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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

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


Authors: O. L. Vinogradov and V. V. Zhuk
Translated by: O. L. Vinogradov
Original publication: Algebra i Analiz, tom 24 (2012), nomer 5.
Journal: St. Petersburg Math. J. 24 (2013), 691-721
MSC (2010): Primary 41A17
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

$\displaystyle A_{\sigma -0}(f)\leq \biggl \{\frac {1}{\binom {2m}{m}} \sum _{k=... ...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

$\displaystyle \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.

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Additional 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

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01261-1
PII: S 1061-0022(2013)01261-1
Keywords: Best approximation, modulus of continuity, Jackson inequalities, sharp constants, Steklov functions
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).
Article copyright: © Copyright 2013 American Mathematical Society