NORM CONVERGENT PARTIAL SUMS OF TAYLOR SERIES

Abstract

It is known that the partial sum of the Taylor series of an holomorphic function of one complex variable converges in norm on $H^p(\mathbb{D})$ for 1 < p < ${\infty}$. In this paper, we consider various type of partial sums of a holomorphic function of several variables which also converge in norm on $H^p(\mathbb{B}_n)$ for 1 < p < ${\infty}$. For the partial sums in several variable cases, some variables could be chosen slowly (fastly) relative to other variables. We prove that in any cases the partial sum converges to the original function, regardlessly how slowly (fastly) some variables are taken.

Keywords

• Taylor series;
• Hardy space

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References

1. W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill Book Co., New York, 1987.
2. K. Zhu, Duality of Bloch spaces and norm convergence of Taylor series, Michigan Math. J. 38 (1991), no. 1, 89-101. https://doi.org/10.1307/mmj/1029004264
3. K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer-Verlag, New York, 2005.