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Multivariate smoothing spline functions. (English) Zbl 0581.65012
Author’s summary: Given data $$z_ i=g(t_ i)+\epsilon_ i$$, $$1\leq i\leq n$$, where g is the unknown function, the $$t_ i$$ are known d-dimensional variables in a domain $$\Omega$$, and the $$\epsilon_ i$$ are i.i.d. random errors, the smoothing spline estimate $$g_{n\lambda}$$ is defined to be the minimizer over h of $$n^{-1}\sum (z_ i-h(t_ i))^ 2+\lambda J_ m(h)$$, where $$\lambda >0$$ is a smoothing parameter and $$J_ m(h)$$ is the sum of the integrals over $$\Omega$$ of the squares of all the m th order derivatives of h. Under the assumptions that $$\Omega$$ is bounded and has a smooth boundary, $$\lambda$$ $$\to 0$$ appropriately, and the $$t_ i$$ become dense in $$\Omega$$ as $$n\to \infty$$, bounds on the rate of convergence of the expected square of p th order Sobolev norm $$(L_ 2$$ norm of p th derivatives) are obtained. These extend known results in the one dimensional case. The method of proof utilizes an approximation to the smoothing spline based on a Green’s function for a linear elliptic boundary value problem. Using eigenvalue approximation techniques, these rate of convergence results are extended to fairly arbitrary domains including $$\Omega ={\mathbb{R}}^ d$$, but only for the case $$p=0$$, i.e. $$L_ 2$$ norm.
Reviewer: Ch.K.Chui

##### MSC:
 65D10 Numerical smoothing, curve fitting 65C99 Probabilistic methods, stochastic differential equations 41A15 Spline approximation 41A25 Rate of convergence, degree of approximation 62J05 Linear regression; mixed models
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