Integrability and L1-convergence of Rees-stanojević Sums with Generalized Semiconvex Coefficients

The problem of L1-convergence of the Fourier cosine series (1.1) has been settled for various special classes of coefficients. Young [6] found that an logn= o(1), n→∞ is a necessary and sufficient condition for cosine series with convex (∆an ≥ 0) coefficients, and Kolmogorov [5] extended this result to the cosine series with quasi-convex ( ∑∞ n=1n|∆an−1| < ∞) coefficients. Later, Garrett and Stanojević [3] using modified cosine sums (1.2), proved the following theorem.