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

Abstract

Fairly general conditions on the coefficients \(\left\{ {a_n } \right\}_{n = 1}^\infty \) of even and odd trigonometric Fourier series under which L-convergence (boundedness) of partial sums of the series is equivalent to the relation \(\sum\nolimits_{k = \left[ {{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} \right]}^{2n} {{{\left| {a_k } \right|} \mathord{\left/ {\vphantom {{\left| {a_k } \right|} {\left( {\left| {n - k} \right| + 1} \right) = o\left( 1 \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\left| {n - k} \right| + 1} \right) = o\left( 1 \right)}}} \left( { = O\left( 1 \right),{\text{ respectively}}} \right)\) are given.