Approximation orders for natural splines in arbitrary dimensions
Let \(\Omega\subset \mathbb{R}^n\) be a bounded domain with Lipschitz boundary, and let \(m\), \(k\in\mathbb{N}_0\), \(p\in [2,\infty]\) be such that \(m>n/2\) and \(H^m(\Omega)\subset W^{k,p}(\Omega)\) hold. The aim of the authors is to improve a result by \textit{J. Duchon} [RAIRO, Anal. Numer. 12, 325-334 (1978; Zbl 0403.41003)] concerning the minimal norm interpolant of a function belonging to the Beppo Levi space \(BL^m(\Omega)\). It is shown that there exist constants \(C\) and \(h_0>0\) having the following property: for any collection \(X=\{x_1, \dots, x_N\} \subset\Omega\) of interpolation points containing a \({\mathcal P}^n_{m-1}\)-unisolvent subset and satisfying NEWLINE\[NEWLINEh=\sup_{x\in\Omega} \inf_{x_i\in X} \|x-x_i\|_2\leq h_0,NEWLINE\]NEWLINE and for all \(f\in BL^{2m} (\Omega)\) with \(\partial^\alpha f=0\) on \(\partial\Omega\) for \(|\alpha |=m, \dots,2m-1\), the function \(s_f \in BL^m (\mathbb{R}^n)\) that minimizes \(\|s_f\|_{BL^m (\mathbb{R}^n)}\) and interpolates \(s_f(x_i)= f(x_i)\), \(i=1,\dots,N\), satisfies \(\sum_{|\alpha |=k} \|\partial^\alpha (f-s_f) \|_{L^p (\Omega)}\leq Ch^{2m-k-n/2+n/p} \|f\|_{BL^{2m} (\Omega)}\).
scientific article