## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Weighted integral representations of entire functions of several complex variables

Arman H. Karapetyan

#### Abstract

In the paper we consider the spaces of entire functions $f(z), z\in C^n$, satisfying the condition $$\int_{R^n}\left(\int_{R^n}|f(x+iy)|^p dx \right)^s |y|^{\alpha}e^{-\sigma |y|^{\rho}}dy <+\infty .$$ For these classes the following integral representation is obtained: $$f(z)=\int_{C^n}f(u+iv)\Phi(z,u+iv)|v|^{\alpha}e^{-\sigma |v|^{\rho}}dudv ,\quad z\in C^n ,$$ where the reproducing kernel $\Phi(z,u+iv)$ is written in an explicit form as a Fourier type integral. Also, an estimate for $\Phi$ is obtained.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 15, Number 2 (2008), 287-302.

Dates
First available in Project Euclid: 8 May 2008