Multivariate Smoothing Spline Functions

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