Approximate interpolation with applications to selecting smoothing parameters. (English) Zbl 1081.65016

An approximation problem can be shortly stated as follows: for a finite set \(X\) of points situated in a bounded set \(\Omega\) and a corresponding data values of an unknown function \(f \in C(\Omega)\), a function \(s_{f,X} \in C(\Omega)\) to produce a good approximation is required. The authors investigate how a small error at the data set influences the global error on \(\Omega\). An estimation of the global error is given and the main application refers to the spline smoothing. It is shown that a specific, a priori choice of the smoothing parameter is possible and leads to the same approximation order as the classical interpolant. An application in stabilizing the interpolation process by splines and positive definite kernels is presented and the final section gives some eloquent numerical examples.


65D05 Numerical interpolation
65D07 Numerical computation using splines
65D10 Numerical smoothing, curve fitting
Full Text: DOI


[1] Anselone, P.M., Laurent, P.J.: A general method for the construction of interpolating or smoothing spline-functions. Numer. Math. 12, 66–82 (1968) · Zbl 0197.13501
[2] Brenner, S., Scott, L.: The mathematical theory of finite element methods. Springer, New York, 1994 · Zbl 0804.65101
[3] Carr, J.C., Beatson, R.K., Cherrie, J.B., Mitchell, T.J., Fright, W.R. McCallum, B.C., Evans, T.R.: Reconstruction and representation of 3D objects with radial basis functions. In: Computer graphics proceedings, annual conference series, Addison Wesley, 2001, pp. 67–76
[4] Cox, D.D.: Multivariate smoothing spline functions. SIAM J. Numer. Anal. 21, 789–813 (1984) · Zbl 0581.65012
[5] Craven, P., Wahba, G.: Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math. 31, 377–403 (1979) · Zbl 0377.65007
[6] Cristianini, N., Shawe-Taylor, J.: An introduction to support vector machines and other kernel-based learning methods. Cambridge University Press, Cambridge, 2000 · Zbl 0994.68074
[7] Cucker, F., Smale, S.: On the mathematical foundation of learning. Bull. Amer. Math. Soc. 39, 1–49 (2001) · Zbl 0983.68162
[8] de Boor, C.: A practical guide to splines. Springer, New York, revised ed., 2001 · Zbl 0987.65015
[9] Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Constructive theory of functions of several variables, Schempp, W., Zeller, K. (eds.) Berlin, Springer, 1977, pp. 85–100,
[10] Duchon, J., Sur l’erreur d’interpolation des fonctions de plusieurs variables par les Dm-splines. Rev. Française Automat. Informat. Rech. Opér. Anal. Numer. 12, 325–334 (1978) · Zbl 0403.41003
[11] Evgeniou, T., Pontil, M., Poggio, T.: Regularization networks and support vector machines. Adv. Comput. Math. 13, 1–50 (2000) · Zbl 0939.68098
[12] Golitschek, M.V., Schumaker, L.L.: Data fitting by penalized least squares. In: Algorithms for approximation II, Mason, C., Cox, M.G. (eds.), London, Chapman and Hall, 1990, pp. 210–227 · Zbl 0749.41004
[13] Kersey, S.N.: Near-interpolation. Numer. Math. 94 523–540 (2003) · Zbl 1107.41002
[14] Kersey, S.N., On the problem of smoothing and near-interpolation. Math. Comput. 72, 1873–1895 (2003) · Zbl 1044.41002
[15] Narcowich, F.J., Ward, J.D. (1991) Norms of inverses and condition numbers for matrices associated with scattered data. J. Approx. Theory 64, 69–94 (1991) · Zbl 0724.41004
[16] Narcowich, F.J., Norms of inverses for matrices associated with scattered data. In: Curves and surfaces, Laurent, P.-J., Méhauté, A.L., Schumaker, L.L. (eds.), Boston, Academic Press, 1991, pp. 341–348 · Zbl 0798.65010
[17] Narcowich, F.J., Norm estimates for the inverse of a general class of scattered-data radial-function interpolation matrices. J. Approx. Theory, 69, 84–109 (1992) · Zbl 0756.41004
[18] Narcowich, F.J., On condition numbers associated with radial-function interpolation. J. Math. Anal. Appl. 186, 457–485 (1994) · Zbl 0813.65005
[19] Narcowich, F.J., Scattered-data interpolation on \(\mathbb{R}\)n: Error estimates for radial basis and band-limited functions. SIAM J. Math. Anal. 36, 284–300 (2004) · Zbl 1081.41014
[20] Narcowich, F.J., Ward, J.D., Wendland, H.: Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting. Math. Comput. 74, 643–763 (2005) · Zbl 1063.41013
[21] Ragozin, D.L.: Error bounds for derivative estimates based on spline smoothing of exact or noisy data. J. Approx. Theory 37, 335–355 (1983) · Zbl 0531.41011
[22] Reinsch, C.H.: Smoothing by spline functions. Numer. Math. 10, 177–183 (1967) · Zbl 0161.36203
[23] Reinsch, C.H.: Smoothing by spline functions II. Numer. Math. 16, 451–454 (1971) · Zbl 1248.65020
[24] Schaback, R.: Error estimates and condition number for radial basis function interpolation. Adv. Comput. Math. 3, 251–264 (1995) · Zbl 0861.65007
[25] Schoenberg, I.J.: Spline functions and the problem of graduation. Proc. Nat. Acad. Sci. (USA) 52, 947–950 (1964) · Zbl 0147.32102
[26] Schölkopf, B., Smola, A.J.: Learning with kernels – support vector machines, regularization, optimization, and beyond. MIT Press, Cambridge, Massachusetts, 2002
[27] Schultz, M.H.: Error bounds for polynomial spline interpolation. Math. Comput. 24, 507–515 (1970) · Zbl 0216.23002
[28] Schumaker, L.L.: Spline functions - basic theory. Wiley-Interscience Publication, New York, 1981 · Zbl 0449.41004
[29] Wahba, G.: Smoothing noisy data by spline functions. Numer. Math. 24, 383–393 (1975) · Zbl 0299.65008
[30] Wahba, G., Spline models for observational data. CBMS-NSF, Regional Conference Series in Applied Mathematics, Siam, Philadelphia, 1990 · Zbl 0813.62001
[31] Wei, T., Hon, Y., Wang, Y.B.: Reconstruction of numerical derivatives from scattered noisy data. Inverse Problems 21, 657–672 (2005) · Zbl 1071.65026
[32] Wendland, H.: Scattered data approximation. Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2005 · Zbl 1075.65021
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.