An improved order of approximation for thin-plate spline interpolation in the unit disc

Summary.

We show that the $L_p$-norm of the error in thin-plate spline interpolation in the unit disc decays like $O(h^{\gamma_p+1/2})$, where $\gamma_p:=\min\{2,1+2/p\}$, under the assumptions that the function to be approximated is $C^\infty$ and that the interpolation points contain the finite grid $\{hj:j\in\mathbb{Z}^2, \vert hj\vert <1-h\}$.