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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Asymptotic behaviour of reproducing kernels of weighted Bergman spaces
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by Miroslav Engliš PDF
Trans. Amer. Math. Soc. 349 (1997), 3717-3735 Request permission

Abstract:

Let $\Omega$ be a domain in $\mathbb {C}^{n}$, $F$ a nonnegative and $G$ a positive function on $\Omega$ such that $1/G$ is locally bounded, $A^{2}_{\alpha }$ the space of all holomorphic functions on $\Omega$ square-integrable with respect to the measure $F^{\alpha }G\,d\lambda$, where $d\lambda$ is the $2n$-dimensional Lebesgue measure, and $K_{\alpha }(x,y)$ the reproducing kernel for $A^{2}_{\alpha }$. It has been known for a long time that in some special situations (such as on bounded symmetric domains $\Omega$ with $G=\mathbf {1}$ and $F=\,$the Bergman kernel function) the formula \begin{equation*}\lim _{\alpha \to +\infty }K_{\alpha }(x,x)^{1/\alpha }=1/F(x) \tag {$*$} \end{equation*} holds true. [This fact even plays a crucial role in Berezin’s theory of quantization on curved phase spaces.] In this paper we discuss the validity of this formula in the general case. The answer turns out to depend on, loosely speaking, how well the function $-\log F$ can be approximated by certain pluriharmonic functions lying below it. For instance, ($*$) holds if $-\log F$ is convex (and, hence, can be approximated from below by linear functions), for any function $G$. Counterexamples are also given to show that in general ($*$) may fail drastically, or even be true for some $x$ and fail for the remaining ones. Finally, we also consider the question of convergence of $K_{\alpha }(x,y)^{1/\alpha }$ for $x\neq y$, which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of $F$: for instance, if $F$ is not real-analytic at some point, then $K_{\alpha }(x,y)$ must have zeroes for all $\alpha$ sufficiently large.
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Additional Information
  • Miroslav Engliš
  • Affiliation: Mathematical Institute of the Academy of Sciences, Žitná 25, 11567 Prague 1, Czech Republic
  • Email: englis@math.cas.cz
  • Received by editor(s): March 22, 1996
  • Additional Notes: The author’s research was supported by GA AV ČR grants C1019601 and 119106 and by GA ČR grant 201/96/0411.
  • © Copyright 1997 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 349 (1997), 3717-3735
  • MSC (1991): Primary 30C40, 32H10; Secondary 31C10, 30E15
  • DOI: https://doi.org/10.1090/S0002-9947-97-01843-6
  • MathSciNet review: 1401769