Convergence of series of Fourier coefficients for multiplicative convolutions

We study the convergence of series with Fourier–Vilenkin coefficients for functions represented as multiplicative convolutions. In the trigonometric case similar results are obtained by C. Onneweer, M. Izumi, and S. Izumi. Moreover, we consider certain analogs of I. Hirshman and W. Rudin transformations of Fourier coefficients. Some results are proved to be unimprovable in a certain sense.