On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives

For 2π-periodic functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x \in L_\infty ^r $$ \end{document} and arbitrary q ∈ [1, ∞] and p ∈ (0, ∞], we obtain the new exact Kolmogorov-type inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$|| x^(k) ||_q \leqslant (\frac{v(x^(k))}{2})^{1/q} \frac{|| \phi_{r-k} ||_q}{||| \phi_r |||_p^\alpha} ||| x |||_p^\alpha || x^(r) ||_\infty^{1- \alpha}, k, r \in N, k < r,$$ \end{document} which takes into account the number of changes in the sign of the derivatives ν(x(k)) over the period. Here, α = (r − k + 1/q)/(r + 1/p), ϕr is the Euler perfect spline of degree r, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{gathered} \left\| {\left| x \right|} \right\|_p : = {\text{sup}}_{a,b \in {\text{R}}} \{ E_0 (x)_{L_p [a,b]} :x'(t) \ne 0{\text{ }}\forall t \in (a,b)\} , \hfill \\ {\text{ }} \hfill \\ {\text{ }}E_0 (x)_{L_p [a,b]} : = {\text{ inf}}_{c \in {\text{R}}} \left\| {x - c} \right\|_{L_p [a,b]} , \hfill \\ \hfill \\ \left\| x \right\|_{L_p [a,b]} : = \left\{ {\int\limits_a^b {\left| {x(t)} \right|^p dt} } \right\}^{1/p} {\text{ for }}0 < p < \infty , \hfill \\ \end{gathered} $$ \end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\| x \right\|_{L_p [a,b]} : = {\text{ sup vrai}}_{t \in \left[ {a,b} \right]} \left| {x(t)} \right|$$ \end{document}. The inequality indicated turns into the equality for functions of the form x(t) = aϕr(nt + b), a, b ∈ R, n ∈ N. We also obtain an analog of this inequality in the case where k = 0 and q = ∞ and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines.