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

  • S. B. Vakarchuk Днепропетр. ун-т им. А. Нобеля

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.
Published
25.05.2017
How to Cite
VakarchukS. B. “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”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, no. 5, May 2017, pp. 599-23, http://umj.imath.kiev.ua/index.php/umj/article/view/1720.
Section
Research articles