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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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