= Solution
The <Fejér kernel> is nonnegative, and preservation of constants gives
$$
\boxed{\pi^{-1}\int_{\mathbb T}|F_n(t)|dt=1}.
$$
By evenness,
$$
\sigma_n(f,x)-f(x)=\frac1{2\pi}\int_{\mathbb T}F_n(t)[f(x-t)-2f(x)+f(x+t)]dt.
$$
Put $\delta=n^{-1/2}$. Use $\omega_2(f,|t|)\leq(|t|/\delta+1)^2\omega_2(f,\delta)$ together with $F_n(t)\leq n/2$ and $F_n(t)\leq C/(nt^2)$. Splitting at $1/n$ and $\delta$ shows that the remaining weighted integral is uniformly bounded, so
$$
\boxed{\lVert\sigma_n(f)-f\rVert_\infty\leq C\omega_2(f,n^{-1/2})}.
$$
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