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Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc

Authors: Gadadhar Misra, Subrata Shyam Roy and Genkai Zhang
Journal: Proc. Amer. Math. Soc. 141 (2013), 2361-2370
MSC (2010): Primary 30H20, 46E22, 47B32
Published electronically: March 13, 2013
MathSciNet review: 3043017
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Abstract: A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions includes the Szegö and the Bergman kernel on the symmetrized polydisc.

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Additional Information

Gadadhar Misra
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India

Subrata Shyam Roy
Affiliation: Indian Institute of Science Education and Research, Kolkata, Mohanpur Campus, Mohanpur (West Bengal) 741 252, India

Genkai Zhang
Affiliation: Department of Mathematics, Chalmers University of Technology and Gothenburg University, S-412 96 Gothenburg, Sweden

Keywords: Symmetrized polydisc, permutation group, sign representation, Schur functions, weighted Bergman space, Hardy space, weighted Bergman kernel, Szeg\"o kernel
Received by editor(s): June 20, 2011
Received by editor(s) in revised form: October 17, 2011
Published electronically: March 13, 2013
Additional Notes: Financial support for the work of the first and third authors was provided by the Swedish Research Links programme entitled “Hilbert modules, operator theory and complex analysis”. The research of the first and second authors was funded by grants from DST and UKIERI
Communicated by: Richard Rochberg
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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