Uniform approximation from Tchebycheff systems

Type of content
Publisher's DOI/URI
Thesis discipline
Degree name
Research Report
Publisher
University of Canterbury. Mathematics
Journal Title
Journal ISSN
Volume Title
Language
Date
1987
Authors
Brookes, Richard Gordon
Abstract

This report is concerned with the study of best uniform approximation to f E C[a,b) from the linear space generated by some finite subset U == {uo,u1, ... ,u} of C[a,b). n p* E span U such that By a best uniform approximation we mean max{jf(x) ~p*(x) j: x E [a,b]} = min{max{jf(x) -p(x) I x E [a,b]} : p E span u}. We explore, firstly, the case U = {l,x, ... ,xn}. It will be shown in Section 4 that in this situation each f E C[a,b) has a unique best approximation and for this best approximation there is a strong characterisation theorem. It is then natural to ask whether these results are true for a more general U = {u 0 ,u 1 , ••• ,u } . n If a strong type of linear independence known as the Haar condition is imposed on U then this will indeed turn out 1. to be the case. We will attempt to develop this condition using an approach more intuitively obvious than those found in many standard texts. When the Haar condition is not satisfied the problem rapidly becomes complicated and it appears that much work remains to be done in this area. A theorem concerning a particularly simple situation is given in Section 8.

Description
Citation
Keywords
Ngā upoko tukutuku/Māori subject headings
ANZSRC fields of research
Field of Research::01 - Mathematical Sciences::0101 - Pure Mathematics
Rights
Copyright Richard Gordon Brookes