Norm approximation by Taylor polynomials in Hardy and Bergman spaces

For positive [Formula: see text] and real [Formula: see text] let [Formula: see text] denote the weighted Bergman spaces of the unit ball [Formula: see text] introduced in [R. Zhao and K. Zhu, Theory of Bergman Spaces on the Unit Ball in [Formula: see text], Mémoires de la Société Mathématique de France, Vol. 115 (2008)]. It is well known that, at least in the case [Formula: see text], all functions in [Formula: see text] can be approximated in norm by their Taylor polynomials if and only if [Formula: see text]. In this paper we show that, for [Formula: see text] with [Formula: see text], we always have [Formula: see text] as [Formula: see text], where [Formula: see text] and [Formula: see text] is the [Formula: see text]th Taylor polynomial of [Formula: see text]. We also show that for every [Formula: see text] in the Hardy space [Formula: see text], [Formula: see text], we always have [Formula: see text] as [Formula: see text], where [Formula: see text]. This generalizes and improves a result in [J. McNeal and J. Xiong, Norm convergence of partial sums of [Formula: see text] functions, Internat. J. Math. 29 (2018) 1850065, 10 pp.].

Approximation numbers of composition operators on the Hardy and Bergman spaces of the ball and of the polydisk

We give general estimates for the approximation numbers of composition operators on the Hardy space on the ball Bd and the polydisk ${\mathbb D}$d and of composition operators on the Bergman space on the polydisk.