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.

- 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