In the case where either r = 2, k = 1 or r = 3, k = 1, 2, for any q, p ≥ 1, β ∈ [0, 2π), and a Lebesgue-measurable set B ⊂ I2π ≔ [−π/2, 3π/2], μB ≤ β, we prove a sharp Kolmogorov–Remez-type inequality \( {\left\Vert {f}^{(k)}\right\Vert}_q\le \frac{\left\Vert \varphi r-k\right\Vert q}{E_0{{\left(\varphi r\right)}_L^{\alpha}}_p\left({I}_{2\uppi}/{B}_{2m}\right)}{\left\Vert f\right\Vert}_{L_p}^{\alpha}\left({I}_{2\uppi}/B\right){\left\Vert {F}^{(r)}\right\Vert}_{\infty}^{1-\alpha },\kern1em f\in {L}_{\infty}^r, \) with 𝛼 = min {1 − k/r, (r − k + 1/q)/(r + 1/p)}, where 𝜑r is the perfect Euler spline of order r, \( {E}_0{(f)}_{L_p(G)} \) is the best approximation of f by constants in Lp(G), \( {B}_{2m}=\left[\frac{\pi -2m}{2},\frac{\pi +2m}{2}\right] \), and m = m(β) 𝜖 [0, 𝜋) is uniquely defined by β. We also establish a sharp Kolmogorov–Remez-type inequality, which takes into account the number of sign changes of the derivatives.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 4, pp. 483–493, April, 2020.
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Kofanov, V.A. Sharp Kolmogorov–Remez-Type Inequalities for Periodic Functions of Low Smoothness. Ukr Math J 72, 555–567 (2020). https://doi.org/10.1007/s11253-020-01800-2
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DOI: https://doi.org/10.1007/s11253-020-01800-2