Generalized absolute convergence of the series of fourier coefficients with respect to Haar type systems

The Haar orthogonal system had been constructed in 1909 [7] as an answer on the question by D. Hilbert: is there exist an orthogonal system on the segment [0,1] such that the Fourier series of any continuous function with respect to this system converges uniformly on [0,1] to this function? The Haar system has many applications in the theory of orthogonal series and applied mathematics (see [9], [11] and survey paper [4]). The absolute convergence of the series of Fourier-Haar coefficients of functions from the spaces L^p[0,1], 1 ≤ p <; ∞, or C[0,1] was studied by Z. Ciesielski and J. Musielak, P.L. Ul'yanov, B.I. Golubov and others (see the survey paper [4]). In our talk we will consider a generalized absolute convergence of the series of Fourier coefficents with respect to the Haar type systems introduced by N.Ya. Vilenkin. In most cases our results generalize ones from [6].