Quasi-interpolation in shift invariant spaces
Let \(\phi:\mathbb{R}^s\to\mathbb{R}\) be a compactly supported function, and \(S(\phi)\) denote the linear span of \(\{\phi(\cdot -{\mathbf k}) :{\mathbf k}\in\mathbb{Z}^s\}\), \(s\in\mathbb{N}.\) The authors consider the problem of approximating a continuous function \(f : \mathbb{R}^s\to \mathbb{R}\) on compact subsets of \(\mathbb{R}^s\) from the classes \(S(\phi(h\cdot))\), \(h > 0\). A construction of quasi-linear operators based on scattered data, with no conditions on their location, is described. The degree of polynomials which are preserved by these operators depends upon the density of the data. Rather than solving a system of equations, the authors solve a quadratic (or linear) programming problem of a size having the same order of magnitude. It is demonstrated how classical Markov inequalities for polynomials lead to the constructions of local, quasi-interpolatory operators.
local functional based on scattered data
scientific article; zbMATH DE number 1545050