On the Best Approximation in the Mean by Algebraic Polynomials with Weight and the Exact Values of Widths for the Classes of Functions
Abstract
The exact value of the extremal characteristic
is obtained on the class L 2 r (D ρ ), where r ∈ ℤ+; \( {D}_{\rho} = \sigma (x)\frac{d^2}{d{ x}^2}+\tau (x)\frac{d}{d x} \) , σ and τ are polynomials of at most the second and first degrees, respectively, ρ is a weight function, 0 < p ≤ 2, 0 < h < 1, λ n (ρ) are eigenvalues of the operator D ρ , φ is a nonnegative measurable and summable function (in the interval (a, b)) which is not equivalent to zero, Ω k,ρ is the generalized modulus of continuity of the k th order in the space L 2,ρ (a, b), and E n (f)2,ρ is the best polynomial approximation in the mean with weight ρ for a function f ∈ L 2,ρ (a, b). The exact values of widths for the classes of functions specified by the characteristic of smoothness Ω k,ρ and the K-functional \( \mathbb{K} \) m are also obtained.
