, and J. D. Ward, “Approximations in Sobolev spaces by kernel expansions,” Journal of Approximation Theory, vol. 114, no. 1, pp. 70–83, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  16. C. Müller, Spherical Harmonics, vol. 17 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1966. View at MathSciNet
  17. K. Y. Wang and L. Q. Li, Harmonic Analysis and Approximation on the Unit Sphere, Science Press, Beijing, China, 2000.
  18. I. J. Schoenberg, “Positive definite functions on spheres,” Duke Mathematical Journal, vol. 9, pp. 96–108, 1942. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. F. J. Narcowich, N. Sivakumar, and J. D. Ward, “Stability results for scattered-data interpolation on Euclidean spheres,” Advances in Computational Mathematics, vol. 8, no. 3, pp. 137–163, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet