In the space of 2π-periodic functions L2, we study the characteristic of smoothness \({\omega }_{\mathcal{M}}^{*}\left(f,t\right) :=\left(1/t\right){\int }_{0}^{t}\omega \mathcal{M}\left(f,\tau \right)d\tau \) obtained as a result of averaging of the generalized modulus of continuity ωℳ(f) formed by using a generalized finite-difference operator \({\Delta }_{h}^{\mathcal{M}}: {L}_{2}\to {L}_{2}.\) We also study some properties of the functions ωℳ(f) and \({\omega }_{\mathcal{M}}^{*}\left(f\right).\) For the classes of functions \(W\left({\omega }_{\mathcal{M}}^{*},\Phi \right),\) where Φ is a majorant, we determine the lower and upper estimates for the values of a series of n-widths and establish the condition for Φ under which the exact values of these estimates are obtained. Several exact results are illustrated by specific examples.
