On $L^{1}$ convergence of certain cosine sums

Abstract:

Rees and Stanojević introduced a new class of modified cosine sums $\{ {g_n}(x) = \tfrac {1} {2}\sum \nolimits _{k = 0}^n {\Delta a(k) + \sum \nolimits _{k = 1}^n {\sum \nolimits _{j = k}^n {\Delta a(j)\cos kx\} } } }$ and found a necessary and sufficient condition for integrability of these modified cosine sums. Here we show that to every classical cosine series $f$ with coefficients of bounded variation, a Rees-Stanojević cosine sum ${g_n}$ can be associated such that ${g_n}$ converges to $f$ pointwise, and a necessary and sufficient condition for ${L^1}$ convergence of ${g_n}$ to $f$ is given. As a corollary to that result we have a generalization of the classical result of this kind. Examples are given using the well-known integrability conditions.