= Fractional-scale Fejér approximation bound
{title2=$\|\sigma_nf-f\|_\infty\leq C_\alpha\omega(f,n^{-\alpha}),\quad0<\alpha<1$}
The positive normalized <Fejér kernel> has a $1/(nt^2)$ tail. Split the error integral at $\delta=n^{-\alpha}$ and use the chaining inequality $\omega(f,t)\leq(1+t/\delta)\omega(f,\delta)$. Its tail contributes at most a constant times $n^{\alpha-1}(1+\alpha\log n)\omega(f,\delta)$. The factor is uniformly bounded for each fixed $\alpha<1$, proving the displayed estimate. Its constant need not remain bounded as $\alpha$ tends to one.
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