Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 62 / 2 / b / Solution

Squaring the finite geometric sum in the Fejér kernel gives its Fourier seriesFor a fixed positive integer $k$ and $n>k$, convolution therefore gives ${\sigma }_n(e^{ikx})=(1-k/n)e^{ikx}$. Taking the real part,The ratio to $1/n$ is the nonzero constant $k$, even though the function is infinitely differentiable. This Fejér saturation on a Fourier mode disproves an $o(n^{-1})$ estimate for all $C^2$ periodic functions.