×

zbMATH — the first resource for mathematics

Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. (English) Zbl 0838.41014
The author constructs a class of positive definite and compactly supported radial functions of even order of smoothness, which are defined – within their supports – by a single univariate polynomial. For given space dimension and smoothness, it is shown that these functions have minimal degree, and that they are uniquely determined up to a constant factor. Finally, connections are made to other compactly supported positive definite radial functions, namely the so-called Wu’s functions and Euclid’s hats.
Reviewer: E.Quak (Schwerte)

MSC:
41A30 Approximation by other special function classes
41A05 Interpolation in approximation theory
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
41A15 Spline approximation
65D07 Numerical computation using splines
65D10 Numerical smoothing, curve fitting
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] R. Askey, Radial characteristic functions, MRC Technical Sum: Report no. 1262, University of Wisconsin (1973). · Zbl 0253.33009
[2] D.S. Broomhead and D. Lowe, Multivariable functional interpolation and adaptive networks, Complex Syst. 2 (1988) 321–355. · Zbl 0657.68085
[3] A.Y. Chanysheva, Positive definite functions of a special form, Vestnik Moskovskogo Universiteta Matematika UDC 517.5; 519.2 (1989). English translation in: Moscow University Math. Bull. 45 (1990) 57–59.
[4] N. Dyn, D. Levin and S. Rippa, Numerical procedures for surface fitting of scattered data by radial functions, SIAM J. Sci. Stat. Comp. 7 (1986) 639–659. · Zbl 0631.65008 · doi:10.1137/0907043
[5] R. Franke, Scattered data interpolation: Tests of some methods, Math. Comp. 38 (1982) 181–200. · Zbl 0476.65005
[6] G. Gasper, Positive integrals of Bessel functions, SIAM J. Math. Anal. 6 (1975) 868–881. · Zbl 0313.33013 · doi:10.1137/0506076
[7] C.A. Micchelli, Interpolation of scattered data: Distance matrices and conditionally positive definite functions, Constr. Approx. 2 (1986) 11–22. · Zbl 0625.41005 · doi:10.1007/BF01893414
[8] F.J. Narcowich and J.D. Ward, Norm estimates for the inverse of a general class of scattered-data radial-function interpolation matrices, J. Approx. Theory 69 (1992) 84–109. · Zbl 0756.41004 · doi:10.1016/0021-9045(92)90050-X
[9] M.J.D. Powell, Truncated Laurent expansions for the fast evaluation of thin plate splines, DAMTP/1992/NA10, University of Cambridge (1992).
[10] R. Schaback, Creating surfaces from scattered data using radial basis functions, in:Mathematical Methods in CAGD III, eds. M. Daehlen, T. Lyche and L.L. Schumaker (1994). · Zbl 0835.65036
[11] R. Schaback, Error estimates and condition numbers for radial basis function interpolation, Adv. Comp. Math. 3 (1995) 251–264. · Zbl 0861.65007 · doi:10.1007/BF02432002
[12] R. Schaback, Multivariate interpolation and approximation by translates of a basis function, in:Approximation Theory VIII, eds. C.K. Chui and L.L. Schumaker (1995). · Zbl 1139.41301
[13] R. Schaback and Z. Wu, Operators on radial functions, preprint (1994). · Zbl 0857.42004
[14] H. Wendland, Ein Beitrag zur Interpolation mit radialen Basisfunktionen, Diplomarbeit, Göttingen (1994).
[15] Z. Wu, Multivariate compactly supported positive definite radial functions, Adv. Comp. Math. 4 (1995) 283–292. · Zbl 0837.41016 · doi:10.1007/BF03177517
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.