A Family of Unitary Operators Satisfying a Poisson-Type Summation Formula
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.
KeywordsFourier Transform Unitary Operator Summation Formula Poisson Summation Formula Isometric Operator
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.
- 1.L. Bàez-Duarte, A class of invariant unitary operators. Adv. Math. 144, 1–12 (1999)Google Scholar
- 2.J.-F. Burnol, On Fourier and Zeta(s). Habilitationsschrift, 2001–2002, Forum Mathematicum 16, 789–840 (2004)Google Scholar
- 3.A. Cordoba, La formule sommatoire de Poisson. C.R. Acad Sci. Paris, Serie I 306, 373–376 (1988)Google Scholar
- 4.H. Davenport, On some infinite series involving arithmetical functions. Q. J. Math. 8(8–13), 313–320 (1937)Google Scholar
- 5.R.J. Duffin, H.F. Weinberger, Dualizing the Poisson summation formula, Proc. Natl. Acad. Sci. USA 88, 7348–7350 (1991)Google Scholar
- 6.D. Faifman, A characterization of Fourier transform by Poisson summation formula. Comptes rendus – Mathematique 348, 407–410 (2010)Google Scholar
- 7.A. Korànyi, The Bergman kernel function for tubes over convex cones. Pac. J. Math. 12(4), 1355–1359 (1962)Google Scholar
- 8.M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (Academic, CA, 1975)Google Scholar