On the moduli of continuity and fractional-order derivatives in the problems of best mean-square approximations by entire functions of the exponential type on the entire real axis

Abstract

The exact Jackson-type inequalities with modules of continuity of a fractional order $\alpha \in (0,\infty )$ are obtained on the classes of functions defined via the derivatives of a fractional order $\alpha \in (0,\infty )$ for the best approximation by entire functions of the exponential type in the space $L_2(R)$. In particular, we prove the inequality $$2^{- \beta /2}\sigma^{- \alpha} (1 - \cos t)^{- \beta /2} \leq \sup \{ \scr {A}_\sigma (f) / \omega_{\beta }(\scr{D}^{\alpha} f, t/\sigma ) : f \in L^{\alpha}_2 (R)\} \leq \sigma^{-\alpha} (1/t^2 + 1/2)^{\beta /2},$$ where $\beta \in [1,\infty ), t \in (0, \pi ], \sigma \in (0,\infty ).$ The exact values of various mean $\nu$ -widths of the classes of functions determined via the fractional modules of continuity and majorant satisfying certain conditions are also determined.