Realization functionals and description of a modulus of smoothness in variable exponent Lebesgue spaces

In variable exponent Lebesgue spaces the equivalence between generalized modulus of smoothness defined with help of one-sided Steklov means and realization functionals using Riesz-Zygmund and Euler means is established. The description of a class of functions which are equivalent to a generalized modulus of smoothness of order $r\in\mathbb N$ is given.