SCHWEITZER 1978 Problem 6

Suppose that the function $g : (0,1) \rightarrow \mathbb{R}$ can be uniformly approximated by polynomials with nonnegative coefficients. Prove that $g$ must be analytic. Is the statement also true for the interval $(-1,0)$ instead of $(0,1)$ ?

J. Kalina, L. Lempert