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A Dimensionally Continued Poisson Summation Formula

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Abstract

We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula, our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, our methods work to establish the (Voronoi) summation formulae associated with functions satisfying (modular) transformations of the Jacobi imaginary type by means of a density argument (as opposed to the usual Mellin transform approach). In particular, we construct a family of generalized theta series from Jacobi theta functions from which these summation formulae can be obtained. This family contains several families of modular forms, but is significantly more general than any of them. Our result also relaxes several of the hypotheses in the standard statements of these summation formulae. The density result we prove for Gaussians in the Schwartz space may be of independent interest.

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Notes

  1. The integral representation for 0 F 1 we used is 07.17.07.0004.01 on the Wolfram Functions Site [41].

  2. These identities are 03.01.03.0004.01 and 07.17.03.0037.01, respectively, on the Wolfram Functions Site [41].

  3. The Maclaurin series for 0 F 1 is 07.17.02.0001.01 on the Wolfram Functions Site [41].

  4. Nota bene: We denote the set of positive integers by ℕ, and the set of nonnegative integers by ℕ0.

  5. Nota bene: Simon defines the h n without the factor of (−1)n (that here comes from our H n ). We have included the (−1)n for notational simplicity (since we use the standard convention for the Hermite polynomials). This does not have any effect on Simon’s Theorem 3, since it simply amounts to a sign change of the odd Hermite coefficients.

  6. This is 05.01.11.0001.01 on the Wolfram Functions Site [41].

  7. Personal communication from John Roe.

  8. This differential equation for 0 F 1 is 07.17.13.0003.01 on the Wolfram Functions Site [41].

  9. The notebook is available at http://gravity.psu.edu/~nathanjm/Dim_cont_PSF_test.nb.

  10. But note that Ryavec characterizes all admissible ϒs (under certain assumptions) for d=1 in [33]. We also call attention to the work of Córdoba [11, 12], who shows that in integer dimensions, large classes of generalized Poisson summation formulae arise from the standard Poisson summation formula applied to the finite disjoint union of (integer dimensional) lattices. (Note that Lagarias makes a slight correction to the statement of Theorem 2 of [11] in Theorem 3.7 of [23].)

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Acknowledgements

It is our pleasure to thank George Andrews, Jingzhi Tie, and Shuzhou Wang for encouragement and useful comments, John Roe for mentioning an alternative proof of the density result, Artur Tšobanjan for helpful remarks about the exposition, and the anonymous referee for various perceptive comments and suggestions. We also thank Ben Owen for comments on the manuscript and for suggesting the physics problem that sparked these investigations. This work was supported by NSF grants PHY-0555628 and PHY-0855589, the Eberly research funds of Penn State, and the DFG SFB/Transregio 7.

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  1. Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität, Max-Wien-Platz 1, 07743, Jena, Germany

    Nathan K. Johnson-McDaniel

  2. Department of Physics, Center for Particle and Gravitational Astrophysics, Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA, 16802, USA

    Nathan K. Johnson-McDaniel

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Correspondence to Nathan K. Johnson-McDaniel.

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Communicated by Hans G. Feichtinger.

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Johnson-McDaniel, N.K. A Dimensionally Continued Poisson Summation Formula. J Fourier Anal Appl 18, 367–385 (2012). https://doi.org/10.1007/s00041-011-9207-0

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