 Solution

Periodicity and the Hölder condition give $|f(x-t)-f(x)|\le M|t{|}^{\alpha }$ for $|t|\le \pi$. The positive normalized Fejér kernel therefore yieldsThe exponential-sum representation of the kernel bounds $F_{n-1}(t)\le n/2$. Also $\sin (t/2)\ge t/\pi$ for $0\le t\le \pi$, so $F_{n-1}(t)\le {\pi }^2/(2n{t}^2)$. Splitting at $1/n$ and using $0<\alpha <1$ givesHence the Hölder error bound for Fejér summation isThe constant depends on the Hölder exponent and $M$, but not on $n$ or $f$ beyond its Hölder seminorm bound.