Classes of 𝐿¹-convergence of Fourier and Fourier-Stieltjes series

Abstract

It is shown that the Fomin class F p ( 1 > p ⩽ 2 ) {\mathcal {F}_p}(1 > p \leqslant 2) is a subclass of C ∩ B V \mathcal {C} \cap \mathcal {B}\mathcal {V} , where C \mathcal {C} is the Garrett-Stanojević class and B V \mathcal {B}\mathcal {V} the class of sequences of bounded variation. Wider classes of Fourier and Fourier-Stieltjes series are found for which a n lg n = o ( 1 ) , n → ∞ {a_n}\;{\text {lg}}\;n = o(1),n \to \infty , is a necessary and sufficient condition for L 1 {L^1} -convergence. For cosine series with coefficients in B V \mathcal {B}\mathcal {V} and n Δ a n = O ( 1 ) n\Delta {a_n} = O(1) , n → ∞ n \to \infty , necessary and sufficient integrability conditions are obtained.