Absolute convergence of Fourier series of functions of class Lip 1 and of functions of bounded variation (Q2889562)

If one does not restrict himself to the trigonometric system but considers a variety of orthonormal systems, certain interesting phenomena occur. On the one hand, Banach proved that given an \(L_2\) function, there is an orthonormal system with respect to which the Fourier series of this function diverges almost everywhere. On the other hand, the Fourier series with respect to a family of orthonormal systems of every function from Lip1 converges absolutely. This picture in whole differs from that for the trigonometric system alone. In the paper under review, the author distinguishes among all orthonormal systems those for which the Fourier series of every function from Lip1 converges absolutely. Functions of bounded variation are also studied in a similar way.