Point evaluation for polynomials on the circle

Abstract

We study the constant ${\mathcal{C}}_{d,p}$ defined as the smallest constant $C$ such that $\Vert P{\Vert }_{\infty }^p\le C\Vert P{\Vert }_p^p$ holds for every polynomial $P$ of degree $d$, where we consider the $L^p$ norm on the unit circle. We conjecture that ${\mathcal{C}}_{d,p}\le dp/2+1$ for all $p\ge 2$ and all degrees $d$. We show that the conjecture holds for all $p\ge 2$ when $d\le 4$ and for all $d$ when $p\ge 6.8$.

16 pages, 8 figures. Several structural changes. The proof of Theorem 7 has been corrected. This paper has been accepted for publication in Collectanea Mathematica