Convergence rates for multivariate smoothing spline functions

Abstract Let Ω be an open bounded subset of R d , the d -dimensional space, and let ƒ be an unknown function belonging to H m (Ω) , where m is an integer ( m > d 2 ). Given the values of ƒ at n scattered data point in Ω known with error, i.e., given z i = ƒ(t i ) + e i , i = 1, …, n , where the e i 's are i.i.d. random errors, we study the error E[¦ƒ − σ λ ¦ k,Ω 2 ], where ¦·¦ k,Ω 2 are the Sobolev semi-norms in H m (Ω) and σ λ is the thin plate smoothing spline with parameter λ, i.e., the unique minimizer of λ¦u¦ m 2 + ( 1 n )∑ i = 1 n (u(t i ) − z i ) 2 . Under the assumption that the boundary of Ω is smooth and the points satisfy a “quasi-uniform” condition, we obtain E[¦ƒ − σ λ ¦ k,Ω 2 ] ⩽ C[λ (m − k) m ¦ƒ¦ m,Ω 2 + D (nλ (2k + d 2m ) )], k = 0, 1,…, m − 1 .