Overview document for: A weight function theory of basis function interpolants and smoothers
aa r X i v : . [ m a t h . NA ] O c t Contents i verview document for:A weight function theory of basis function interpolants andsmoothers. Phillip Y. Williams
E-mail address : [email protected], [email protected] ey words and phrases. interpolant, smoother, basis, convergence, non-parametric, Hilbert,reproducing-kernel, errorI would like to thank the University of Canberra for their generous assistance..Also, thanks to my Masters degree supervisors Dr Markus Hegland and Dr Steve Roberts of the CMAat the Australian National University. Abstract.
This document is a brief overview of two documents which continue to develop the weightfunction theory of basis function smoothers and interpolants. One document considers the zero ordertheory and one considers the positive order theory.
Created this document using the abstracts from the version 1 documents.
Altered this document using the abstracts from the consolidated version 2 docu-ments i.e. version 2 of arXiv:0708.0780 and version 2 of arXiv:0708.0795. . POSITIVE ORDER DOCUMENT (ARXIV:0708.0795) 1
1. Overview
This work currently consists of two documents which continue to develop the
Light weightfunction theory of basis function smoothers and interpolants . One document considers the zero order theory and one considers the positive order theory .In brief, some important general features:(1) Extends the positive order work of Light and Wayne [ ], [ ] and [ ] to the zero order case andextends the positive order case to tensor product weight functions.For both the positive and zero order cases:(2) A weight function is first defined and then used to define a continuous basis function and a datafunction Hilbert space are defined using the Fourier transform. This technique is illustrated byseveral examples.(3) The standard minimal norm and seminorm interpolants are defined and pointwise orders ofconvergence are derived on a bounded set.(4) We define the well known variational non-parametric smoother which stabilizes the interpolantusing a smoothing parameter - I call this the Exact smoother. Orders of uniform pointwiseconvergence are derived on a bounded open set.(5) A scalable smoother is derived which I call the Approximate smoother. Orders of uniformpointwise convergence are derived on a bounded open set.(6) For the zero order case numeric examples are given which compare the theoretical and actualerrors w.r.t. the data function.
2. Zero order document (arXiv:0708.0780)
Here is a short description of the document (the abstract).In this document I develop a weight function theory of zero order basis function interpolants andsmoothers.In
Chapter 1 the basis functions and data spaces are defined directly using weight functions. Thedata spaces are used to formulate the variational problems which define the interpolants and smoothersdiscussed in later chapters. The theory is illustrated using some standard examples of radial basisfunctions and a class of weight functions I will call the tensor product extended B-splines.In
Chapter 2 the theory of Chapter 1 is used to prove the pointwise convergence of the minimal normbasis function interpolant to its data function and to obtain orders of convergence. The data functionsare characterized locally as Sobolev-like spaces and the results of several numerical experiments usingthe extended B-splines are presented.In
Chapter 3 a large class of tensor product weight functions will be introduced which I call thecentral difference weight functions. These weight functions are closely related to the extended B-splinesand have similar properties. The theory of this document is then applied to these weight functions toobtain interpolation convergence results. To understand the theory of interpolation and smoothing it isnot necessary to read this chapter.In
Chapter 4 a non-parametric variational smoothing problem will be studied using the theory ofthis document with special interest in its order of pointwise convergence of the smoother to its datafunction. This smoothing problem is the minimal norm interpolation problem stabilized by a smoothingcoefficient.In
Chapter 5 a non-parametric, scalable, variational smoothing problem will be studied, again withspecial interest in its order of pointwise convergence to its data function. We discuss the
SmoothOperator software (freeware) package which implements the Approximate smoother algorithm. It has a full usermanual which describe several tutorials and data experiments.
3. Positive order document (arXiv:0708.0795)
Here is a short description of the document (the abstract).In this document I develop a weight function theory of positive order basis function interpolants andsmoothers.In
Chapter 1 the basis functions and data spaces are defined directly using weight functions. Thedata spaces are used to formulate the variational problems which define the interpolants and smoothers
CONTENTS discussed in later chapters. The theory is illustrated using some standard examples of radial basisfunctions and a class of weight functions I will call the tensor product extended B-splines.
Chapter 2 shows how to prove functions are basis functions without using the awkward space of testfunctions So,n which are infinitely smooth functions of rapid decrease with several zero-valued derivativesat the origin. Worked examples include several classes of well-known radial basis functions.The goal of
Chapter 3 is to derive ‘modified’ inverse-Fourier transform formulas for the basisfunctions and the data functions and to use these formulas to obtain bounds for the rates of increase ofthese functions and their derivatives near infinity.In
Chapter 4 we prove the existence and uniqueness of a solution to the minimal seminorm inter-polation problem. We then derive orders for the pointwise convergence of the interpolant to its datafunction as the density of the data increases.In
Chapter 5 a well-known non-parametric variational smoothing problem will be studied withspecial interest in the order of pointwise convergence of the smoother to its data function. This smoothingproblem is the minimal norm interpolation problem stabilized by a smoothing coefficient.In
Chapter 6 a non-parametric, scalable, variational smoothing problem will be studied, again withspecial interest in its order of pointwise convergence to its data function. ibliography
1. W. Light and H. Wayne,
Error estimates for approximation by radial basis functions. , Approximation theory, waveletsand applications (Maratea, 1994), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 454, Kluwer Acad. Publ., Dordrecht,1995, pp. 215–246.2. ,
On power functions and error estimates for radial basis function interpolation , J. Approx. Th. (1998), no. 2,245–266.3. , Spaces of distributions, interpolation by translates of a basis function and error estimates , Numer. Math. (1999), no. 3, 415–450.(1999), no. 3, 415–450.