On Bernstein's inequality for polynomials

Hervé Queffélec (1) , Rachid Zarouf (2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 ADEF - Apprentissage, Didactique, Evaluation, Formation

Bernstein's classical inequality asserts that given a trigonometric polynomial $T$ of degree $n\geq1$, the sup-norm of the derivative of $T$ does not exceed $n$ times the sup-norm of $T$. We present various approaches to prove this inequality and some of its natural extensions/variants, especially when it comes to replacing the sup-norm with the $L^p-norm$.

Domaines: Analyse classique [math.CA]

Hervé Queffélec, Rachid Zarouf. On Bernstein's inequality for polynomials. Analysis and Mathematical Physics, 2019. ⟨hal-02077659⟩