An estimate of an optimal argument in the sharp multidimensional Jackson-Stechkin L2-inequality

An estimate of an optimal argument in the sharp Jackson-Stechkin inequality in the space L2(ℝn) is proved in the case of a generalized modulus of continuity; its special case is the classical modulus of continuity. Similar statements hold for the torus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T}^n $\end{document}. The obtained results agree with Chernykh’s classical one-dimensional theorems and refine some results by S.N. Vasil’ev, A.I. Kozko, and A.V. Rozhdestvenskii.