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Ukrainian Mathematical Journal

, Volume 69, Issue 11, pp 1710–1726 | Cite as

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

  • A. E. Gaidabura
  • V. A. Kofanov
Article
  • 16 Downloads

For any p ϵ [1,∞], 𝜔 > 0, β ϵ (0, 2𝜔), and any measurable set BI d := [0,d], μBβ, 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 := [(ωβ)/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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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • A. E. Gaidabura
    • 1
  • V. A. Kofanov
    • 1
  1. 1.Dnepropetrovsk National UniversityDneprUkraine

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