Jackson — Stechkin type inequalities with generalized moduli of continuity and widths of some classes of functions

In the Hilbert space $L_{2,\mu}[-1,1]$ with Chebyshev weight $\mu(x):=1/\sqrt{1-x^{2}}$, we obtain Jackson–Stechkin type inequalities between the value $E_{n-1}(f)_{L_{2,\mu}}$ of the best approximation of a function $f(x)$ by algebraic polynomials of degree at most $n-1$ and the $m$th-order generalized modulus of continuity $\Omega_{m}({\mathcal D}^{r}f;t)$, where ${\mathcal D}$ is some second-order differential operator. For classes of functions $W^{(2r)}_{p,m}(\Psi)$ ($m,r\in\mathbb{N}$, $1/(2r)$<$p\le2$) defined by the mentioned modulus of continuity and a given majorant $\Psi(t)$ ($t\ge0$), which satisfies certain constraints, we calculate the values of various $n$-widths in the space $L_{2,\mu}[-1,1]$.