Multivariate interpolation of arbitrarily spaced data by moving least squares methods (Q1098221)

Given a function f in scattered data points \(x_ 1,...,x_ n\in {\mathbb{R}}^ s\), we solve the least squares problem \[ \sum^{n}_{i- 1}(\sum^{q}_{j=0}a_ j(x)b_ j(x_ i)-f(x_ i))^ 2v_ i(x)=\min \] in order to generate interpolants \(\sum a_ j(x)b_ j(x)\). Here the \(b_ j\) denote basis functions, e.g. polynomials, and the \(v_ i(x)\) are inverse distance weight functions. We express the unknowns \(a_ j(x)\), the interpolant and the interpolation error in terms of moving convex combinations of functions corresponding to those of classical interpolation at \(q+1\) points with respect to these basis functions. Further, we discuss properties of moving least squares methods, and in the case of polynomial basis functions we test various methods and give perspective plots.