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Applied Functional Analysis
Applied Functional Analysis

Kernel-based meshless approximation methods

Frank Filbir

A typical problem in science is the development of a theoretical model for a hidden process from observational data. More precisely, we are given a set of measurements A = C×ℂ = {(xj,fj) : j=1,..., M}, where we assume that the sampling nodes C={xj : j=1..., M} are a finite subset of a metric space (X,d). We suppose that there exists a function p which generated the observed data. This assumption obviously leads to the equations p(xj)=fj, j=1,..., M or, even more realistic, p(xj) ≈ fj since the data are usually corrupted with noise or measurement errors. The function p will then be considered as a model for the underlying process. Dealing with real world problems we can hardly expect that the sampling nodes are equally spaced or lie on a regular grid. Indeed, due to experimental constraints the sampling nodes are frequently scattered points. Consequently we have to deal with interpolation respectively approximation of functions from scattered data.
Various aspects of this approximation resp. interpolation problem will be considered.