Generalized absolute convergence of series from Fourier coeficients by systems of Haar type

For orthogonal systems of Haar type introduced by N.Ya. Vilenkin in 1958 we study absolute convergence of series from Fourier coefficients raised to a positive power with multiplicators from Gogoladze–Meskhia class. The conditions for convergence of the series mentioned above are given in terms of best approximations of functions in $L^p$ spaces by polynomials with respect to Haar type systems or in terms of fractional modulus of continuity of functions from Wiener spaces $V_p$, $p>1$. We establish the sharpness of obtained results.