Norm approximation by Taylor polynomials in Hardy and Bergman spaces

For positive p and real alpha let A(alpha)(p) denote the weighted Bergman spaces of the unit ball B-n introduced in [R. Zhao and K. Zhu, Theory of Bergman Spaces on the Unit Ball in C-n, Memoires de la Societe Mathematique de France, Vol. 115 (2008)]. It is well known that, at least in the case n = 1, all functions in A(alpha)(p) can be approximated in norm by their Taylor polynomials if and only if p > 1. In this paper we show that, for f is an element of A(alpha)(p) with 0 < p <= 1, we always have parallel to S(N)f - f parallel to(p,beta) -> 0 as N -> infinity, where beta > alpha + n(1 - p) and S(N)f is the Nth Taylor polynomial of f. We also show that for every f in the Hardy space Hp, 0 < p <= 1, we always have parallel to S(N)f - f parallel to(p,beta) -> 0 as N -> infinity, where beta > n(1 - p) - 1. This generalizes and improves a result in [J. McNeal and J. Xiong, Norm convergence of partial sums of H-1 functions, Internat. J. Math. 29 (2018) 1850065, 10 pp.].