Approximating a strictly increasing non-negative function on a non-negative domain by polynomials with non-negative coefficients

Question

Let $f:[0,2]\rightarrow [0,\infty)$ be a strictly increasing smooth function. The Weierstrass approximation theorem says that we can uniformly approximate $f$ by polynomials. But my concern is

Question: Can we uniformly approximate $f$ by polynomials with non-negative coefficients?

All the proofs of the Weierstrass approximation theorem I have seen is either existential or the continuous function $f:[0,1]\rightarrow \mathbb{R}$ has been approximated by the sequence of functions $$B_n(x)=\sum_{k=0}^n{n \choose k}x^k(1-x)^{n-k}f(\frac{k}{n}).$$ But there is no assurance that all the $B_n(x)$s are polynomials with non-negative coefficients.

P.s.- For my particular case, I don't care about the constant term of the polynomials, I only need the coefficients of $x^k$ for $k>0$ to be non-negative.

Answer 1

This is just my comment reposted as an answer.

The problem of which functions $f:[0,1]\to\mathbb{R}$ can be approximated by polynomials with non negative coefficients has already been studied, eg by Robert Whitley: as theorem 2 of this paper states, a continuous function $f:[0,1]\to\mathbb{R}$ is a pointwise limit of polynomials with non negative coefficients iff $f(x)=\sum_{n=0}^\infty a_nx^n$ for some non negative sequence $a_n$ such that $\sum a_n<\infty$.