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Berezin Transform in Polynomial Bergman Spaces

Ameur, Yacin and Hedenmalm, Håkan and Makarov, Nikolai (2010) Berezin Transform in Polynomial Bergman Spaces. Communications on Pure and Applied Mathematics, 63 (12). pp. 1533-1584. ISSN 0010-3640.

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Fix a smooth weight function Q in the plane, subject to a growth condition from below. Let K_(m,n) denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n - 1 of finite L^2-norm with respect to the measure e^(-mQ) dA. Here dA is normalized area measure, and m is a positive real scaling parameter. The (polynomial) Berezin measure dB^(<Z0>)_(m,n)(z)= K_(m,n)(z_0,z_0)^(-1) │K_(m,n)(z,z_0)│^2e^(-mQ(z)) dA(z) for the point z_0 is a probability measure that defines the (polynomial) Berezin transform B_(m,n f)(z_0)= ʃC f dB^(<z0>)_(m,n) for continuous f є L^∞(C). We analyze the semiclassical limit of the Berezin measure (and transform) as m → +∞ while n = m τ + o(1), where τ is fixed, positive, and real. We find that the Berezin measure for z_0 converges weak-star to the unit point mass at the point z_0 provided that ΔQ(z_0) > 0 and that z_0 is contained in the interior of a compact set S_ τ, defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane. For points z_0 є C\S _τ , the Berezin measure cannot converge to the point mass at z_0. In the model case Q(z)= │z│^2, when S_ τ is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z_0 relative to C\S_ τ. Our results have applications to the study of the eigenvalues of random normal matrices. The auxiliary results include weighted L^2 -estimates for the equation ∂[overscore]u = f when f is a suitable test function and the solution u is restricted by a polynomial growth bound at ∞.

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Additional Information:© 2010 Wiley Periodicals, Inc. Received October 2009. Article first published online: 4 Aug 2010. This research was partially supported by grants from the Swedish Science Council (VR) and from the the Göran Gustafsson Foundation.
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Swedish Science CouncilUNSPECIFIED
Göran Gustafsson FoundationUNSPECIFIED
Issue or Number:12
Record Number:CaltechAUTHORS:20101111-144213512
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Official Citation:Ameur, Y., Makarov, N. and Hedenmalm, H. (2010), Berezin transform in polynomial bergman spaces. Communications on Pure and Applied Mathematics, 63: 1533–1584. doi: 10.1002/cpa.20329
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20765
Deposited By: Ruth Sustaita
Deposited On:15 Nov 2010 16:46
Last Modified:03 Oct 2019 02:14

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