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Convergence rates for multivariate smoothing spline functions. (English) Zbl 0646.41006
Given the set $$\{t_ i\}$$ $$n_ 1$$ of scattered data points in an open bounded subset $$\Omega$$ of the d-dimensional space $${\mathbb{R}}^ d$$and the values $$\{z_ i\}$$ $$n_ 1$$, the thin plate smoothing spline $$\sigma_{\lambda}$$ is defined as the unique minimizer of the expression $$\lambda | u|^ 2_{m,\Omega}+(1/n)\sum^{n}_{i-1}(u(t_ i)-z_ i)^ 2,$$ Here $$| \cdot |$$ $$2_{m,\Omega}$$ is the Sobolev semi-norm in H m($$\Omega)$$. The author studies the approximation of functions $$f\in H$$ m($$\Omega)$$ by a thin plate smoothing spline $$\sigma_{\lambda}$$ based on the values $$z_ i=f(t_ i)+\epsilon_ i$$, $$i=1,...,n$$ of f dt n scattered points in $$\Omega$$ known with random errors $$\epsilon_ i$$. He obtains the following estimate $E[| f- \sigma_{\lambda}|^ 2_{k,\Omega}]\leq C[\lambda^{(m-k)/m}| f|^ 2_{m,\Omega}+D/(n\lambda^{(2k+d)/2m})],\quad k=0,...,m-1$ provided the boundary of $$\Omega$$ is smooth and the points satisfy a “quasi-uniform” condition.
Reviewer: B.Bojanov

##### MSC:
 41A15 Spline approximation 41A25 Rate of convergence, degree of approximation 65D10 Numerical smoothing, curve fitting
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