Estimates for functionals with a known, finite set of moments, in terms of moduli of continuity, and behavior of constants, in the Jackson-type inequalities
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O. L. Vinogradov and V. V. Zhuk
Translated by: O. L. Vinogradov - St. Petersburg Math. J. 24 (2013), 691-721
- DOI: https://doi.org/10.1090/S1061-0022-2013-01261-1
- Published electronically: July 24, 2013
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Abstract:
A new technique is developed for estimating functionals by moduli of continuity. The generalized Jackson inequality \[ A_{\sigma -0}(f)\leq \biggl \{\frac {1}{\binom {2m}{m}} \sum _{k=0}^{m-1}\frac {{\mathcal K}_{2k}}{(\gamma \pi )^{2k}} \nu _m^{k}+\frac {{\mathcal K}_{2m}}{(\gamma \pi )^{2m}} \frac {\nu _m^m}{2^{2m}}\biggr \} \omega _{2m}\Bigl (f,\frac {\gamma \pi }{\sigma }\Bigr ) \] is an example of such an estimate. Here $r,m\in \mathbb N$, $\sigma ,\gamma >0$, a function $f$ is uniformly continuous and bounded on $\mathbb R$, $A_{\sigma -0}$ is the best uniform approximation by entire functions of type less than $\sigma$, $\omega _{2m}$ is a uniform modulus of continuity of order $2m$, ${\mathcal K}_s$ are the Favard constants, and \[ \nu _m=\frac {8}{\binom {2m}{m}}\sum _{l=0}^{\lfloor (m-1)/2\rfloor }\frac {\binom {2m}{m-2l-1}}{(2l+1)^2}, \] where $\lfloor x\rfloor$ is the entire part of $x$. Similar inequalities are obtained for best approximations of periodic functions by splines. In some cases, the constants in inequalities are close to optimal.References
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Bibliographic Information
- O. L. Vinogradov
- Affiliation: Department of mathematics and mechanics, St. Petersburg State University, Universitetskii pr. 28, Staryi Peterhof, St. Petersburg 198504, Russia
- Email: olvin@math.spbu.ru
- V. V. Zhuk
- Affiliation: Department of mathematics and mechanics, St. Petersburg State University, Universitetskii pr. 28, Staryi Peterhof, St. Petersburg 198504, Russia
- Email: zhuk@math.spbu.ru
- Received by editor(s): September 22, 2011
- Published electronically: July 24, 2013
- Additional Notes: The authors were supported by the Federal Target Program (FTP) of the Ministry of Education and Science of Russian Federation (project no. 2010-1.1-111-128-033).
- © Copyright 2013 American Mathematical Society
- Journal: St. Petersburg Math. J. 24 (2013), 691-721
- MSC (2010): Primary 41A17
- DOI: https://doi.org/10.1090/S1061-0022-2013-01261-1
- MathSciNet review: 3087819