On functions which are the second modulus of continuity (Q1589391)

For \(f\in C_{2\pi}\) and \(\delta\in [0,\pi]\), let \[ \omega_2(f, \delta)= \sup_{|h|\leq\delta} \max_x|\Delta^2_h f(x)|, \] where \(\Delta^2_h f(x)\) means the second difference of \(f\) with stepsize \(h\). Then a function \(\varphi\) defined on \([0,\pi]\) is called second modulus of continuity if for certain \(f_\varphi\in C_{2\pi}\) \[ \omega_2(f_\varphi, \delta)= \varphi(\delta). \] The author presents several sufficient conditions for second moduli of continuity.