Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants

Abstract

We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials $p$ minimizing Dirichlet-type norms $\Vert pf-1{\Vert }_{\alpha }$ 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.

22 pages, 4 figures. Submitted for publication