Existence of best rational Tchebycheff approximations

Barry William Boehm

Vol. 15 (1965), No. 1, 19–28

DOI: 10.2140/pjm.1965.15.19

Abstract

Some conditions are given which guarantee the existence of best Tchebycheff approximations to a given function $f$ by generalized rational functions of the form

$$r\left(x\right)=\frac{a_1{g}_1\left(x\right)+\cdots +a_n{g}_n\left(x\right)}{b_1{h}_1\left(x\right)+\cdots +b_m{h}_m\left(x\right)}$$

The principal theorem states that such a best Tchebycheff approximation exists whenever $f,g_1,\cdots ,g_n,h_1,\cdots ,h_m$ are bounded continuous functions, defined on an arbitrary topological space $X$, and the set $\left\{h_1,\cdots ,h_m\right\}$ has the dense nonzezo property on $X$: if $b_1,\cdots ,b_n$ are real numbers not all zero, then the function $b_1{h}_1+\cdots +b_m{h}_m$ is different from zero on a set dense in $X$. An equivalent statement is that the set $\left\{h_1,\cdots ,h_m\right\}$ is linearly independent on every open subset of $X$.

Further theorems assure the existence of best weighted Tchebycheff approximations and best constrained Tchebycheff approximations by generalized rational functions and by approximating functions of other similar forms.

Mathematical Subject Classification

Primary: 41.17

Secondary: 41.40

Milestones

Received: 12 March 1964

Published: 1 March 1965