Bernstein basis ($b_{k,n}(x)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)x^k(1-x{)}^{n-k}$)

These $n+1$ nonnegative polynomials form a basis of polynomials of degree at most $n$, and sum to one on $[0,1]$. For $0\le j\le n$,where ratios for $j=0$ equal one and terms with $k<j$ vanish. The identity follows from $\left(\genfrac{}{}{0ex}{}{n}{k}\right)(k{)}_j/(n{)}_j=\left(\genfrac{}{}{0ex}{}{n-j}{k-j}\right)$ and the binomial theorem. It gives explicit basis coefficients using the falling factorial.

Write $p(x)=a_1{x}^2+2a_{2}x(1-x)+a_3(1-x{)}^2$ on $[0,1]$. Endpoint evaluation gives $a_1=p(1)$ and $a_3=p(0)$, while midpoint evaluation gives $a_2=2p(1/2)-[p(0)+p(1)]/2$. Thus $\Vert p{\Vert }_{\infty }\le 1$ implies $|a_1|,|a_3|\le 1$ and $|a_2|\le 3$. These bounds are sharp: $p(x)=8x^2-8x+1$ has uniform norm one and coefficients $(1,-3,1)$.

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.

Write $f={\sum }_k{\beta }_{k,n}{b}_{k,n}$ in the Bernstein basis. Representing the same polynomial at degrees $n$ and $n+1$ gives the displayed convex combinations for internal indices; the endpoint coefficients are copied. Therefore the minimum coefficient cannot decrease under degree elevation. In the unnormalized basis $x^k(1-x{)}^{n-k}$, the elevated coefficients are $c_k^{\prime }=c_{k-1}+c_k$ with the corresponding endpoint convention, so nonnegativity certificates are preserved.