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

Vinogradov, O.; Zhuk, V.

doi:10.1090/S1061-0022-2014-01297-6

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
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by O. L. Vinogradov and V. V. Zhuk
Translated by: O. L. Vinogradov

St. Petersburg Math. J. 25 (2014), 421-446

DOI: https://doi.org/10.1090/S1061-0022-2014-01297-6

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, $|E|$ 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 {\nu _m}{\gamma ^{2}}\frac {{\mathcal K}_2}{4}+\sum _{k=2}^{m-1} \frac {{\mathcal K}_{2k}}{2^{2k}}\frac {(2m-2k)! (2m)^{2k}}{(2m)!} \frac {\nu _m^k}{\gamma ^{2k}}\bigg ) \\ +\frac {{\mathcal K}_{2m}}{2^{2m}}\frac {(2m)^{2m}}{(2m)!} \frac {\nu _m^m}{4^m\gamma ^{2m}}\bigg ) (2^{2m}-1){\mathcal W}_{2m}\omega _{2m}\Big (f,\frac {\gamma |E|}{n}\Big ). \end{multline*}

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