Remarks on Mean Convergence (Boundedness) of Partial Sums of Trigonometric Series

Fairly general conditions on the coefficients \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\{ {a_n } \right\}_{n = 1}^\infty $$ \end{document} of even and odd trigonometric Fourier series under which L-convergence (boundedness) of partial sums of the series is equivalent to the relation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sum\nolimits_{k = \left[ {{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} \right]}^{2n} {{{\left| {a_k } \right|} \mathord{\left/ {\vphantom {{\left| {a_k } \right|} {\left( {\left| {n - k} \right| + 1} \right) = o\left( 1 \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\left| {n - k} \right| + 1} \right) = o\left( 1 \right)}}} \left( { = O\left( 1 \right),{\text{ respectively}}} \right)$$ \end{document} are given.