Approximation by Classical Orthogonal Polynomials with Weight in Spaces L2,γ(a,b) and Widths of Some Functional Classes

We investigate approximations of functions of classes W2r(Dγ;(a,b)), r = 2, 3, …, by classical orthogonal polynomials with a weight γ in the spaces L2,γ(a,b). We obtain upper and lower estimates for different widths on the classes W2r(Ωm,γ, Ψ; (a,b)), where r ∈ ℤ+, m ∈ ℕ, Ψ is a majorant, Ωm,γ is a generalized modulus of continuity of m-th order. We find the condition on majorant, which enable us to compute the exact values of widths, and give certain examples of these values. In all mentioned above classes we obtain bounds (including the least upper bounds) for the Fourier coefficients.