Bulletin of the Belgian Mathematical Society - Simon Stevin

Weighted integral representations of entire functions of several complex variables

Arman H. Karapetyan

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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

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

First available in Project Euclid: 8 May 2008

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Zentralblatt MATH identifier

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

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


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

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