Convergence rates of "thin plate" smoothing splines wihen the data are noisy

We study the use of "thin plate" smoothing splines for smoothing noisy d dimensional data. The model is $$z_i = u(t_i ) + \varepsilon _i ,i = 1,2,...,n,$$ where u is a real valued function on a closed, bounded subset Ω of Euclidean d-space and the ei are random variables satisfying Eei=0, Eeiej=σ2, i=j, =0, i≠j, tieΩ. The zi are observed. It is desired to estimate u, given zl, ..., zn. u is only assumed to be "smooth", more precisely we assume that u is in the Sobolev space Hm(Ω) of functions with partial derivatives up to order m in L2(Ω), with m>d/2. u is estimated by un,m,λ, the restriction to Ω of ũn,m,λ, where ũn,m,λ is the solution to: Find ũ (in an appropriate space of functions on Rd) to minimize $$\frac{1}{n}\sum\limits_{i = 1}^n {(\tilde u(t_i ) - z_i ) + \lambda _i } ,\sum\limits_{1,...,i_m = 1_R d}^d {(\frac{{\partial ^m \tilde u}}{{\partial x_{i_1 } \partial x_{i_2 } ...\partial x_{i_m } }})^2 dx_1 ,dx_2 ,...,dx_d }$$