Modified Poisson Summation Formula

• Modified Poisson summation formula is a generalization of the classical formula that extends summation over lattices to include weights, non-integer dimensions, and singular spaces.
• It leverages advanced analytic methods and operator theory to derive generalized Fourier transforms and modular corrections for various applications.
• The framework significantly impacts number theory, spectral analysis, and mathematical physics by unifying analytic, arithmetic, and geometric perspectives.

The modified Poisson summation formula encompasses a diverse family of extensions, generalizations, and reformulations of the classical Poisson summation formula, arising from both analytic and arithmetic considerations, operator theory, and applications in geometry, physics, and representation theory. These modifications are motivated by the need to handle more general summation domains, include additional data (weights, indices, modular or arithmetic factors), allow for continuous or non-integer parameters, or adapt to singular spaces. The following sections systematically survey key formulations, methodological frameworks, structural themes, and contemporary research directions for the modified Poisson summation formula.

1. Classical and Dimensionally Continued Generalizations

The seminal generalization consists of extending the Poisson summation formula from lattices in $\mathbb{R}^d$ and their standard duals to the level of theta series and to spaces of non-integer (even continuous or complex) dimension parameters. In this context, the modified Poisson summation formula operates via a “dimensionally continued” Fourier transform. For a radial function $f(r)$ and real parameter $d > 0$, this transform is defined as

$$\hat{f}(p) = 2\pi\, p^{-(d-2)/2} \int_0^\infty f(r) J_{(d-2)/2}(2\pi p r)\, r^{d/2} dr,$$

or, equivalently, using the confluent hypergeometric function,

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

The theta series of a lattice $\Lambda$ is expressed as $$\Theta_{\Lambda}(q) = \sum_{k \in \Lambda} q^{|k|^2}$$, and the dimensionally continued summation formula reads

$$\sum_{l=0}^\infty N_l\, f(\sqrt{A_l}) = (1/\sqrt{\det \Lambda}) \sum_{l=0}^\infty N_l^*\, \hat{f}(\sqrt{A_l^*}),$$

where $N_l$ (resp. $N_l^*$) and $A_l$ (resp. $A_l^*$) are coefficients and exponents in the theta series for $\Lambda$ and its dual. This identity admits analytic continuation in $d$ and is established for all even Schwartz functions via density arguments leveraging the mapping properties of Gaussians under the dimensionally continued transform (Johnson-McDaniel, 2010).

These ideas are further generalized to families of “generalized theta series,” which are finite linear combinations of products of Jacobi theta functions with arbitrary non-negative real exponents, enabling construction of summation identities that extend modular transformation behavior to non-integer and even “non-lattice” situations. The analytic proof avoids the Mellin transform in favor of density of Gaussians and properties of the Jacobi imaginary transformation.

2. Operator, Weighted, and Möbius-Modulated Formulations

The operator-theoretic perspective introduces modified summation operators $T_{(a_n)}$ defined by

$T_{(a_n)} f(x) = \sum_{n \geq 1} a_n f(n x),$

with $(a_n)$ a complex weight sequence subject to decay and convolution-inverse conditions. The family of unitary “Fourier-Poisson operators” is then given by

$$\mathcal{F}_{(a_n)} f (x) = T_{(a_n)}^{-1} S T_{(a_n)} f(x)$$

where $S f(x) = f(1/x)$. For suitable $(a_n)$ and their Dirichlet series $L(s; a_n)$, the operator admits a diagonalization via isometries mapping $L^2([0, \infty))$ to $L^2(\mathbb{R})$, with the diagonal form

$$\left(w \mathcal{F}_{(a_n)} w^{-1} h \right)(x) = \frac{L(\frac{1}{2} - ix; a_n)}{L(\frac{1}{2} + ix; a_n)} h(-x).$$

This structure yields weighted Poisson summation formulas, with pointwise or operator-interpreted identities depending on decay conditions on $(a_n)$ (Faifman, 2011). The classical Fourier transform and involution $S$ appear as special cases.

Using the “mother-equation” formalism, summation formulae including Möbius-modulated variants are unified. The Möbius-Poisson summation formula,

$\sum_{n\geq 1} \mu(n)F(ny) = (\text{asymptotic expansion in derivatives of %%%%20%%%% at %%%%21%%%%}) + R(y),$

arises systematically as a residue computation for a suitable kernel in the contour-integral formulation, revealing intrinsic connections between analytic and arithmetic corrections (Jean-Christophe, 2016).

