Second modulus of smoothness (${\omega }_2(f,h)$)

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.

For a twice continuously differentiable function and $h\ge 0$, the fundamental theorem of calculus givesThe nonnegative triangular weight has integral $h^2$, so its supremum norm estimate directly controls the second modulus of smoothness.