Several questions of approximation by polynomials with respect to multiplicative systems in weighted $L^p$ spaces

In this paper we study approximation by Vilenkin polynomials in weighted $L^p$ spaces. We prove the Butzer–Scherer type result on equivalence between the rate of best approximation of a function $f$ and the growth of generalized derivatives and approximating properties of the best approximation polynomial $t_n(f)$. Some applications to the approximation by linear means of the Fourier–Vilenkin series are given.