On the Jackson theorem for periodic functions in metric spaces with integral metric. II

Abstract

In the spaces $L_{\psi}(T^m)$ of periodic functions with metric $\rho(f, 0)_{\psi} = \int_{T^m}\psi(|f(x)|)dx$ , where $\psi$ is a function of the type of modulus of continuity, we study the direct Jackson theorem in the case of approximation by trigonometric polynomials. It is proved that the direct Jackson theorem is true if and only if the lower dilation index of the function $\psi$ is not equal to zero.