In this paper we study approximation by Vilenkin polynomials in weighted Lp 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 tn(f). Some applications to the approximation by linear means of the Fourier – Vilenkin series are given.
The direct approximation theorem by algebraic polynomials is proved for Riemann–Liouville integrals of order r>0. As a corollary, we obtain asymptotic equalities for ε-entropy of the image of a Hölder type class under Riemann–Liouville integration operator.
