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

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Abstract

We investigate approximations of functions of classes Wr2(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 Wr2(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.

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Authors and Affiliations

  1. Alfred Nobel University, 18 Sicheslavska Seafront, Dnipro, 49000, Ukraine

    S. B. Vakarchuk

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Correspondence to S. B. Vakarchuk.

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Russian Text © The Author(s), 2019, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2019, No. 12, pp. 37–51.

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Vakarchuk, S.B. Approximation by Classical Orthogonal Polynomials with Weight in Spaces L2,γ(a,b) and Widths of Some Functional Classes. Russ Math. 63, 32–44 (2019). https://doi.org/10.3103/S1066369X19120041

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  • DOI: https://doi.org/10.3103/S1066369X19120041

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