Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 2 / c / Solution

Twice applying the fundamental theorem of calculus giveshence $\boxed{{\omega }_2(f,t)\le t^2\Vert f^{\prime \prime }{\Vert }_{\infty }}$. Part (b) gives $\Vert {\sigma }_n(f)-f{\Vert }_{\infty }=O(n^{-1})$. This cannot be little-$o$ for every $C^2$ function: for $f_0(x)=\cos x$,