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 Mat. Zametki, 2014, Volume 96, Issue 6, Pages 827–848 (Mi mz10372)

Best Mean-Square Approximations by Entire Functions of Exponential Type and Mean $\nu$-Widths of Classes of Functions on the Line

S. B. Vakarchuk

Alfred Nobel University Dnepropetrovsk

Abstract: For the classes $L^r_2(\mathbb{R})$, $r\in \mathbb{Z}_{+}$, we establish the upper and lower bounds for the quantities
$$\chi_{\sigma,k,r,\mu,p}(\psi,t):=\sup\{\mathcal{A}_{\sigma} (f^{(r-\mu)})/(\int_0^t \omega^p_k(f^{(r)},\tau) \psi(\tau) d\tau)^{1/p}:f \in L^r_2(\mathbb{R})\},$$
where $\mu, r \in \mathbb{Z}_{+}$, $\mu \le r$, $k \in \mathbb{N}$, $0< p \le 2$, $0< \sigma <\infty$, $0<t \le \pi/\sigma$, and $\psi$ is a nonnegative, measurable function summable on the closed interval $[0,t]$ and not equivalent to zero. In the cases $\chi_{\sigma,k,r,\mu,p}(1,t)$, where $\mu\in \mathbb{N}$, $1/\mu\le p \le 2$, and $\chi_{\sigma,k,r,\mu,2/k}(1,t)$, where $0<t \le \pi/(2 \sigma)$, we obtain the exact values of these quantities. We also obtain the exact values of the average $\nu$-widths of classes of functions defined in terms of the modulus of continuity $\omega^{*}$ and the majorant $\Psi$.

Keywords: entire function of exponential type, best mean-square approximation, average $\nu$-width, modulus of continuity, Jackson-type inequality, Fourier transform, Plancherel's theorem, Paley–Wiener theorem, Hölder's inequality, majorant, Kolmogorov width, Bernstein width, Bernstein's inequality.

DOI: https://doi.org/10.4213/mzm10372

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English version:
Mathematical Notes, 2014, 96:6, 878–896

Bibliographic databases:

UDC: 517.5
Revised: 10.12.2013

Citation: S. B. Vakarchuk, “Best Mean-Square Approximations by Entire Functions of Exponential Type and Mean $\nu$-Widths of Classes of Functions on the Line”, Mat. Zametki, 96:6 (2014), 827–848; Math. Notes, 96:6 (2014), 878–896

Citation in format AMSBIB
\Bibitem{Vak14} \by S.~B.~Vakarchuk \paper Best Mean-Square Approximations by Entire Functions of Exponential Type and Mean $\nu$-Widths of Classes of Functions on the Line \jour Mat. Zametki \yr 2014 \vol 96 \issue 6 \pages 827--848 \mathnet{http://mi.mathnet.ru/mz10372} \crossref{https://doi.org/10.4213/mzm10372} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3343642} \zmath{https://zbmath.org/?q=an:06435055} \elib{https://elibrary.ru/item.asp?id=22834448} \transl \jour Math. Notes \yr 2014 \vol 96 \issue 6 \pages 878--896 \crossref{https://doi.org/10.1134/S000143461411025X} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000347032700025} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84919897432} 

• http://mi.mathnet.ru/eng/mz10372
• https://doi.org/10.4213/mzm10372
• http://mi.mathnet.ru/eng/mz/v96/i6/p827

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. B. Vakarchuk, “On the moduli of continuity and fractional-order derivatives in the problems of best mean-square approximation by entire functions of the exponential type on the entire real axis”, Ukr. Math. J., 69:5 (2017), 696–724
2. R. Akgun, A. Ghorbanalizadeh, “Approximation by integral functions of finite degree in variable exponent Lebesgue spaces on the real axis”, Turk. J. Math., 42:4 (2018), 1887–1903
3. S. B. Vakarchuk, “Generalized characteristics of smoothness and some extreme problems of the approximation theory of functions in the space l-2(). Ii”, Ukr. Math. J., 70:10 (2019), 1550–1584
4. S. B. Vakarchuk, “Generalized characteristics of smoothness and some extreme problems of the approximation theory of functions in the space l-2(). I”, Ukr. Math. J., 70:9 (2019), 1345–1374
5. S. B. Vakarchuk, “On Estimates in $L_2(\mathbb{R})$ of Mean $\nu$-Widths of Classes of Functions Defined via the Generalized Modulus of Continuity of $\omega_{\mathcal{M}}$”, Math. Notes, 106:2 (2019), 191–202
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