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Nikol’skii-Stechkin inequality for trigonometric polynomials in \(L_0\). (English. Russian original) Zbl 1117.41017
Math. Notes 80, No. 3, 403-409 (2006); translation from Mat. Zametki 80, No. 3, 421-428 (2006).
Let \(P_n(z),z\in \mathbb{C}\), be an algebraic polynomial of degree \(n\) with complex coefficients. The ”zero norm” or the Mahler measure of \(P_n\) is defined as follows \[ \| P_n\| _0=\exp\left(\frac{1}{2\pi}\int^\pi_{-\pi} \ln| P_n(e^{it})| dt\right). \] The author proves a generalization of the Bernshtein-Nikol’skiĭ–Stechkin inequality for the ”0-norm”: \[ \| P_n'\| _0\leq A_{n,h}\| P_n(z)-P_n(ze^{ih})\| _0,\quad 0<h<\frac{2\pi}{n}, \] and it is shown that this inequality is sharp. Here \(A_{n,h}\) is the ”0-norm” of a certain function, and it is shown that \[ A_{n,h}\leq\frac{n}{\sin(\frac{nh}{2})}\| \sup_{0\leq t\leq 1}| 1+tz| ^{n-1}\| _0, \] where the second factor is equivalent to \(2^{\frac{n-1}{2}}\) as \(n\to\infty\). A similar result is established for trigonometric polynomials.
MSC:
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
42A05 Trigonometric polynomials, inequalities, extremal problems
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