Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 318 / 1 / b / Solution

Because the Fejér kernel has normalized integral one,For $f\in \mathrm{Lip}\alpha$ this impliesOn $0\le t\le \pi$, the inequalities $\sin (t/2)\ge t/\pi$ and $|\sin (nt/2)|\le n|\sin (t/2)|$ giveSplitting the integral at $1/n$ givesFor $0<\alpha <1$, both terms are $O(n^{-\alpha })$. For $\alpha =1$, the second is $O((\log n)/n)$. Uniformly in $x$,where the constants absorb the Lipschitz continuity constant $M$.