A Family of Unitary Operators Satisfying a Poisson-Type Summation Formula

• Dmitry Faifman
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)

Abstract

We consider a weighted form of the Poisson summation formula. We prove that under certain decay rate conditions on the weights, there exists a unique unitary Fourier–Poisson operator which satisfies this formula. We next find the diagonal form of this operator, and prove that under weaker conditions on the weights, a unique unitary operator still exists which satisfies a Poisson summation formula in operator form. We also generalize the interplay between the Fourier transform and derivative to those Fourier–Poisson operators.

Keywords

Fourier Transform Unitary Operator Summation Formula Poisson Summation Formula Isometric Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

I am indebted to Bo’az Klartag for the idea behind Sect. 5, and also for the motivating conversations and reading the drafts. I am grateful to Nir Lev, Fedor Nazarov, Mikhail Sodin and Sasha Sodin for the illuminating conversations and numerous suggestions. Also, I’d like to thank my advisor, Vitali Milman, for the constant encouragement and stimulating talks. Finally, I would like to thank the Fields Institute for the hospitality during the final stages of this work.

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