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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. (English. Russian original) Zbl 1275.41016
St. Petersbg. Math. J. 24, No. 5, 691-721 (2013); translation from Algebra Anal. 24, No. 5, 1-43 (2012).
Summary: A new technique is developed for estimating functionals by moduli of continuity. The generalized Jackson inequality
$A_{\sigma -0}(f)\leq\left \{\frac {1}{\binom {2m}{m}} \sum _{k=0}^{m-1}\frac{K_{2k}}{(\gamma\pi)^{2k}}\nu ^k_m+\frac{K_{2m}}{(\gamma\pi)^{2m}}\frac{\nu^m_ m}{2^{2m}}\right\}\omega _{2m}\left (f,\frac {\gamma \pi }{\sigma }\right )$ is an example of such an estimate. Here $$r, m\in\mathbb N$$, $$\sigma, \gamma >0$$, $$f$$ is a uniformly continuous and bounded function 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 $$\lfloor x\rfloor$$ denotes the entire part of $$x$$. Similar inequalities are obtained for best approximations of periodic functions by splines. In some cases, the constants in the inequalities are close to optimal.

##### MSC:
 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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##### References:
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