A dimensionally continued Poisson summation formula

TL;DR

This paper extends the Poisson summation formula to noninteger dimensions using a dimensionally continued Fourier transform, providing new proofs and broader applicability for theta series and modular forms.

Contribution

It introduces a generalized Poisson summation formula operating on theta series with noninteger dimensions, expanding the theoretical framework and relaxing previous hypotheses.

Findings

Generalized Poisson summation formula for noninteger dimensions

New proof of the classical Poisson summation formula for dimensions > 2

Construction of a broad family of generalized theta series

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.