Estimates of functionals by the second moduli of continuity of even derivatives

We establish an expansion of a function in terms of the second order differences of its derivatives. This expansion generalizes the well-known expansion in terms of the first order differences. Then, with the help of this expansion, we estimate some functionals by the second moduli of continuity. As particular cases of the estimates obtained, we have Jackson-type inequalities for approximations by entire fuctions of exponential type, trigonometric polynomials and splines in various function spaces. The constants in the new inequalities are smaller than those known before.