A weight function theory of zero order basis function interpolants and smoothers

TL;DR

This paper develops a comprehensive weight function framework for zero order basis function interpolants and smoothers, establishing convergence, error estimates, and applications to B-splines and Sobolev spaces.

Contribution

It introduces a new weight function theory for basis functions, proves convergence and error bounds, and applies these to B-splines and smoothing problems with practical algorithms.

Findings

Proves pointwise convergence of minimal norm basis function interpolants.

Derives local error estimates for basis function interpolants.

Shows convergence of smoothing algorithms to data functions.

Abstract

I develop a weight func theory of zero order basis func interpolants and smoothers.**Ch1 Basis funcs and data spaces are defined using wt funcs. Data (native)spaces are used to formulate the variational problems which define our interpolants /smoothers. Introduce the tensor prod extended B-splines.**Ch2 Prove the p'twise convergence of the minimal norm basis func interpol to its data func and obtain orders of converg. Data func spaces for the B-splines are locally Sobolev spaces.**Ch3 Another set of error estims for basis func interpol. Use distrib'n Taylor expansion of exp(i(a,x)).**Ch4 Derive local interpol errors for data funcs with bounded first derivs.**Ch5 Introduce class of tensor prod wt funcs which I call the central diff wt funcs - related to the B-splines. Apply theory to these wt funcs to obtain interpol converge results. The data func spaces are locally Sobolev spaces. **Ch6 A non-param variat smoothing problem studied with special interest in converge of smoother to its data func. This smoother is the min norm interpol stabilized by a smoothing coeff.**Ch7: A non-parametric, scalable, smoothing problem shown to converge to its data func. We discuss the software which implements these algorithms.**Ch8: Characterizes bounded linear functionals on data space.**Ch9 Bilinear form used to characterize the bounded linear functionals on the data spaces generated by the wt funcs.**Ch10 We derive an upper bound for the deriv of the 1-dim (scaled) hat basis func smoother assuming the data func has bounded derivs and large supp wrt. the data region.**Ch11: In one dim only; the local data funcs are assumed to have bounded derivs on the data region and consider a scaled hat basis func. If the basis func has large enough supp wrt. the data region then we show the order of converg of the interpol is 1. **Ch12: Exten ops derived based on Wloka; assumes rectangle condit.