Jackson-type inequalities with generalized modulus of continuity and exact values of the $n$-widths for the classes of $(ψ,β)$-differentiable functions in $L_2$. III - Vakarchuk, B. S. (Ukrains’kyi Matematychnyi Zhurnal, 2016)
In the classes $L^{ψ}_{β ,2}$ of $2π$ -periodic $(ψ , β)$-differentiable functions for which $f^{ψ}_{β} \in L_2$, we determine the exact constants inJackson-type inequalities for the characteristic of smoothness $Λ_{γ} (f, t) = \biggl\{\frac1t \int^t_0 \| Δ^{γ}_h(f)\|^2dh \biggr\}^{1/2}$, $t > 0$, deternined by averaging the norm of the generalized difference relation $Δ_{ γ}h(f)$. For the classes of $(ψ,β)$ -differentiable functions definedby using the characteristic of smoothness $Λ_{γ}$ and the majorant $Φ$, satisfying numerous conditions, we find the exact valuesof some $n$-widths in $L_2$.