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On the Moduli of Continuity and Fractional-Order Derivatives in the Problems of Best Mean-Square Approximation by Entire Functions of the Exponential Type on the Entire Real Axis

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Ukrainian Mathematical Journal Aims and scope

The exact Jackson-type inequalities with modules of continuity of a fractional order β ∈ (0, ∞) are obtained on the classes of functions defined via the derivatives of a fractional order α ∈ (0, ∞) for the best approximations by entire functions of the exponential type in the space L 2(ℝ). In particular, we prove the inequality \( {2}^{-\beta /2}{\sigma}^{-\alpha }{\left(1-\cos t\right)}^{-\beta /2}\le \sup \left\{{A}_{\sigma }(f)/{\omega}_{\beta}\left({D}^{\alpha }f,t/\sigma \right):f\in {L}_2^{\alpha}\left(\mathbb{R}\right)\right\}\le {\sigma}^{-\alpha }{\left(1/{t}^2+1/2\right)}^{\beta /2} \), where β ∈ [1, ∞), t ∈ (0, π], and σ ∈ (0, ∞). We also determined the exact values of various mean 𝜈 -widths of the classes of functions determined via the fractional modules of continuity and majorants satisfying certain conditions.

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  1. Nobel Dnepropetrovsk University, Dnepropetrovsk, Ukraine

    S. B. Vakarchuk

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 5, pp. 599–623, May, 2017.

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Vakarchuk, S.B. On the Moduli of Continuity and Fractional-Order Derivatives in the Problems of Best Mean-Square Approximation by Entire Functions of the Exponential Type on the Entire Real Axis. Ukr Math J 69, 696–724 (2017). https://doi.org/10.1007/s11253-017-1389-4

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  • DOI: https://doi.org/10.1007/s11253-017-1389-4

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