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

The Schoenberg spline operator is positive and $V_{n}1=1$. If ${\xi }_i=a_{1,i}$, then both ${\tau }_i$ and ${\xi }_i$ lie in $[t_i,t_{i+k}]$, soBecause $a_{2,i}$ averages products of knots in the same interval and all points lie in $[0,1]$,The Korovkin theorem now provesfor every $f\in C[0,1]$.