Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1011.3790

A dimensionally continued Poisson summation formula

Nathan K. Johnson-McDaniel
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.CAmath-phmath.MPmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nathan K. Johnson-McDaniel. 2011-11-04. A dimensionally continued Poisson summation formula. https://doi.org/10.1007/s00041-011-9207-0

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fragment-wise differentiable structures

The $p$-modulus of curves, test plans, upper gradients, charts, differentials, approximations in energy and density of directions are all concepts associated to the theory of Sobolev functions in metric measure spaces. The purpose of this paper is to give an analogous geometric and ``fragment-wise'' theory for Lipschitz functions and Weaver derivations, where $\infty$-modulus of curve fragments, $\ast$-upper gradients and Alberti representations play a central role. We give a new definition of fragment-wise charts and prove that they exists for spaces with finite Hausdorff dimension. We give a replacement for $p$-duality in terms of Alberti representations and $\infty$-modulus and present the theory of $\ast$-upper gradients. Further, we give new and sharper results for approximations of Lipschitz functions, which yields the density of directions. Our results are applicable to all complete and separable metric measure spaces. In the process, we show that there are strong parallels between the Sobolev and Lipschitz worlds.

math.CA↗
arXiv2026

Tensor Derivatives, Unified Tensor-Form Differential Equations, and Model Reduction via Partial Tucker Decomposition

This paper develops a unified tensor calculus for matrix-valued functions and their derivatives, and leverages this framework to construct efficient model reduction techniques for high-dimensional tensor differential equations. We first establish a systematic theory of tensor differentiation, wherein the derivative of a matrix with respect to another matrix is represented as a fourth-order tensor. Building on this calculus, we recast linear ordinary differential equations (ODEs) and partial differential equations(PDEs) into a compact tensor-matrix form $\frac{dX}{dt} = \A\ast X$. The general solution is expressed as $X = \exp(t\A)\ast C$, extending the matrix exponential to the tensor setting. Conditions under which the solution admits this exponential form are characterized in terms of the commutativity of the associated matrix slices. We introduce the partial Tucker decomposition (parTuckerD) to address the computational challenges posed by high-order tensor systems. On a synthetic electronic health record (EHR) tensor, parTuckerD achieves a relative reconstruction error of $0.0992$ with a $136.3\times$ compression ratio, matching the accuracy of the full TuckerD while preserving patient-level similarity structure. The results demonstrate that the proposed tensor calculus and parTuckerD framework provide a principle and computationally efficient approach for analyzing and solving high-dimensional tensor differential equations arising in data-intensive applications.

math.CA↗
arXiv2026

Distance preservers for Lobachevsky space

We obtain a complete description of the class of entrywise preservers of Lorentz-Gram matrices. This resolves, for the case of constant negative curvature, the classification of entrywise preservers obtained by Schoenberg in the zero-curvature (Euclidean) and constant-positive-curvature (spherical) settings. These preservers admit a Lévy--Khintchine-type representation and their asymptotic characteristics are related to Krein's classification of screw lines in Lobachevsky space. Connections with complete Nevanlinna--Pick kernels and Bochner subordination are also obtained.

math.CA↗