The <Korovkin theorem> states that positive linear operators $L_n:C[0,1]\to C[0,1]$ converge uniformly to the identity on every continuous function if they do so on the three test functions $1,t,t^2$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 4 / a / Solution

The <Schoenberg spline operator> is positive and $V_n1=1$. If $\xi_i=a_{1,i}$, then both $\tau_i$ and $\xi_i$ lie in $[t_i,t_{i+k}]$, so
$$
\lVert V_nt-t\rVert_\infty\leq k|\Delta_n|\to0.
$$
Because $a_{2,i}$ averages products of knots in the same interval and all points lie in $[0,1]$,
$$
\lVert V_nt^2-t^2\rVert_\infty\leq2k|\Delta_n|\to0.
$$
The <Korovkin theorem> now proves
$$
\boxed{\lVert V_n(f)-f\rVert_{C[0,1]}\to0}
$$
for every $f\in C[0,1]$.

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 4 / c / Solution

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Uniform convergence of positive linear operators on $1,x,x^2$ implies uniform convergence on every continuous function on a compact interval.


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 Incoming links: Korovkin theorem