Sharp hypotheses and the maximal generalized theta-series class for the dimensionally continued Poisson summation formula

Johnson-McDaniel's dimensionally continued Poisson summation formula (J. Fourier Anal. Appl. 18 (2012) 367-385) lifts the classical Poisson summation formula for lattice sums to the level of theta series, so that the dimension parameter $d$ of the lattice can be taken non-integer: using a dimensionally continued (Hankel-type) Fourier transform for radial functions, it proves an identity between the theta-series-weighted sum of a test function and the corresponding sum of its transform, first for the cubic lattice ($\vartheta_3^d$) and then for a broad family of generalized theta series built from products of rationally rescaled Jacobi theta functions with arbitrary nonnegative real exponents. The author states that the hypotheses — an even Schwartz test function and a real dimension parameter — are surely not optimal, and that the physically relevant slowly decaying kernels of Ewald-type lattice sums are not covered, while numerical evidence supports validity for complex dimension parameters with $\operatorname{Re}d>0$, for test functions with a cusp at the origin such as $e^{-|r|^3}$, and for slowly decaying kernels; the maximal admissible class of generalized theta series is likewise left uncharacterized. This problem asks for the sharp form of the theorem: the weakest hypotheses on the test function and dimension parameter (or counterexamples), a rigorous treatment of slowly decaying Ewald-type kernels, and a characterization of the maximal class of admissible generalized theta series, extending Ryavec's $d=1$ characterization. No resolution of these parts is known in the literature to date.

Background

The classical Poisson summation formula states that for a lattice $\Lambda\subset\mathbb{R}^n$ (a discrete subgroup of Euclidean space whose fundamental Voronoi cells have volume $\det\Lambda$) and a test function $F$ in the Schwartz space $\mathscr{S}(\mathbb{R}^n)$ of smooth functions whose derivatives all decay faster than any power, one has $\sum_{k\in\Lambda}F(k)=\frac{1}{\sqrt{\det\Lambda}}\sum_{p\in\Lambda^*}\tilde F(p)$, where $\Lambda^*$ is the dual lattice and $\tilde F$ is the Fourier transform. The formula underlies lattice-point counting, the mathematical theory of diffraction, and the Ewald summation method of computational physics, which rewrites slowly convergent electrostatic lattice sums as rapidly convergent pairs of real-space and reciprocal-space series. In some physical settings — for example the elastic constants of the crystalline mixed phases conjectured to occur in neutron-star interiors — the effective lattice geometry varies with a physical parameter, and it is natural to treat the dimension $d$ of the lattice sum as a non-integer (continued) parameter, in the spirit of dimensional regularization in quantum field theory, where integrals are analytically continued to complex space-time dimension.

For a radial test function $F(x)=f(|x|)$ the lattice sum can be organized through the theta series of the lattice, $\Theta_\Lambda(q)=\sum_{k\in\Lambda}q^{|k|^2}$: writing the power-series expansion $\Theta_\Lambda(q)=\sum_l N_l q^{A_l}$, one has $\sum_{k\in\Lambda}f(|k|)=\sum_l N_l f(\sqrt{A_l})$, the coefficients $N_l$ counting lattice points on spheres. The theta series of the cubic lattice $\mathbb{Z}^d$ is $\vartheta_3(q)^d$ with $\vartheta_3(q)=\sum_{n\in\mathbb{Z}}q^{n^2}$ a Jacobi theta function, and this expression is meaningful for any complex exponent $d$ — this is the sense in which the lattice itself is dimensionally continued. The Jacobi imaginary transformation $\vartheta_3(-1/z)=(z/i)^{1/2}\vartheta_3(z)$, which has proofs independent of Poisson summation, supplies the lattice duality; and the radial Fourier transform admits the dimensionally continued form

$$\hat f(p)=\frac{2\pi^{d/2}}{\Gamma(d/2)}\int_0^\infty f(r)\,{}_0F_1\!\left(d/2;-\pi^2p^2r^2\right)\,r^{d-1}\,dr,$$

where ${}_0F_1$ is the confluent hypergeometric limit function (equivalently, this is a Hankel-type Bessel transform of order $(d-2)/2$); for integer $d$ it is the ordinary Fourier transform of a radial function, and the expression makes sense for other values of $d$ as well.

