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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 high order moduli of continuity in spaces of functions defined on a segment


Authors: O. L. Vinogradov and V. V. Zhuk
Translated by: O. L. Vinogradov
Original publication: Algebra i Analiz, tom 25 (2013), nomer 3.
Journal: St. Petersburg Math. J. 25 (2014), 421-446
MSC (2010): Primary 41A15, 41A17, 41A35
Published electronically: May 16, 2014
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Abstract: A new technique is developed for estimating functionals in terms of the quantities mentioned in the title. The constants in estimates are indicated explicitly. As examples, Jackson type inequalities for approximations by polynomials and splines can be mentioned, along with estimates of error terms for interpolation formulas and for formulas of numerical differentiation and integration. One of results can be stated as follows. Let $ E$ be a segment, $ \vert E\vert$ its length, $ E_{n-1}$ the best uniform approximation by polynomials of degree at most $ n-1$, and $ \omega _{2m}$ the uniform modulus of continuity of order $ 2m$. Let $ {\mathcal K}_r=\frac {4}{\pi }\sum _{\nu =0}^{\infty } \frac {(-1)^{\nu (r+1)}}{(2\nu +1)^{r+1}}$ be the Favard constants, $ {\mathcal W}_{2m}$ the Whitney 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}$. Let $ m\geq 2$, $ n\geq 2m$, $ \gamma >0$, $ f\in C(E)$. Then

\begin{multline*}E_{n-1}(f)\leq \bigg (\frac {1}{\binom {2m}{m}}\bigg ( 1+\frac ... ...W}_{2m}\omega _{2m}\Big (f,\frac {\gamma \vert E\vert}{n}\Big ). \end{multline*}


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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/S1061-0022-2014-01297-6
PII: S 1061-0022(2014)01297-6
Keywords: Best approximation, modulus of continuity, sharp constants, numerical differentiation and integration
Received by editor(s): January 10, 2013
Published electronically: May 16, 2014
Dedicated: Dedicated to Boris Mikhaĭlovich Makarov
Article copyright: © Copyright 2014 American Mathematical Society