Multiplicative Convolutions of Functions from Lorentz Spaces and Convergence of Series of Fourier-Vilenkin Coefficients

摘要

Let f and g be functions from different Lorentz spaces L~(p,q)[0, 1), h be theirmultiplicative convolution and 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 |h(k)|~a with multiplicators of type k~b in terms of the best approximations of f and g in the corresponding Lorentz spaces. We establish sharpness of this result and of its corollaries for the Lebesgue spaces.