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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
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

$\displaystyle {S_k}(t) = {\left( {\frac{\partial } {{\partial t}}\quad \frac{1} {t}} \right)^k}S(t).$

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?)

  • [1] I. M. Gelfand and G. E. Shilov, Generalized functions. Vol. I, Academic Press, New York, 1964. MR 0166596 (29:3869)
  • [2] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N. J., 1971. MR 0304972 (46:4102)
  • [3] R. S. Strichartz, Fourier transforms and non-compact rotation groups, Indiana Univ. Math. J. 24 (1974), 499-526; Errata 30 (1981), 479-480. MR 0380278 (52:1178)
  • [4a] M. Vergne, A Plancherel formula without group representations, Lecture, O.A.G.R. Conference, Bucharest, Roumania, 1980.
  • [4b] -, 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

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