Berezin transform in polynomial bergman spaces

By Y. Ameur, Nikolai G. Makarov, H. Hedenmalm. 2008

52 citations recorded by the source corpus.

Fix a smooth weight function Q in the plane, subject to a growth condition from below. Let Km,n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n − 1 of finite L2‐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 ${\rm d}B_{m,n}^{\langle z_0\rangle}(z)=K_{m,n}(z_0,z_0)^{-1}|K_{m,n}(z,z_0)|^2{\rm e}^{-mQ(z)}\,{\rm d}A(z)$ for the point z0 is a probability measure that defines the (polynomial) Berezin transform $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}{\rm B}_{m,n}f(z_0)=\int\limits_{\C} f\,{\rm d}B_{m,n}^{\langle z_0\rangle}$ for continuous $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}f\in L^\infty(\C)$. 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 z0 converges weak‐star to the unit point mass at the point z0 provided that Δ Q(z0) > 0 and that z0 is contained in the interior of a compact set ${\cal S}_\tau$, 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 $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}z_0\in \C\setminus{\cal S}_\tau$, the Berezin measure cannot converge to the point mass at z0. In the model case Q(z) = |z|2, when ${\cal S}_\tau$ is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z0 relative to $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}\C\setminus{\cal S}_\tau$.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cec8fa6b26482035ff042419a6bdbd770e217484

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In Applied Mathematics

  1. Y. Ameur
  2. H. Hedenmalm
  3. Nikolai G. Makarov
  4. J. A. Pel'aez
  5. Alexei Poltoratski
  6. Mishko Mitkovski

In Geometry and Topology

  1. Y. Ameur
  2. H. Hedenmalm
  3. Nikolai G. Makarov

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