Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 12F (Topics in Analysis)

For fixed $t$, let $B$ have the binomial distribution with parameters $m,t$. The Bernstein polynomial is $f_m(t)=\mathbb{E}[f(B/m)]$. Continuity on the compact interval gives uniform continuity and a finite bound $M=\Vert f{\Vert }_{\infty }$.

Given $ϵ>0$, choose $\delta >0$ so $|f(s)-f(t)|<ϵ$ whenever $|s-t|<\delta$. Since $\mathbb{E}[B/m]=t$ and $\mathrm{Var}(B/m)=t(1-t)/m\le 1/(4m)$, Chebyshev's inequality givesThe bound is independent of $t$, including both endpoints. Taking the supremum and then letting $m$ tend to infinity proves $f_m\to f$ uniformly on $[0,1]$. This provides the required polynomial-approximation theorem with its proof.

For each node define the Lagrange interpolation polynomialThe nodes are distinct, so this is well defined, has degree $n-1$, and satisfies $L_j(x_{k,n})={\delta }_{jk}$. Its square has degree $2n-2$, within the assumed exactness range. ThereforeThe strict inequality follows because $L_j$ is not the zero polynomial. Exactness for the constant polynomial also gives ${\sum }_{j=1}^n{a}_{j,n}=1$. Thus the quadrature is a positive weighted average; the requested nonnegativity follows in stronger form.

Choose a polynomial $P$ with $\Vert f-P{\Vert }_{\infty }<ϵ$, using part (i). For all sufficiently large $n$, its degree is at most $2n-1$, so $I_n(P)={\int }_0^{1}P$. Positive weights summing to one give $|I_n(f-P)|\le ϵ$, and $|{\int }_0^1(f-P)|\le ϵ$. HenceSince $ϵ$ is arbitrary, the quadratures converge to the integral of every continuous $f$. The positivity established in part (ii), not just polynomial exactness on its own, controls the approximation error uniformly in $n$.