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 ∈ [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 distDα(1, f · Pn), where Pn 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.