Boundary Behavior of Optimal Polynomial Approximants

Abstract

In this paper, we provide an efficient method for computing the Taylor coefficients of 1−pnf, where pn denotes the optimal polynomial approximant of degree n to 1/f in a Hilbert space H2ω of analytic functions over the unit disc D, and f is a polynomial of degree d with d simple zeros. As a consequence, we show that in many of the spaces H2ω, the sequence {1−pnf}n∈N is uniformly bounded on the closed unit disc and, if f has no zeros inside D, the sequence {1−pnf} converges uniformly to 0 on compact subsets of the complement of the zeros of f in D, and we obtain precise estimates on the rate of convergence on compacta. We also obtain the precise constant in the rate of decay of the norm of 1−pnf in the previously unknown case of a function with a single zero of multiplicity greater than 1, when the weights are given by ωk=(k+1)α for α≤1.