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
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...
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| Datum: | 2017 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Russisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2017
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1720 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | 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. |
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