Abstract
In the paper, we study inequalities for the best trigonometric approximations and fractional moduli of smoothness involving the Weyl and Liouville-Grunwald derivatives in Lp, 0 < p < 1. We extend known inequalities to the whole range of parameters of smoothness as well as obtain several new inequalities. As an application, the direct and inverse theorems of approximation theory involving the modulus of smoothness ωβ(f(α), δ)p, where f(α) is a fractional derivative of the function f, are derived. A description of the class of functions with the optimal rate of decrease of a fractional modulus of smoothness is given.
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