The Bergman kernel and a generalized Fourier-Borel transform (Q2712248)

Let \(P= (p_m)_{m\geq 1}\) be a decreasing sequence of continuous functions \(p_m:\mathbb{C}^n\to \mathbb{R}\). Let \(h_{p_m}\) be the Hilbert space of integer functions \(f\) such that NEWLINE\[NEWLINE\|f\|_m:= \Biggl[\int_{\mathbb{C}^n}|f(z)|^2 \exp(-2p_m(z) d\lambda(z))\Biggr]^{{1\over 2}}< \infty,NEWLINE\]NEWLINE where \(\lambda\) denotes the Lebesgue measure on \(\mathbb{C}^n\). Let \({\mathcal F}_p\) be the intersection of the Hilbert spaces \(H_{p_m}\), \(m= 1,2,\dots\), with its ordinary Fréchet spaces structure. Let \(p_0(z):= \lim_m p_m(z)\), \(z\in\mathbb{C}^n\). If \(H_{p_0}\) is dense in \({\mathcal F}_p\) and Bergman's kernel \(K_{p_0}(z,w)\) satisfies a certain condition, in this paper the dual space of \({\mathcal F}_p\) is represented as a space of entire functions.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00034].