Fractional-scale Fejér approximation bound ($\Vert {\sigma }_{n}f-f{\Vert }_{\infty }\le C_{\alpha }\omega (f,n^{-\alpha }),0<\alpha <1$)

The positive normalized Fejér kernel has a $1/(n{t}^2)$ tail. Split the error integral at $\delta =n^{-\alpha }$ and use the chaining inequality $\omega (f,t)\le (1+t/\delta )\omega (f,\delta )$. 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.