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Frames in Hermite-Bergman and special Hermite-Bergman spaces

  • A. Antony Selvan
  • R. Radha
Article
  • 113 Downloads

Abstract

It is well known that the image of \(L^2(\mathbb {R}^n)\) under the Hermite semigroup \(e^{-tH}\) is a Bergman space \(\mathcal {H}_t(\mathbb {C}^n)\) with the reproducing kernel \(L_t\), for \(t>0\). In this paper, it is shown that the discrete system \(\{e_{ap+ibq, t}:p,q\in \mathbb {Z}^n\}\), for \(a,b>0\), arising out of \(L_t\) turns out to be a frame for \(\mathcal {H}_t(\mathbb {C}^n)\) iff \(ab<\pi \sinh 4t\). A similar type of problem is also discussed for Bessel sequences and frames in holomorphic Sobolev spaces associated with the Hermite semigroup. An attempt is also made to study a similar type of problem for the Bergman space \(\mathcal {B}_t^*(\mathbb {C}^{2n})\), which is the image of \(L^2(\mathbb {C}^n)\) under the special Hermite semigroup.

Keywords

Bergman spaces Bessel sequence Frames Gabor frames Hermite functions Hermite semigroup Sobolev spaces Special Hermite functions 

Mathematics Subject Classification

30H20 33C45 42C15 

Notes

Acknowledgments

We thank the referee for meticulously reading our manuscript and giving us several valuable suggestions in revising the manuscript.

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

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