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

At the cusp of $f_0(t)=|t|$, positivity and evenness of the Fejér kernel giveSince $\sin (t/2)\le t/2$,Summing the supplied lower bound over $I_k$, $1\le k\le n^{\prime }=\lfloor n/2\rfloor$, yieldsThe harmonic series satisfies ${\sum }_{k=1}^{n^{\prime }}{k}^{-1}\ge c^{\prime }\log n$, so

The modulus of continuity of the periodic function $f_0$ obeys $\omega (f_0,1/n)=1/n$. If a universal Jackson-type estimateheld for all continuous periodic $f$, it would give $O(1/n)$ for $f_0$, contradicting the lower bound. Thus