Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants

TL;DR

This paper explores the relationships between orthogonal polynomials, reproducing kernels, and optimal polynomial approximants in various function spaces, revealing zero distributions and providing methods for their computation.

Contribution

It establishes new connections between extremal polynomials and orthogonal polynomial theory across different weighted spaces, including zero location insights.

Findings

Extremal polynomials are non-vanishing in the closed unit disk for certain spaces.

Zeros of extremal polynomials can move inside the disk for negative alpha.

Provides formulas and methods for computing approximation distances using orthogonal polynomials.

Abstract

We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials $p$ minimizing Dirichlet-type norms $\|pf-1\|_{\alpha}$ for a given function $f$. For $\alpha\in [0,1]$ (which includes the Hardy and Dirichlet spaces of the disk) and general $f$, we show that such extremal polynomials are non-vanishing in the closed unit disk. For negative $\alpha$, the weighted Bergman space case, the extremal polynomials are non-vanishing on a disk of strictly smaller radius, and zeros can move inside the unit disk. We also explain how $\mathrm{dist}_{D_{\alpha}}(1,f\cdot \mathcal{P}_n)$, where $\mathcal{P}_n$ is the space of polynomials of degree at most $n$, can be expressed in terms of quantities associated with orthogonal polynomials and kernels, and we discuss methods for computing the quantities in question.