Quadratic Bernstein coefficient bound
= Quadratic Bernstein coefficient bound
{title2=$(|a_1|,|a_2|,|a_3|)\leq(1,3,1)$}
Write $p(x)=a_1x^2+2a_2x(1-x)+a_3(1-x)^2$ 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 $\|p\|_\infty\leq1$ implies $|a_1|,|a_3|\leq1$ and $|a_2|\leq3$. These bounds are sharp: $p(x)=8x^2-8x+1$ has <uniform norm> one and coefficients $(1,-3,1)$.