The Stone-Weierstrass Theorem guarantees that every continuous function defined on a closed interval [a,b] can be approximated as closely as desired by a polynomial.
I'm curious in polynomials of the form
$P(x) = a_0 + a_1 f(x)^1 + a_2 f(x)^2 + a_3 f(x)^3$ where $f(x)$ is some continuous function f(x) could equal $e^x$ for example. What can be said for polynomials of this form? Does something similar to the Stone-Weierstrass Theorem hold?
Best Answer
If $f$ is injective and continuous, then the Stone-Weierstrass allows us to deduce that the functions of the form $\sum_{k=0}^na_kf^k$ are dense in $C\bigl([a,b]\bigr)$. Actually, assuming that $f$ is continuous, the set that I described is dense if and only if $f$ is injective.