Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 69 / 1 / Solution

Use the modulus of continuity on the circle, $\omega (f,\delta )={\sup }_{|h|\le \delta }\Vert f(\cdot +h)-f{\Vert }_{\infty }$. Splitting a displacement into steps of length at most $\delta$ and applying the triangle inequality givesThe Fejér kernel is nonnegative and has integral one, soWe therefore need its first absolute moment. The inequalities $|\sin (nt/2)|\le n|\sin (t/2)|$ and $\sin (|t|/2)\ge |t|/\pi$ on $[-\pi ,\pi ]$ implySplitting at $1/n$ givesTaking $\delta ={\delta }_n=(\ln n)/n$ in the preceding error bound proves the Fejér first-moment approximation bound:The constants do not depend on $f$ or $n$. By the Heine-Cantor theorem, a continuous function on the compact circle is uniformly continuous, hence $\omega (f,{\delta }_n)\to 0$ as ${\delta }_n\to 0$. Thus the Fejér sums converge uniformly to $f$. The index restriction is necessary: at $n=1$ the printed scale is zero, while ${\sigma }_{1}f$ is the mean of $f$ and generally differs from it.