Best Polynomial Approximations in L 2 and Widths of Some Classes of Functions

Abstract

We obtain the exact values of extremal characteristics of a special form that connect the best polynomial approximations of functions f(x) ∈ L2r(r ∈ ℤ+) and expressions containing moduli of continuity of the kth order ωk(f(r), t). Using these exact values, we generalize the Taikov result for inequalities that connect the best polynomial approximations and moduli of continuity of functions from L2. For the classes \(\mathcal{F}\) (k, r, Ψ*) defined by ω k(f(r), t) and the majorant \(\Psi _ (t) = t^{4k/\pi ^2 }\), we determine the exact values of different widths in the space L2.