Asymptotic behaviour of reproducing kernels, Berezin quantization and mean-value theorems (Q2712245)

Let \(\Omega\) be a domain in \(\mathbb{C}^n\), \(F\) and \(G\) positive measurable functions on \(\Omega\) such that \(1/F\) and \(1/G\) are locally bounded, \(A^2_{\alpha}\) the space of all holomorphic functions on \(\Omega\) square-integrable with respect to the measure \(F^\alpha G dm\), where \(dm\) is the \(2n\)-dimensional Lebesgue measure, \(K_\alpha(x,y)\) the reproducing kernel for \(A^2_\alpha\) (if it exists), and NEWLINE\[NEWLINEB_\alpha f(y) = K_\alpha(y,y)^{-1} \int_{\Omega} f(x) |K_\alpha(x,y)|^2 F(x)^\alpha G(x) dm(x)NEWLINE\]NEWLINE the Berezin operator on \(\Omega\). NEWLINENEWLINENEWLINEIn this paper the author gives some results on the asymptotic behavior of \(K_\alpha\) and \(B_\alpha\) as \(\alpha \to + \infty\). For instance, if \(-\log F\) is convex then \(\lim_{\alpha \to +\infty} K_\alpha (x,x)^{1/\alpha} = 1/F(x)\) for any integrable \(G\), and \(K_\alpha(x,y) \) has a zero for all sufficiently large \(\alpha\) whenever \(F\) is not real-analytic. Applications to mean value theorems and to quantization on curved phase spaces are also discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00034].