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.
