Averaged characteristics of smoothness in $L_2$ and estimations for the widths values of function classes - Vakarchuk, S.; Zabutna , V.; Vakarchuk, M.; Вакарчук, Сергій; Забутна, Валентина; Вакарчук, Михайло (Institute of Mathematics, NAS of Ukraine, 2025)
UDC 517.5 In the space of $2π$-periodic functions $L_2,$ we investigate the characteristic of smoothness $ω^{*}_{\mathcal{M}}(f,t):=$ $(1/t) \displaystyle\int\nolimits_0^t ω_{\mathcal{M}}(f,τ) d τ $ obtained as a result of averaging of the generalized modulus of continuity $ω_{\mathcal{M}}(f)$ formed by using a generalized finite-difference operator $Δ^{\mathcal{M}}_h \colon L_2 → L_2.$ We also study some properties of the functions $ω_{\mathcal{M}}(f)$ and $ω^{*}_{\mathcal{M}}(f).$ For the classes of functions $W(ω^{*}_{\mathcal{M}}, Φ),$ where $Φ$ is a majorant, we determine the lower and upper estimates for the values of a series of $n$-widths and the establish the condition for $Φ$ under which we obtain the exact values of these estimates. Several exact results are illustrated by specific examples.