Fejér first-moment approximation bound ($\Vert {\sigma }_{n}f-f{\Vert }_{\infty }\le C\omega (f,\log n/n)$)

The normalized Fejér kernel satisfies $0\le F_n(t)\le \min \{n/(2\pi ),\pi /(2n{t}^2)\}$ on $0<|t|\le \pi$. Splitting its first absolute moment at $1/n$ gives $\int |t|F_n(t)dt\le 1/(2\pi n)+(\pi /n)\log (\pi n)$. 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 $n\ge 2$ and hence uniform convergence for continuous periodic functions. The logarithm vanishes at $n=1$, so the displayed estimate is not intended at that index.

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.