Solution

Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 69 / 4 / Solution

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{title2=$\|\sigma_{n-1}f-f\|_\infty\le C_\alpha M n^{-\alpha}$}

For a periodic <Hölder continuous function> of <Hölder exponent> $0<\alpha<1$ and <Hölder seminorm> at most $M$, the degree-$n-1$ <Fejér sum> has the displayed <supremum norm> error bound. In the half-kernel normalization $F_{n-1}=(2n)^{-1}\sin^2(nt/2)/\sin^2(t/2)$, use $F_{n-1}(t)\le\min\{n/2,\pi^2/(2nt^2)\}$. Splitting the convolution error integral at $1/n$ gives the valid constant $C_\alpha=1/[\pi(\alpha+1)]+\pi/(1-\alpha)$. At the endpoint $\alpha=1$ this argument instead gives $O(M\log n/n)$.

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 Incoming links: Hölder error bound for Fejér summation