Approximation by classical orthogonal polynomials with weight in spaces $L_{2,\gamma}(a,b)$ and widths of some functional classes

Questions of the approximation of functions from classes $W^r_2(D_{\gamma};(a,b))$, $r=2,3,\ldots,$ by classical orthogonal polynomials have been analyzed in the spaces $L_{2,\gamma}(a,b)$ with a weight $\gamma$. For different widths estimates above and below were obtained on the classes $W^r_2(\Omega_{m,\gamma}, \Psi; (a,b))$, where $r\in \mathbb{Z}_{+}$, $m \in \mathbb{N}$, $\Psi$ is a majorant, $\Omega_{m,\gamma}$ is a generalized modulus of continuity of $m$-th order. The condition on majorant has been indicated when we can to compute the exact values of widths if it be fulfilled. Some concrete examples of the obtained exact results are reduced. Estimates (including exact) of the supremums of Fourier coefficients were obtained on the all indicated classes.