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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A Poisson summation formula for integrals over quadratic surfaces


Author: Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 270 (1982), 163-173
MSC: Primary 42B10; Secondary 22E30, 43A85
DOI: https://doi.org/10.1090/S0002-9947-1982-0642335-7
MathSciNet review: 642335
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Abstract: Let $S(t)$ denote Lebesgue measure on the sphere of radius $t > 0$ in ${{\mathbf {R}}^n}$, and \[ {S_k}(t) = {\left ( {\frac {\partial } {{\partial t}}\quad \frac {1} {t}} \right )^k}S(t).\] Let $P{\sum _k} = {S_k}(0) + 2\sum _{m = 1}^\infty {S_k}(m)$. Theorem. If $n$ is odd and $j$ and $k$ are nonnegative integers with $j + k = (n - 1) / 2$, then the Fourier transform of $P{\sum _j}$ is ${(2\pi )^{j - k}}P{\sum _k}$. There is an analogous, although slightly different, identity involving integrals over hyperboloids in odd dimensions. These results were inspired by recent work of M. Vergne.


References [Enhancements On Off] (What's this?)

  • I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Academic Press, New York-London, 1964. Translated by Eugene Saletan. MR 0166596
  • Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
  • Robert S. Strichartz, Fourier transforms and non-compact rotation groups, Indiana Univ. Math. J. 24 (1974/75), 499–526. MR 380278, DOI https://doi.org/10.1512/iumj.1974.24.24037
  • M. Vergne, A Plancherel formula without group representations, Lecture, O.A.G.R. Conference, Bucharest, Roumania, 1980. ---, A Poisson-Plancherel formula for semi-simple Lie groups, preprint.

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Keywords: Poisson summation formula
Article copyright: © Copyright 1982 American Mathematical Society