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Ukrainian Mathematical Journal

, Volume 65, Issue 12, pp 1774–1792 | Cite as

On the Best Approximation in the Mean by Algebraic Polynomials with Weight and the Exact Values of Widths for the Classes of Functions

  • S. B. Vakarchuk
  • A. V. Shvachko
Article
The exact value of the extremal characteristic

is obtained on the class L 2 r (D ρ ), where r ∈+; \( {D}_{\rho} = \sigma (x)\frac{d^2}{d{ x}^2}+\tau (x)\frac{d}{d x} \) , 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.

Keywords

Fourier Series Orthogonal Polynomial Approximation Theory Hermite Polynomial Jacobi Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • S. B. Vakarchuk
    • 1
  • A. V. Shvachko
    • 2
  1. 1.Alfred Nobel Dnepropetrovsk UniversityDnepropetrovskUkraine
  2. 2.Dnepropetrovsk State Agrarian UniversityDnepropetrovskUkraine

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