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.


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