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Poisson Summation Formula in Hardy Spaces \(H^p(T_\Gamma )\), \(p\in (0,1]\)

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Abstract

The Poisson summation formula for Hardy spaces \(H^p\left( T_\Gamma \right) \) in tubes \(T_\Gamma \subset \mathbb {C}^n\) for \(p\in \left( 0,1\right] \) is obtained. Unlike the case of \(L^p\left( \mathbb {R}^n\right) \) spaces, the formula holds everywhere in \(T_\Gamma \) without any additional assumptions. To the best of our knowledge, the result is new even for the univariate case—Hardy spaces in the upper half-plane.

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Acknowledgments

The authors thank the anonymous referees for their valuable suggestions helping to improve the article. In particular, they led to including references to several interesting results on Poisson summation formula we had missed. The authors also thank Dr. John Paul Ward (University of Central Florida) for useful interactions during the presentation of the proof at the Analysis seminar at the University of Central Florida.

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  1. Department of Mathematics, University of Central Florida, 4393 Andromeda Loop N, Orlando, FL, 32816, USA

    Xin Li & Alexander V. Tovstolis

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  1. Xin Li
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  2. Alexander V. Tovstolis
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Correspondence to Alexander V. Tovstolis.

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Communicated by Dmitry Khavinson.

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Li, X., Tovstolis, A.V. Poisson Summation Formula in Hardy Spaces \(H^p(T_\Gamma )\), \(p\in (0,1]\) . Comput. Methods Funct. Theory 16, 689–697 (2016). https://doi.org/10.1007/s40315-016-0170-2

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