= Fejér second-modulus approximation bound
{c}
{title2=$\|\sigma_nf-f\|_\infty\le(1+\pi^2)\omega_2(f,n^{-1/2})$}
The even positive <Fejér kernel> of total mass one gives $\sigma_nf-f=\tfrac12\int F_n(t)[f(\cdot+t)-2f+f(\cdot-t)]dt$. The scaling property of the <second modulus of smoothness> bounds this by $\omega_2(f,\delta)(1+\delta^{-2}\int t^2F_n(t)dt)$. On $[-\pi,\pi]$, $\sin(|t|/2)\ge |t|/\pi$ gives $\int t^2F_n(t)dt\le\pi^2/n$. Choose $\delta=n^{-1/2}$. For twice continuously differentiable functions, the <second-difference integral formula> makes the error $O(n^{-1})$.
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