= Fejér first-moment approximation bound
{c}
{title2=$\|\sigma_nf-f\|_\infty\le C\omega(f,\log n/n)$}
The normalized <Fejér kernel> satisfies $0\le F_n(t)\le\min\{n/(2\pi),\pi/(2nt^2)\}$ on $0<|t|\le\pi$. Splitting its first absolute moment at $1/n$ gives $\int|t|F_n(t)dt\le1/(2\pi n)+(\pi/n)\log(\pi n)$. The chaining inequality for the <modulus of continuity>, $\omega(f,t)\le(1+t/\delta)\omega(f,\delta)$, then bounds the <Fejér sum> error by $\omega(f,\delta)[1+\delta^{-1}\int|t|F_n]$. Taking $\delta=\log n/n$ proves the stated bound for $n\ge2$ and hence <uniform convergence> for continuous periodic functions. The logarithm vanishes at $n=1$, so the displayed estimate is not intended at that index.
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