Asymptotic Behaviour of Reproducing Kernels, Berezin Quantization and Mean-Value Theorems

  • Miroslav Engliš
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 3)


Let Ω be a domain in C n , F and G positive measurable functions on Ω such that 1/F and 1/G are locally bounded, A α 2 the space of all holomorphic functions on Ω square-integrable with respect to the measure F α G dm, where dm is the 2n-dimensional Lebesgue measure, K α (x,y) the reproducing kernel for A α 2 (if it exists), and \({B_\alpha }f(y) = {K_\alpha }{(y,y)^{ - 1}}\int {_\Omega } f(x){\text{|}}{{\text{K}}_\alpha }{\text{(x,y)}}{{\text{|}}^{\text{2}}}F{(x)^\alpha }G(x)dm(x)\) the Berezin operator on Ω. In this paper we present some results on the asymptotic behavior of K α and B α ,as α → +∞. 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 ,(x,y) has a zero for all sufficiently large α whenever F is not real-analytic. Applications to mean value theorems and to quantization on curved phase spaces are also discussed.


Bergman Space Planar Domain Bergman Kernel Hankel Operator Weighted Bergman Space 
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© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Miroslav Engliš
    • 1
  1. 1.Mathematical Institute of the Academy of SciencesPrague 1Czech Republic

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