Park, Inyoung; Zhao, Jian; Zhu, Kehe Norm approximation by Taylor polynomials in Hardy and Bergman spaces. (English) Zbl 1475.32006 Int. J. Math. 32, No. 6, Article ID 2150037, 13 p. (2021). Let \(0<p< \infty\) and \(\alpha \in \mathbb R.\) Fix a non-negative integer \(k\) such that \(pk + \alpha > -1.\) Generalizing the standard weighted Bergman spaces the authors introduce the space \(A^p_\alpha\) as the space of holomorphic functions \(f\) on the unit ball \(\mathbb B_n\) such that \((1-|z|^2)^k R^kf(z) \in L^p(\mathbb B_n, dv_\alpha),\) where \(Rf(z)= z_1 \frac{\partial f}{\partial z_1} + \dots + z_n \frac{\partial f}{\partial z_n}\) and \(dv_\alpha (z) = (1-|z|^2)^\alpha \, dv(z).\) They endow \(A^p_\alpha\) with the “norm” \(\|f\|^p_{p,\alpha} = |f(0)|^p + \int_{\mathbb B_n} \left | (1-|z|^2)^k R^kf(z) \right |^p\, dv_\alpha (z),\) indicating that for \(0<p<1\) one has only an F-space. Consider the \(N\)th Taylor polynomial \(S_N f\) of \(f\) and let \(0<p<\infty\) and \(\alpha \in \mathbb R.\) It is shown that \(\|S_Nf -f \|_{p,\alpha} \to 0 \) as \(N\to \infty\) for every \(f\in A^p_\alpha\) if and only if \(p>1,\) and \(\|S_Nf -f \|_{H^p} \to 0\) as \(N\to \infty\) for every \(f\) in the Hardy space \(H^p\) if and only if \(p>1.\) In addition, they prove that for \(\beta > \alpha\) and \(f\in A^1_\alpha\) one has \(\|S_Nf -f \|_{1,\beta} \to 0 \) as \(N\to \infty.\) The last part of the paper is devoted to interesting results in the case \(0<p<1.\) Reviewer: Fritz Haslinger (Wien) Cited in 1 ReviewCited in 2 Documents MSC: 32A36 Bergman spaces of functions in several complex variables 32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables 41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) Keywords:weighted Bergman spaces; Hardy spaces; Taylor polynomials; embedding theorems PDFBibTeX XMLCite \textit{I. Park} et al., Int. J. Math. 32, No. 6, Article ID 2150037, 13 p. (2021; Zbl 1475.32006) Full Text: DOI References: [1] McNeal, J. and Xiong, J., Norm convergence of partial sums of \(H^1\) functions, Internat. J. Math.29 (2018) 1850065, 10 pp. · Zbl 1408.30046 [2] Rudin, W., Functional Analysis, 2nd edn. (McGraw-Hill, New York, 1991). · Zbl 0867.46001 [3] R. Zhao and K. Zhu, Theory of Bergman Spaces on the Unit Ball in \(\mathbb{C}_n\), Mémoires de la Société Mathématique de France, Vol. 115 (2008). [4] Zhu, K., Duality of Bloch functions and norm convergence of Taylor series, Michigan Math. J.38 (1991) 89-101. · Zbl 0728.30026 [5] Zhu, K., Spaces of Holomorphic Functions in the Unit Ball, , Vol. 226 (Springer-Verlag, New York, 2005). · Zbl 1067.32005 [6] Zhu, K., Operator Theory in Function Spaces, 2nd edn., , Vol. 138 (American Mathematical Society, Providence, RI, 2007). · Zbl 1123.47001 [7] Zygmund, A., Trignometric Series, 2nd edn. (Cambridge University Press, Cambridge, 1968). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.