 Second modulus of smoothness

For a continuous periodic function setIt measures smoothness through second differences without requiring a derivative. For $h>0$, factoring the translation difference gives ${\omega }_2(f,t)\le (1+t/h{)}^2{\omega }_2(f,h)$. The second-difference integral formula gives ${\omega }_2(f,h)\le h^2\Vert f^{\prime \prime }{\Vert }_{\infty }$ for twice continuously differentiable functions.