On One Generalized Translation and the Corresponding Inequality of Different Metrics

In this paper, we discuss the properties of the generalized translation operator generated by the system of functions superscriptsubscript2𝑘1𝜋2𝑡𝑘1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{\cos\left(\frac{(2k-1)\pi}{2}t\right)\right\}_{k=1}^{\infty}$$\end{document} in the spaces superscript𝐿𝑝01\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}(0,1)$$\end{document}, 𝑝1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\geq 1$$\end{document}. The translation operator is applied to the study of the Nikol’skii inequality between the uniform norm and the superscript𝐿𝑝\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}$$\end{document}-norm of polynomials in this system.