Error bounds for derivative estimates based on spline smoothing of exact or noisy data. (English) Zbl 0531.41011

Author’s summary: ”Estimates are found for the \(L_ 2\) error in approximating the jth derivative of a given smooth function f by the corresponding derivative of the 2mth order smoothing spline based on an n-point sample from the function. The results cover both the case of an exact sample from f and the case when the sample is subject to some random noise. In the noisy case, the estimates are for the expected value of the approximation error. These bounds show that, even in the presence of noise, the derivatives of the smoothing splines of order less than m can be expected to converge to those of f as the number of (uniform) sample points increases, and the smoothing parameter approaches zero at a rate appropriately related to m, n and the order of differentiability of f.”
Reviewer: E.Neumann


41A15 Spline approximation
41A25 Rate of convergence, degree of approximation
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