## Annals of Functional Analysis

- Ann. Funct. Anal.
- Volume 8, Number 2 (2017), 259-269.

### Hadamard gap series in weighted-type spaces on the unit ball

#### Abstract

We give a sufficient and necessary condition for an analytic function $f\left(z\right)$ on the unit ball $\mathbb{B}$ in ${\mathbb{C}}^{n}$ with Hadamard gaps, that is, for $f\left(z\right)={\sum}_{k=1}^{\infty}{P}_{{n}_{k}}\left(z\right)$ where ${P}_{{n}_{k}}\left(z\right)$ is a homogeneous polynomial of degree ${n}_{k}$ and ${n}_{k+1}/{n}_{k}\ge c>1$ for all $k\in \mathbb{N}$, to belong to the weighted-type space ${H}_{\mu}^{\infty}$ and the corresponding little weighted-type space ${H}_{\mu ,0}^{\infty}$ under some condition posed on the weighted funtion $\mu $. We also study the growth rate of those functions in ${H}_{\mu}^{\infty}$.

#### Article information

**Source**

Ann. Funct. Anal., Volume 8, Number 2 (2017), 259-269.

**Dates**

Received: 9 August 2016

Accepted: 16 September 2016

First available in Project Euclid: 1 March 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.afa/1488358821

**Digital Object Identifier**

doi:10.1215/20088752-0000011X

**Mathematical Reviews number (MathSciNet)**

MR3619321

**Zentralblatt MATH identifier**

1361.32007

**Subjects**

Primary: 30H99: None of the above, but in this section 46B99: None of the above, but in this section

**Keywords**

weighted-type space Hadamard gaps homogeneous polynomial

#### Citation

Hu, Bingyang; Li, Songxiao. Hadamard gap series in weighted-type spaces on the unit ball. Ann. Funct. Anal. 8 (2017), no. 2, 259--269. doi:10.1215/20088752-0000011X. https://projecteuclid.org/euclid.afa/1488358821