Vol. 15, No. 1, 1965

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Existence of best rational Tchebycheff approximations

Barry William Boehm

Vol. 15 (1965), No. 1, 19–28
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 $f,{g}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{g}_{n},{h}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{h}_{m}$ are bounded continuous functions, defined on an arbitrary topological space $X$, and the set $\left\{{h}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{h}_{m}\right\}$ has the dense nonzezo property on $X$: if ${b}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{b}_{n}$ 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 $\left\{{h}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{h}_{m}\right\}$ 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.

Primary: 41.17
Secondary: 41.40