Sharp Kolmogorov-type inequalities for moduli of continuity and best approximations by trigonometric polynomials and splines

O. L. Vinogradov ; V. V. Zhuk

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 30, Tome 290 (2002), pp. 5-26

Résumé

In what follows $C$ is the space of $2\pi$-periodic continuous functions; $P$ is a seminorm defined on $C$, shift-invariant, and majorized by the uniform norm; $\omega_m(f, h)_P$ is the $m$th modulus of continuity of a function $f$ with step $h$ and calculated with respect to $P$; $\mathscr K_r=\frac4\pi\sum\limits^{\infty}_{l=0}\frac{(-1)^{l(r+1)}}{(2l+1)^{r+1}}$, $B_r(x)=-\frac{r!}{2^{r-1}\pi^r}\sum\limits^{\infty}_{k-1}\frac{\cos(2k\pi x-r\pi/2)}{k^r}$ $(r\in\mathbb N)$, $B_0(x)=1$, $\gamma_r=\frac{B_r(\frac12)}{r!}$; $(k)=k_1+\cdots+k_m$, \begin{gather*} K_{r,m}=\{k\in\mathbb Z^m_+:0\le k_{\nu}\le r+\nu-2-k_1-\dots-k_{\nu-1}\}, \\ A_{r,0}=\frac2{r!}\int^{1/2}_0\left|B_r(t)-B_r\left(\frac12\right)\right|\,dt, \\ A_{r, m}=\sum_{k\in K_{r,m}}\left(\prod^m_{j=1}|\gamma_{k_j}|\right)A_{r+m-(k), 0}, \quad \Sigma_{r, m}=\sum^{m-1}_{\nu=0}2^{\nu}A_{r,\nu}, \\ M_{r, m}(f, h)_P=\begin{cases} \Sigma^{-1}_{r,m}\sum\limits^{m-1}_{\nu=0}A_{r,\nu}\omega_{\nu}(f,h)_P,\text{