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

Abstract

A new technique is developed for estimating functionals by moduli of continuity. The generalized Jackson inequality \[ A_{\sigma -0}(f)\leq \biggl \{\frac {1}{\binom {2m}{m}} \sum _{k=0}^{m-1}\frac {{\mathcal K}_{2k}}{(\gamma \pi )^{2k}} \nu _m^{k}+\frac {{\mathcal K}_{2m}}{(\gamma \pi )^{2m}} \frac {\nu _m^m}{2^{2m}}\biggr \} \omega _{2m}\Bigl (f,\frac {\gamma \pi }{\sigma }\Bigr ) \] 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 \[ \nu _m=\frac {8}{\binom {2m}{m}}\sum _{l=0}^{\lfloor (m-1)/2\rfloor }\frac {\binom {2m}{m-2l-1}}{(2l+1)^2}, \] 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.