Within this framework Johnson-McDaniel proved two theorems. Theorem 1: for $f$ an even Schwartz function and real $d\ge1$, $\sum_l N_l f(\sqrt l)=\sum_l N_l\hat f(\sqrt l)$, where $\vartheta_3(q)^d=\sum_l N_l q^l$. Theorem 2: the same identity with $\vartheta_3^d$ replaced by any generalized theta series — a finite linear combination of products $\prod_m\vartheta_2(q^{s_m})^{\lambda_m}\vartheta_3(q^{t_m})^{\rho_m}\vartheta_4(q^{u_m})^{\sigma_m}$ of rationally rescaled Jacobi theta functions with nonnegative real exponents summing to $d$ — with the dual series $\Upsilon^*$ obtained from the Jacobi transformation with weight $(i/z)^{d/2}$, so that $\sum_l N_l f(\sqrt{A_l})=\sum_l N_l^*\hat f(\sqrt{A_l^*})$ (the proof of the general theorem excludes $d=1$ and parts of the derivation use $d>1$). The proof verifies the identity on Gaussians (which the continued transform permutes) and extends by a density argument, using polynomial boundedness of the coefficients $N_l$; the admissible class contains all classical even-weight modular forms (real exponents replace integral weights) but is far larger. This recovers, in a new guise, Bochner's classical theory of modular relations, which derives summation formulae from transformation laws of Jacobi type: Bochner's version already yields the basic formula for real $d>0$ and test functions of only finite differentiability, as the source itself notes, but it does not cover complex $d$ or functions with a cusp at the origin, and the source's density proof needs rapid decay. The only precedent for a maximal-class statement is Ryavec's characterization of all admissible summation series for $d=1$. Neighboring frameworks fix integer dimension: Voronoi-type summation formulae from automorphic forms and Dirichlet-series functional equations, a distributional radial Poisson formula in diffraction theory (Baake, Frettlöh, and Grimm, 2007), Poisson summation in Hardy spaces of tubes (Li and Tovstolis, 2016), and Poisson-type formulas for Fourier eigenmeasures on spheres (Alfes, Kiefer, and Mazáč, 2025). None of these continues the dimension parameter itself, which is precisely what the present problem concerns.

Problem Statement

Determine the maximal scope of the dimensionally continued Poisson summation formula of Johnson-McDaniel (2012), within that framework: the dimensionally continued Fourier transform

$$\hat f(p)=\frac{2\pi^{d/2}}{\Gamma(d/2)}\int_0^\infty f(r)\,{}_0F_1\!\left(d/2;-\pi^2p^2r^2\right)\,r^{d-1}\,dr$$

for radial test functions, the theta-series expansion $\Upsilon(q)=\sum_l N_l q^{A_l}$ of a generalized theta series, its dual $\Upsilon^*$ defined through the Jacobi imaginary transformation with weight $(i/z)^{d/2}$, and the identity $\sum_l N_l f(\sqrt{A_l})=\sum_l N_l^*\hat f(\sqrt{A_l^*})$. The problem has three jointly supporting parts, each of which the source states as unresolved; establish or refute each, delimiting by proof and by counterexample how far the extension reaches:

1. Parameter and regularity domain. The published theorems assume an even Schwartz test function and a real dimension parameter ($d\ge1$; the general proof excludes $d=1$ and parts of the derivation use $d>1$). Does the identity remain valid for complex dimension parameters with $\operatorname{Re}d>0$ — where a branch of $(i/z)^{d/2}$ must be fixed in the duality — and for test functions of lower regularity or decay, in particular for functions with a cusp at the origin such as $f(r)=e^{-|r|^3}$, for which the source reports numerical evidence of validity? Which differentiability, decay-at-infinity, and origin-regularity conditions are actually necessary?

2. Slowly decaying physical kernels. The formula was motivated by Ewald-type electrostatic lattice sums, but the theorem does not cover the physically relevant slowly decaying kernels, for which the two sides may only converge conditionally under a stated summation convention. Does the identity extend to such kernels (with an explicitly stated, physically faithful convention), or is there a counterexample — a specific kernel and parameter value for which the two sides diverge or disagree under every reasonable convention?

3. Maximal theta class. Characterize the most general class of admissible generalized theta series — q-series with convergent power series in the unit disk and suitably well-behaved coefficients satisfying a Jacobi-type modular transformation — for which the dimensionally continued summation formula holds, extending to noninteger $d$ the characterization of admissible series that Ryavec obtained for $d=1$.

A complete answer comprises the positive theorems covering the extendable part of this range, explicit counterexamples showing where the extension fails, and the characterization of the admissible theta-series class.

References

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