ON ONE INEQUALITY OF DIFFERENT METRICS FOR TRIGONOMETRIC POLYNOMIALS

Vitalii V. Arestov     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russia Ural Federal University, 51 Lenin ave., Ekaterinburg, 620000, Russian Federation)
Marina V. Deikalova     (Ural Federal University, 51 Lenin ave., Ekaterinburg, 620000, Russia Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)

Abstract


We study the sharp inequality between the uniform norm and \(L^p(0,\pi/2)\)-norm of polynomials in the system \(\mathscr{C}=\{\cos (2k+1)x\}_{k=0}^\infty\) of cosines with odd harmonics. We investigate the limit behavior of the best constant in this inequality with respect to the order \(n\) of polynomials as \(n\to\infty\) and provide a characterization of the extremal polynomial in the inequality for a fixed order of polynomials.


Keywords


Trigonometric cosine polynomial in odd harmonics; Nikol’skii different metrics inequality

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References


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DOI: http://dx.doi.org/10.15826/umj.2022.2.003

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