An application of MVBV condition in real sense for \(L^1\)-convergence of trigonometric series (Q484535)

A real sequence \(\{a_n\}_n\) is said to satisfy the mean value bounded variation condition (MVBV) if there are \(\lambda\geq2\) and \(M>0\) such that \(\sum_{k=n}^{2n}| a_k-a_{k+1}| \leq(M/n)\sum_{n/\lambda\leq k\leq\lambda n}| a_k| \) for all \(n\). Classical results in Fourier analysis on convergences of trigonometric and Fourier series often require positivity and monotonicity of coefficients. The MVBV condition is a generalization of the monotonicity and positivity requirements that allows improvements of such results. Recently the authors proved that MVBV condition guarantee the necessary and sufficient condition for uniform convergence. In the paper under the review the authors investigate the problem of \(L^1\)-convergence. They prove that if \(S(x)=\sum_{n=1}^\infty a_n\sin nx\) is the Fourier series of an integrable function \(f\in L_{2\pi}\) such that the real sequence \(\{a_n\}_n\) satisfies the MVBV condition, then \(S\) converges to \(f\) in \(L^1\)-norm if and only if \(\lim_{n\to\infty}a_n\log n=0\).