Extremal polynomials on the $n$-grid

TL;DR

This paper investigates the asymptotic behavior of extremal polynomials on evenly spaced grids, revealing limits related to stable approximation constraints, with implications for polynomial approximation theory.

Contribution

It determines the limit of the normalized logarithmic norm of extremal polynomials on the $n$-grid for all $\, ext{α} \, ext{in}\, (0,1)$, connecting to stability in approximation.

Findings

Established the limit of $rac{1}{n} \,\log \| p_n^* \|_{[-1,1]}$ for all $\,\alpha \,\in (0,1)$.

Linked the asymptotic behavior to impossibility results in stable polynomial approximation.

Provided insights into extremal polynomial growth on discrete grids.

Abstract

The $n$-grid $E_n$ consists of $n$ equally spaced points in $[-1,1]$ including the endpoints $\pm 1$. The extremal polynomial $p_n^*$ is the polynomial that maximizes the uniform norm $\| p \|_{[-1,1]}$ among polynomials $p$ of degree $\leq \alpha n$ that are bounded by one on $E_n$. For every $\alpha \in (0,1)$, we determine the limit of $\frac{1}{n} \log \| p_n^*\|_{[-1,1]}$ as $n \to \infty$. The interest in this limit comes from a connection with an impossibility theorem on stable approximation on the $n$-grid.