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

TL;DR

This paper develops a comprehensive weight function theory for positive order basis function interpolants and smoothers, providing new formulas, convergence results, and analysis of basis functions using weight functions and variational problems.

Contribution

It introduces a weight function framework for basis functions and smoothers, deriving new formulas, proving existence and uniqueness, and analyzing convergence and bounded functionals.

Findings

Derived modified inverse-Fourier transform formulas for basis functions.

Proved existence and uniqueness of minimal seminorm interpolants.

Analyzed pointwise convergence rates of interpolants and smoothers.

Abstract

In this document I develop a weight function theory of positive order basis function interpolants and smoothers. **In Chapter 1 the basis functions and data spaces are defined directly using weight functions. The data spaces are used to formulate the variational problems which define the interpolants and smoothers discussed in later chapters. The theory is illustrated using some standard examples of radial basis functions and two classes of weight functions I will call the tensor product extended B-splines and the central difference weight functions. **In Chapter 2 I derive modified inverse-Fourier transform formulas for the basis functions and the data functions (native spaces) and to use these formulas to obtain bounds for the rates of increase of these functions and their derivatives near infinity. **Chapter 3 shows how to prove functions are basis functions without using the awkward space of test functions $S_{0,n}$ which are infinitely smooth functions of rapid decrease with several zero-valued derivatives at the origin. Worked examples include several classes of well-known radial basis functions. **In Chapter 4 we prove the existence and uniqueness of a solution to the minimal seminorm interpolation problem. We then derive orders for the pointwise convergence of the interpolant to its data function as the density of the data increases. **In Chapter 5 a well-known non-parametric variational smoothing problem will be studied with special interest in the order of pointwise convergence of the smoother to its data function. This smoothing problem is the minimal norm interpolation problem stabilized by a smoothing coefficient. **In Chapter 6 a non-parametric, scalable, variational smoothing problem will be studied, with special interest in its order of pointwise convergence to its data function. **In Chapter 7 we study the bounded linear functionals on the data spaces.