Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants

We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials $p$ minimizing Dirichlet-type norms $\|pf-1\|_α$ for a given function $f$. For $α\in [0,1]$ (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 $α$, 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 $\mathrm{dist}_{D_α}(1,f\cdot \mathcal{P}_n)$, 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.

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