Multiplicative convolutions of functions from Lorentz spaces and convergence of series from Fourier–Vilenkin coefficients

Let $f$ and $g$ be functions from different Lorentz spaces $L^{p,q}[0,1)$, $h$ be their multiplicative convolution and $\widehat{h}(k)$ be Fourier coefficients of $h$ with respect to a multiplicative system with bounded generating sequence. We estimate the remainder of the series of $|\widehat{h}(k)|^a$ with multiplicators of type $k^b$ in terms of best approximations of $f$ and $g$ in corresponding Lorentz spaces. We establish the sharpness of this result and its corollaries for Lebesgue spaces.