## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

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

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1210254826

**Mathematical Reviews number (MathSciNet)**

MR2424114

**Zentralblatt MATH identifier**

1147.32007

**Subjects**

Primary: 32A15: Entire functions 32A25: Integral representations; canonical kernels (Szego, Bergman, etc.) 32A37: Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) [See also 46Exx] 26D15: Inequalities for sums, series and integrals 30D10: Representations of entire functions by series and integrals 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx] 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 44A10: Laplace transform

**Keywords**

weighted spaces of entire functions Paley-Wiener type theorems reproducing kernels weighted integral representations

#### Citation

Karapetyan, Arman H. Weighted integral representations of entire functions of several complex variables. Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 2, 287--302. https://projecteuclid.org/euclid.bbms/1210254826