Multivariate Smoothing Spline Functions

Given data zᵢ = g (tᵢ) + ᵢ, 1 i n, where g is the unknown function, the tᵢ are known d-dimensional variables in a domain, and the ᵢ are i. i. d. random errors, the smoothing spline estimate gnu is defined to be the minimizes over h of n^ - 1 (zᵢ - h (tᵢ) ) ² + Jₘ (h), where > 0 is a smoothing parameter and Jₘ (h) is the sum of the integrals over of the squares of all the mth order derivatives of h. Under the assumptions that is bounded and has a smooth boundary, 0 appropriately, and the tᵢ become dense in as n, bounds on the rate of convergence of the expected square of pth order Sobolev norm (L₂ norm of pth 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 = Rᵈ, but only for the case p = 0, i. e. L₂ norm.