Some estimates of the rate of summability of orthogonal series by linear means

An estimate of the rate of convergence of linear means of the orthogonal series of an $L^2$ function $f(x)$ generated by a summation function is established. This estimate depends on the rate of growth of factors which do not destroy the convergence of the series of squares of coefficients of $f(x)$ with respect to the orthonormal system under consideration.