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

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Ukrainian Mathematical Journal Aims and scope

The exact value of the extremal characteristic

is obtained on the class L r2 (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.

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Authors and Affiliations

  1. Alfred Nobel Dnepropetrovsk University, Dnepropetrovsk, Ukraine

    S. B. Vakarchuk

  2. Dnepropetrovsk State Agrarian University, Dnepropetrovsk, Ukraine

    A. V. Shvachko

Authors
  1. S. B. Vakarchuk
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  2. A. V. Shvachko
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 12, pp. 1604–1621, December, 2013.

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Vakarchuk, S.B., Shvachko, A.V. On the Best Approximation in the Mean by Algebraic Polynomials with Weight and the Exact Values of Widths for the Classes of Functions. Ukr Math J 65, 1774–1792 (2014). https://doi.org/10.1007/s11253-014-0897-8

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  • DOI: https://doi.org/10.1007/s11253-014-0897-8

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