Hölder error bound for Fejér summation ($\Vert {\sigma }_{n-1}f-f{\Vert }_{\infty }\le C_{\alpha }M{n}^{-\alpha }$)

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 $F_{n-1}(t)\le \min \{n/2,{\pi }^2/(2n{t}^2)\}$. 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 $O(M\log n/n)$.