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Error estimates in Sobolev spaces for moving least square approximations. (English) Zbl 1001.65014
Let $\Omega$ be a convex set in $\Bbb{R}^N$ 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 $\{ z|\|z\|\le R \}$. Denote by $\cal{P}_{m}$ the set of polynomials of degree $m$ or less and $s$ its dimension. Let $p_{1},\dots,p_{s}$ be a basis of $\cal{P}_{m}$. 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 $$\sum_{j=1}^n \Phi_{R}(x-\xi_{j})\left( u(\xi_{j})-\sum_{k=1}^s p_{k}(\xi_{j})\alpha_{k}(x) \right)^2 \Rightarrow\min.$$ 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 $L^2$.

65D10Smoothing, curve fitting
65D05Interpolation (numerical methods)
41A10Approximation by polynomials
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