Approximation by polynomials with respect to multiplicative systems in weighted $L^p$-spaces

We study best approximations of polynomials with respect to multiplicative systems in the $L^p$-spaces with Muckenhoupt weights. Using Jackson's and Bernstein's inequalities, we obtain the direct and inverse approximation theorems in terms of the $K$-functional and the inverse theorem of the Timan–Besov type. In the case of a power weight, we give a criterion for the membership of a function in the weighted $L^p$-space in terms of the Fourier coefficients with respect to multiplicative systems.