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 $$\alpha \le 1$$ .