Boundary Behavior of Optimal Polynomial Approximants

Abstract

In this paper, we provide an efficient method for computing the Taylor coefficients of $1-p_n f$, where $p_n$ denotes the optimal polynomial approximant of degree n to 1/f in a Hilbert space $H^2_\omega$ of analytic functions over the unit disc $\mathbb {D}$, and f is a polynomial of degree d with d simple zeros. As a consequence, we show that in many of the spaces $H^2_\omega$, the sequence $\{1-p_nf\}_{n\in \mathbb {N}}$ is uniformly bounded on the closed unit disc and, if f has no zeros inside $\mathbb {D}$, the sequence $\{1-p_nf \}$ converges uniformly to 0 on compact subsets of the complement of the zeros of f in $\overline{\mathbb {D}},$ 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 $1 - p_n f$ in the previously unknown case of a function with a single zero of multiplicity greater than 1, when the weights are given by $\omega _k = (k+1)^{\alpha }$ for $\alpha \le 1$.