Order Estimates for the Best Orthogonal Trigonometric Approximations of the Classes of Convolutions of Periodic Functions of Low Smoothness

We establish order estimates for the best uniform orthogonal trigonometric approximations on the classes of 2π-periodic functions whose (ψ,β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \beta $$\end{document})-derivatives belong to unit balls in the spaces Lp, 1 ≤ p < ∞, in the case where the sequence ψ(k) is such that the product ψ(n)n1/p may tend to zero slower than any power function and ∑k=1∞ψp′kp′−2<∞for1<p<∞,1p+1p′=1,or∑k=1∞ψk<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\displaystyle {\sum}_{k=1}^{\infty }}{\psi}^{p\prime }{(k)}^{p\prime -2}<\infty \kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em 1<p<\infty, \frac{1}{p}+\frac{1}{p^{\prime }}=1,\kern0.5em \mathrm{o}\mathrm{r}{\displaystyle {\sum}_{k=1}^{\infty }}\psi (k)<\infty $$\end{document} for p = 1. Similar estimates are also established in the Ls-metrics, 1 < s ≤ ∞, for the classes of summable (ψ,β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \beta $$\end{document})-differentiable functions such that ‖fβψ‖1 ≤ 1.