Conditions on Taylor series coefficients of a positive function

Question

Consider a continuous function $f$ on the interval $[0,1]$, defined by a convergent series expansion $$f(x) = \sum_{n=0}^\infty c_n x^n .$$ What are necessary (and ideally, sufficient) conditions on the coefficients $\{ c_n \}$ for $f$ to be positive on $[0,1]$, $f(x) > 0$ for all $x \in [0,1]$? If this is too difficult or unknown, can an answer be given when $f$ is a polynomial?