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Asymptotic expansion of polyanalytic Bergman kernels
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-4971-7147
2014 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 267, no 12, 4667-4731 p.Article in journal (Refereed) Published
Abstract [en]

We consider the q-analytic functions on a given planar domain Omega, square integrable with respect to a weight. This gives us a q-analytic Bergman kernel, which we use to extend the Bergman metric to this context. We recall that f is q-analytic if (partial derivative) over bar (q) f = 0 for the given positive integer q. Polyanalytic Bergman spaces and kernels appear naturally in time-frequency analysis of Gabor systems of Hermite functions as well as in the mathematical physics of the analysis of Landau levels.

We obtain asymptotic formulae in the bulk for the q-analytic Bergman kernel in the setting of the power weights e(-2mQ), as the positive real parameter m tends to infinity. This is only known previously for q = 1, by the work of Tian, Yau, Zelditch, and Catlin. Our analysis, however, is inspired by the more recent approach of Berman, Berndtsson, and Sjostrand, which is based on ideas from microlocal analysis.

We remark here that since a q-analytic function may be identified with a vector-valued holomorphic function, the Bergman space of q-analytic functions may be understood as a vector-valued holomorphic Bergman space supplied with a certain singular local metric on the vectors. Finally, we apply the obtained asymptotics for q = 2 to the bianalytic Bergman metrics, and after suitable blow-up, the result is independent of Q for a wide class of potentials Q. We interpret this as an instance of geometric universality.

Place, publisher, year, edition, pages
2014. Vol. 267, no 12, 4667-4731 p.
Keyword [en]
Polyanalytic functions, Bergman kernel, Asymptotic expansion, Bulk universality
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-122142DOI: 10.1016/j.jfa.2014.09.002ISI: 000346226500003Scopus ID: 2-s2.0-84921969223OAI: oai:DiVA.org:kth-122142DiVA: diva2:620977
Funder
Göran Gustafsson Foundation for Research in Natural Sciences and MedicineSwedish Research Council, 2012-3122
Note

QC 20150220. Updated from submitted to published.

Available from: 2013-05-13 Created: 2013-05-13 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Polyanalytic Bergman Kernels
Open this publication in new window or tab >>Polyanalytic Bergman Kernels
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The thesis consists of three articles concerning reproducing kernels ofweighted spaces of polyanalytic functions on the complex plane. In the first paper, we study spaces of polyanalytic polynomials equipped with a Gaussianweight. In the remaining two papers, more general weight functions are considered. More precisely, we provide two methods to compute asymptotic expansions for the kernels near the diagonal and then apply the techniques to get estimates for reproducing kernels of polyanalytic polynomial spaces equipped with rather general weight functions.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. vii, 25 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 2013:02
Keyword
Polyanalytic function, determinantal point process, Bergman kernel
National Category
Natural Sciences
Identifiers
urn:nbn:se:kth:diva-122073 (URN)
Public defence
2013-05-28, F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20130513

Available from: 2013-05-13 Created: 2013-05-08 Last updated: 2013-05-13Bibliographically approved

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Hedenmalm, Håkan

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