= Second modulus of smoothness
{title2=$\omega_2(f,h)$}
For a continuous periodic function set
$$
\omega_2(f,h)=\sup_{|u|\leq h}\|f(\cdot+u)-2f+f(\cdot-u)\|_\infty.
$$
It measures smoothness through second differences without requiring a derivative. For $h>0$, factoring the translation difference gives $\omega_2(f,t)\leq(1+t/h)^2\omega_2(f,h)$. The <second-difference integral formula> gives $\omega_2(f,h)\leq h^2\|f''\|_\infty$ for twice continuously differentiable functions.
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