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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)

Abstract

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.

Keywords

Bergman Space Planar Domain Bergman Kernel Hankel Operator Weighted Bergman Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© 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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