Toward approximate moving least squares approximation with irregularly spaced centers.

*(English)*Zbl 1060.65041Summary: By combining the well-known moving least squares approximation method and the theory of approximate approximations due to V. Maz’ya and G. Schmidt [J. Approximation Theory 110, No. 2, 125–145 (2001; Zbl 0976.41004)] we are able to present an approximate moving least squares method which inherits the simplicity of D. Shepard’s method [A two-dimensional interpolation function for irregularly spaced data, in: Proc. 23rd Nat. Conf., ACM. New York, 517–523 (1968)] along with the accuracy of higher-order moving least squares approximations.

In this paper we focus our interest on practical implementations for irregularly spaced data sites. The two schemes described here along with some first numerical experiments are to be viewed as exploratory work only. These schemes apply to centers that are obtained from gridded centers via a smooth parametrization. Further work to find a robust numerical scheme applicable to arbitrary scattered data is needed.

In this paper we focus our interest on practical implementations for irregularly spaced data sites. The two schemes described here along with some first numerical experiments are to be viewed as exploratory work only. These schemes apply to centers that are obtained from gridded centers via a smooth parametrization. Further work to find a robust numerical scheme applicable to arbitrary scattered data is needed.

##### MSC:

65F20 | Numerical solutions to overdetermined systems, pseudoinverses |

65D15 | Algorithms for approximation of functions |

##### Keywords:

Moving least squares; Approximate approximation; Irregularly spaced centers; Numerical experiments; scattered data##### Software:

ANN
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\textit{G. E. Fasshauer}, Comput. Methods Appl. Mech. Eng. 193, No. 12--14, 1231--1243 (2004; Zbl 1060.65041)

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##### References:

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