 Positive Bernstein coefficients for a strictly positive polynomial

If a polynomial $f$ is strictly positive on $[0,1]$, take $g_n=B_n^{-1}f$. By inverse Bernstein approximation on a fixed-degree polynomial space, eventually $g_n>0$ on the interval. Thereforehas strictly positive coefficients. The converse implication to nonnegativity follows immediately because each basis term is nonnegative. Strict positivity is essential for the general existence result: a nonzero polynomial vanishing at an interior point cannot have a nonnegative-coefficient representation in this basis, whose individual terms are positive throughout the open interval.