## Journal of the Mathematical Society of Japan

### Fourier-Borel transformation on the hypersurface of any reduced polynomial

#### Abstract

For a polynomial $p$ on $\mathbf{C}^{n}$, the variety $V_{p} = \{ z \in \mathbf{C}^{n} ; p(z)=0 \}$ will be considered. Let $\mathrm{Exp}(V_{p})$ be the space of entire functions of exponential type on $V_{p}$, and $\mathrm{Exp}^{\prime}(V_{p})$ its dual space. We denote by $\partial p$ the differential operator obtained by replacing each variable $z_{j}$ with $\partial / \partial z_{j}$ in $p$, and by $\mathcal{O}_{\partial p}(\mathbf{C}^{n})$ the space of holomorphic solutions with respect to $\partial p$. When $p$ is a reduced polynomial, we shall prove that the Fourier-Borel transformation yields a topological linear isomorphism: $\mathrm{Exp}^{\prime}(V_{p}) \to \mathcal{O}_{\partial p}(\mathbf{C}^{n})$. The result has been shown by Morimoto, Wada and Fujita only for the case $p(z) = z_{1}^{2} + \cdots + z_{n}^{2} + \lambda \, (n \geq 2)$.

#### Article information

Source
J. Math. Soc. Japan, Volume 60, Number 1 (2008), 65-73.

Dates
First available in Project Euclid: 24 March 2008

https://projecteuclid.org/euclid.jmsj/1206367955

Digital Object Identifier
doi:10.2969/jmsj/06010065

Mathematical Reviews number (MathSciNet)
MR2392003

Zentralblatt MATH identifier
1131.42009

#### Citation

KOWATA, Atsutaka; MORIWAKI, Masayasu. Fourier-Borel transformation on the hypersurface of any reduced polynomial. J. Math. Soc. Japan 60 (2008), no. 1, 65--73. doi:10.2969/jmsj/06010065. https://projecteuclid.org/euclid.jmsj/1206367955

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