The approximation power of moving least-squares. (English) Zbl 0911.41016

The author investigates a general method to approximate linear functionals on spaces of multivariate functions, namely the so-called moving least-squares method. Given a data set \(\{L_i(f) \}_{i=1}^l\) (for example, \(f(x_i)\) for sample points \(x_i\in \mathbb{R}^d\)), an unknown functional \(L(f)\) (for example, \(f(x)\) for another point \(x\)) is approximated by \(\sum_{i=1}^l a_iL_i(f)\), where the coefficients are obtained by minimizing the quadratic form \(\sum_{i=1}^l w(L,L_i)a_i^2\) under the linear constraints \(\sum_{i=1}^l a_iL_i(p_j)\), \(j=1,\dots,J\). This means that the elements \(p_j\) (typically polynomials of low degree) are reproduced, while the nonnegative weight \(w(L,L_i)\) is a separation measure between the functionals \(L_i\) and \(L\) (such as a radial function of the distance of the sample point \(x_i\) from \(x\)). It is shown how this constrained minimization can be solved in a matrix formulation. For the multivariate interpolation problem and for specific choices of the weights, \(C^\infty\)-smoothness in \(x\) and the approximation order on suitable data sets are established. The paper concludes with several numerical examples, including some in a bi- and trivariate setting as well as with data-dependent separation measures.
Reviewer: E.Quak (Oslo)


41A45 Approximation by arbitrary linear expressions
41A25 Rate of convergence, degree of approximation
65D15 Algorithms for approximation of functions
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