2019/01/03 by Bénéteau, Catherine, Manolaki, Myrto, Seco, Daniel · 1 citation
#Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1901.00694
In this paper, we provide an efficient method for computing the Taylor coefficients of 1-pn f, 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 \mathbbD, 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 \mathbbD, the sequence \1-pnf \ converges uniformly to 0 on compact subsets of the complement of the zeros of f in \mathbbD, and we obtain precise estimates on the rate of convergence on compacta. We also treat the previously unknown case of a single zero with higher multiplicity.