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Abstract: We generalize the standard Poisson summation formula for lattices so that itoperates on the level of theta series, allowing us to introduce nonintegerdimension parameters using the dimensionally continued Fourier transform.When combined with one of the proofs of the Jacobi imaginary transformation oftheta functions that does not use the Poisson summation formula, our proof ofthis generalized Poisson summation formula also provides a new proof of thestandard Poisson summation formula for dimensions greater than 2 withappropriate hypotheses on the function being summed. In general, our methodswork to establish the Voronoi summation formulae associated with functionssatisfying modular transformations of the Jacobi imaginary type by means of adensity argument as opposed to the usual Mellin transform approach. Inparticular, we construct a family of generalized theta series from Jacobi thetafunctions from which these summation formulae can be obtained. This familycontains several families of modular forms, but is significantly more generalthan any of them. Our result also relaxes several of the hypotheses in thestandard statements of these summation formulae. The density result we provefor Gaussians in the Schwartz space may be of independent interest.



Author: Nathan K. Johnson-McDaniel

Source: https://arxiv.org/







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