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Bernstein inequalities in \({L_ p}\), \(0\leq p\leq+\infty\). (English) Zbl 0738.42003

Let \(\|\cdot\| _ p\) denote the usual \(L_ p\) norm or quasinorm when \(0<p\leq \infty\), \(\| f\| _ 0=\exp {1 \over 2\pi} \int_ T \log| f(t)| dt\). The authors prove that for every trigonometric polynomial \(T_ n\) of degree \(\leq n\) with complex coefficients one has \[ \| T_ n \cos \alpha+{1 \over n} T_ n' \sin \alpha\| _ p\leq \| T_ n\| _ p \] for arbitrary \(\alpha\) and every \(p\) with \(0\leq p\leq \infty\). Location of the zeros of polynomials with extremal properties are also obtained.

MSC:

42A05 Trigonometric polynomials, inequalities, extremal problems
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
26D05 Inequalities for trigonometric functions and polynomials
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