Error estimates for interpolation by compactly supported radial basis functions of minimal degree.(English)Zbl 0904.41013

The author considers interpolation at $$N$$ distinct points $$x_i\in\mathbb{R}^d$$, using radial basis functions introduced by him in the paper [Adv. Comput. Math. 4, No. 4, 389-396 (1995; Zbl 0838.41014)]. After identifying the associated Hilbert spaces of these functions as norm-equivalent to certain Sobolev spaces, error estimates for the corresponding interpolants are derived by computing upper and lower bounds for the Fourier transforms of these radial basis functions. Upper bounds for the condition numbers of the interpolation matrices, in terms of the separation distance of the centers $$x_i$$, conclude the paper.
Reviewer: E.Quak (Oslo)

MSC:

 41A30 Approximation by other special function classes 41A05 Interpolation in approximation theory 41A63 Multidimensional problems

Zbl 0838.41014
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References:

 [1] Askey, R., MRC Technical Sum Report (1973) [2] Buhmann, M. D., New developments in the theory of radial basis functions interpolation, (Jetter, K.; Utreras, F. I., Multivariate Approximations: From CAGD to Wavelets (1993)), 35-75 [3] Cooke, R. G., A monotonic property of Bessel functions, J. London Math. Soc. (2), 12, 279-284 (1937) · JFM 63.0329.02 [4] Duchon, J., Sur l’erreur d’interpolation des fonctions de plusieurs variables par les$$D^m$$-splines, RAIRO Modél. Math. Anal. Numér., 12, 325-334 (1978) · Zbl 0403.41003 [5] Dyn, N., Interpolation and approximation by radial and related functions, (Chui, C. K.; Schumaker, L. L.; Ward, J. D., Approxi- mation Theory VI (1983)), 211-234 · Zbl 0705.41006 [6] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Tables of integral transforms Vol. II (1954), McGraw-Hill: McGraw-Hill New York · Zbl 0055.36401 [7] Gasper, G., Positive integrals of Bessel functions, SIAM J. Math. Anal., 6, 868-891 (1975) · Zbl 0313.33013 [8] Madych, W. R.; Nelson, S. A., Multivariate interpolation and conditionally positive definite functions, Approx. Theory Appl. (N.S.), 44, 77-89 (1988) · Zbl 0703.41008 [9] Madych, W. R.; Nelson, S. A., Multivariate interpolation and conditionally positive definite functions II, Math. Comp., 54, 211-230 (1990) · Zbl 0859.41004 [10] Narcowich, F.; Ward, J., Norms of inverses and condition numbers for matrices associated with scattered data, J. Approx. Theory, 64, 84-109 (1991) [11] Powell, M. J.D., The theory of radial basis function approximation in 1990, (Light, W., Advances in Numerical Analysis (1992), Clarendon: Clarendon Oxford), 105-210 · Zbl 0787.65005 [12] Schaback, R., Error estimates and condition numbers for radial basis function interpolation, AICM, 3, 251-264 (1995) · Zbl 0861.65007 [13] Schaback, R., Creating surfaces from scattered data using radial basis functions, (Daehlen, M.; Lyche, T.; Schumaker, L. L., Mathematical Methods for Curves and Surfaces (1995)), 477-496 · Zbl 0835.65036 [14] Schaback, R., Multivariate interpolation and approximation by translates of a basis function, (Chui, C. K.; Schumaker, L. L., Approximation Theory VIII. Approximation Theory VIII, Approximation and Interpolation, 1 (1995)), 491-514 · Zbl 1139.41301 [15] Schoenberg, I. J., Metric spaces and completely monotone functions, Ann. of Math. (2), 39, 811-841 (1938) · Zbl 0019.41503 [16] Watson, G. N., A Treatise on the Theory of Bessel Functions (1966), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0174.36202 [17] Wendland, H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, AICM, 4, 389-396 (1995) · Zbl 0838.41014 [18] Wu, Z., Multivariate compactly supported positive definite radial functions, AICM, 4, 283-292 (1995) · Zbl 0837.41016 [19] Wu, Z.; Schaback, R., Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal., 13, 13-27 (1993) · Zbl 0762.41006
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