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Mathematics of Computation

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The approximation power of moving least-squares

Author: David Levin
Journal: Math. Comp. 67 (1998), 1517-1531
MSC (1991): Primary 41A45; Secondary 41A25
MathSciNet review: 1474653
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Abstract: A general method for near-best approximations to functionals on $\mathbb {R}^d$, using scattered-data information is discussed. The method is actually the moving least-squares method, presented by the Backus-Gilbert approach. It is shown that the method works very well for interpolation, smoothing and derivatives’ approximations. For the interpolation problem this approach gives Mclain’s method. The method is near-best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation. The interpolation approximation in $\mathbb {R}^d$ is shown to be a $C^\infty$ function, and an approximation order result is proven for quasi-uniform sets of data points.

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Additional Information

David Levin
Affiliation: School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel

Received by editor(s): September 7, 1995
Received by editor(s) in revised form: September 4, 1996, and March 28, 1997
Article copyright: © Copyright 1998 American Mathematical Society