- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 206265, 5 pages
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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- X. Sun and E. W. Cheney, “Fundamental sets of continuous functions on spheres,” Constructive Approximation, vol. 13, no. 2, pp. 245–250, 1997.
- 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.
- 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.
- C. Müller, Spherical Harmonics, vol. 17 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1966.
- K. Y. Wang and L. Q. Li, Harmonic Analysis and Approximation on the Unit Sphere, Science Press, Beijing, China, 2000.
- I. J. Schoenberg, “Positive definite functions on spheres,” Duke Mathematical Journal, vol. 9, pp. 96–108, 1942.
- 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.