Norm approximation by Taylor polynomials in Hardy and Bergman spaces

Abstract

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.].