We prove a sharp Remez-type inequality of different metrics
for 2π -periodic functions x ∈ \( {L}_{\infty}^r \) satisfying the condition
where \( L{(x)}_p\coloneq \sup \left\{{\left\Vert x\right\Vert}_{L_p\left[a,b\right]}:\left[a,b\right]\subset \left[0,2\uppi \right],\kern0.5em \left|x(t)\right|>0,\kern0.5em t\in \left(a,b\right)\right\} \), B ⊂ [0, 2π], μB ≤ β/λ (λ is chosen so that \( {\left\Vert x\right\Vert}_p={\left\Vert {\varphi}_{\uplambda, r}\right\Vert}_{L_p\left[0,2\uppi /\uplambda \right]} \)), φ r is the ideal Euler’s spline of order r, and \( {B}_1\coloneq \left[\frac{-\uppi -\beta /2}{2},\frac{-\uppi +\beta /2}{2}\right]\cup \left[\frac{\uppi -\beta /2}{2},\frac{\uppi +\beta /2}{2}\right] \).
As a special case, we establish sharp Remez-type inequalities of different metrics for trigonometric polynomials and polynomial splines satisfying the condition (*).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 2, pp. 173–188, February, 2017.
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Kofanov, V.A. Sharp Remez-Type Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines. Ukr Math J 69, 205–223 (2017). https://doi.org/10.1007/s11253-017-1357-z
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DOI: https://doi.org/10.1007/s11253-017-1357-z