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Sharp Remez-type Inequalities of Various Metrics in the Classes of Functions with Given Comparison Function

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Ukrainian Mathematical Journal Aims and scope

For any p ϵ [1,∞], 𝜔 > 0, β ϵ (0, 2𝜔), and any measurable set B ⊂ I d := [0,d], μB ≤ β, we establish a sharp Remez-type inequality of various metrics

$$ {E}_0{(x)}_{\infty}\le \frac{{\left\Vert \varphi \right\Vert}_{\infty }}{E_0{{{\left(\varphi \right)}_L}_p}_{\left({I}_{2\omega}\backslash {B}_1\right)}}{\left\Vert x\right\Vert}_{Lp\;\left({I}_d\backslash B\right)} $$

in the classes S𝜑(ω) of d-periodic (d ≥ 2ω) functions x with a given sine-shaped 2ω-periodic comparison function 𝜑, where B1 := [(ω − β)/2, (ω+β)/2] and E0(f)Lp(G) is the best approximation of the function f by constants in the metric of the space Lp(G). In particular, we prove sharp Remez-type inequalities of various metrics in the Sobolev spaces of differentiable periodic functions. We also obtain inequalities of this type in the spaces of trigonometric polynomials and splines.

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Authors and Affiliations

  1. Dnepropetrovsk National University, Dnepr, Ukraine

    A. E. Gaidabura & V. A. Kofanov

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  1. A. E. Gaidabura
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  2. V. A. Kofanov
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 11, pp. 1472–1485, November, 2017.

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Gaidabura, A.E., Kofanov, V.A. Sharp Remez-type Inequalities of Various Metrics in the Classes of Functions with Given Comparison Function. Ukr Math J 69, 1710–1726 (2018). https://doi.org/10.1007/s11253-018-1465-4

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  • DOI: https://doi.org/10.1007/s11253-018-1465-4

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