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

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{</nomathmode><mathmode>$r$ is even},
\Sigma^{-1}_{r, m}(\dfrac{A_{r, 0}}2\omega_1(f, h)_P+\sum\limits^{m-1}_{\nu=1}A_{r,\nu}\omega_{\nu}(f,h)_P),&\text{$r$ is odd}. \end{cases} \end{gather*}
</mathmode><nomathmode>
Theorem 1. \textit{Let $r,m\in\mathbb N$, $n,\lambda>0$, $f\in C^{(r+m)}$. Then}
$$ \begin{gathered} P(f^{(m)})\le\lambda^r\left\{\Sigma_{r, m}+2^m\sum_{k\in K_{r, m}}\left(\prod^m_{j=1}|\gamma_{k_j}|\right)\frac{\mathscr K_{r+m-(k)}}{\lambda^{r+m-(k)}}\right\} \\ \times\max\left\{\left(\frac{\omega_m(f,\tfrac{\lambda}n)_P}{\mathscr K_{r+m}2^m}\right)^{\frac r{r+m}}M^{\frac m{r+m}}_{r, m},\left(f^{(r+m)},\frac{\lambda}n\right), \frac{n^m\omega_m(f,\frac{\lambda}n)_P}{\mathscr K_{r+m}2^m}\right\}. \end{gathered} $$

For some values of $\lambda$ and seminorms related to best approximations by trigonometric polynomials and splines in the uniform and integral metrics, the inequalities are sharp.