Convergence rates for multivariate smoothing spline functions
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.
scientific article; zbMATH DE number 4054226
scientific article