Error estimates in Sobolev spaces for moving least square approximations (Q2706402)

Let \(\Omega\) be a convex set in \(\mathbb{R}^N\) and let \(\xi_{1}, \xi_{2},\dots,\xi_{n}\) be given points in \(\Omega\). Furthermore let \(\Phi_{R}\) be a function with values in \([0,1]\) and support in the ball \(\{ z|\|z\|\leq R \}\). Denote by \(\mathcal{P}_{m}\) the set of polynomials of degree \(m\) or less and \(s\) its dimension. Let \(p_{1},\dots,p_{s}\) be a basis of \(\mathcal{P}_{m}\). The moving least squares approximation is defined by \(\hat{u}:=P^{*}(x,x)\), where \(P^{*}(x,y)= \sum_{{k=1}}^s p_{k}(y)\alpha_{k}(x)\) and for each \(x \in \Omega\) the \(\alpha_{k}(x)\) are the solution of the weighted least square problem NEWLINE\[NEWLINE\sum_{j=1}^n \Phi_{R}(x-\xi_{j})\left( u(\xi_{j})-\sum_{k=1}^s p_{k}(\xi_{j})\alpha_{k}(x) \right)^2 \Rightarrow\min.NEWLINE\]NEWLINE This problem is solvable under the condition imposed in the paper that for each \(x\) there exist \(s\) points in the set \(\{ \xi_{1}, \dots,\xi_{n} \}\) such that Lagrange interpolation is possible and \(\Phi_{R}(x-\xi_{j})>0\) in these points. The author proves optimal order error estimates for the function and its gradient in \(L^\infty\) and \(L^2\).