Lebesgue's test for general Dirichlet's integrals

Abstract

It is well-known the Lebesgue \cite{Lebesgue, Zygmund} test for trigonometric Fourier series. Taberski \cite{Taberski1, Taberski2} considered real-valued Lebesgue locally integrable functions $f$, such that \begin{equation*} \lim_{T \to \infty} \frac{1}{T} \int_{T}^{T+c} |f(t)| \, dt\ =0; \quad \lim_{T \to \infty} \frac{1}{T} \int_{-T-c}^{-T} |f(t)| \, dt \ =0 \end{equation*} for every fixed $c>0$. For this class of functions, he defined generalized Dirichlet's integrals. Besides, Taberski \cite{Taberski1,Taberski2} investigated problems of convergence and $(C,1)$-summability of these integrals. In this paper, the analogous of the Lebesgue test for the generalized Dirichlet's integrals is proved.