Jackson-type inequalities with generalized modulus of continuity and exact values of the $n$-widths of the classes of $(ψ,β)$-differential functions in $L_2$. I

Abstract

For the generalized moduli of continuity, including the ordinary moduli of continuity and various their modifications, we establish the exact constants for Jackson-type inequalities in the classes of $2\pi$ -periodic functions in the space $L_2$ with $(\psi , \beta)$-derivatives, introduced by Stepanets. These results take into account the classification of $(\psi , \beta)$-derivatives and enable us to consider the major part of Jackson-type inequalities obtained earlier in the classes of differentiable functions $L_2^r,\; r \in N$, from the common point of view.