3. Modified Domains and Structures: Quadrics, Index and Operator Transforms

Modified Poisson summation appears in settings where the summation domain is not a full lattice but a subvariety, frequently singular or stratified, such as the zero-locus $X$ of a quadratic form $Q$ in even dimension. The summation formula then takes the form

$$\sum_{i=1}^\ell \Big[ c_i(d_{\ell, i}(f)) + \sum_{\xi \in X^\circ_i} I(d_{\ell,i}f)(\xi) \Big] + \kappa d_{\ell,0}f(0)$$

$$= \sum_{i=1}^\ell \Big[ c_i(d_{\ell, i}(\mathcal{F}_X(f))) + \sum_{\xi \in X^\circ_i} I(d_{\ell,i}\mathcal{F}_X(f))(\xi) \Big] + \kappa d_{\ell,0}\mathcal{F}_X(f)(0),$$

where $d_{\ell,i}$ are descent maps and $c_i$, $I$ are explicit functionals or integrals over smaller quadrics or boundary strata. Such formulas capture the precise “boundary terms” necessary for analytic continuation, notably relevant in the analytic theory of quadratic forms and in the spectral theory of automorphic representations, via the Arthur-truncated theta lift and the Schrödinger model (Getz, 2022).

In the context of special function index transforms, such as series in the imaginary index of the Whittaker function $W_{\mu, i n}(x)$, the Poisson summation formula is adapted to the index variable: $$\sum_{n \in \mathbb{Z}} W_{p, i n}(x) = \sum_{n \in \mathbb{Z}} \widehat{W}_{p}(2\pi n, x),$$ where $\widehat{W}_{p}$ denotes the Fourier transform in the index. The derivation employs a weak variant of the classical Poisson formula justified via analytic continuation and residue calculation, extending classical identities (e.g., for $K_\nu(x)$) (Ribeiro et al., 19 Jul 2024).

For box splines and parameterized semi-discrete convolutions, the modified Poisson summation involves the application of a Todd differential operator and a sum over vertex sets,

$$f(\lambda) = \sum_{s \in \mathcal{V}} s^{-\lambda} \lim_{t \to 0^+} (\operatorname{Todd}([q], s, y)(\partial)\, P(s, y, f))(\lambda + t\epsilon)|_{q=1},$$

thus inverting the convolution with the box spline and yielding precise deconvolution formulas even in non-unimodular, parameterized settings (Vergne, 2013).

4. Arithmetic, Modular, and Trace Formula Modifications

The arithmetic refinement of Poisson summation integrates Dirichlet or Möbius weights, and modular/arithmetic data, as in

$$\sum_{n \geq 1} f(n x) = \frac{1}{2\pi i} \int_{(\sigma)} \frac{\zeta(s)}{\zeta(2s)} f^*(s)x^{-s} ds,$$

with $f^*(s)$ the Mellin transform and arithmetic implications following from Ramanujan identities for $\zeta(s)$. Such formulas (Poisson–Müntz–Möbius–Voronoi) systematically interpolate between analytic and arithmetic summation behavior (Yakubovich, 2014), leading to transformation and summation formulae for modular forms and theta functions and even to explicit criteria for the Riemann hypothesis.

In the spectral theory of automorphic forms on $GL(2)$, “modified” Poisson summation achieves isolation/cancellation of the trivial representation in the Arthur–Selberg trace formula. By reindexing the regular elliptic contribution onto the Steinberg–Hitchin base and applying a completed Poisson formula for the function $\theta_f(a)$ (the orbital integral along the fiber of the characteristic polynomial), one obtains

$$J^G_\text{ell}(f) = (1(f)) + \sum_{a \in \mathbb{Q}^\times} \widehat{\theta}_f(a) - \sum_{a \in \mathbb{Q}_{\mathrm{nell}}} \theta_f(a),$$

with $(1(f))$ the trivial representation’s contribution and the sums over $\theta_f$ and its Fourier transform $\widehat{\theta}_f$ capturing regular elliptic and complementary terms. The adelic setting ensures convergence and correct measure normalization; this analytic maneuver is crucial for advancing the Beyond Endoscopy program (Wong, 2 Oct 2025).

Modified Poisson summation also underlies Poisson-type trace formulas for nonstandard test functions (e.g., Whittaker or Kuznetsov trace formulas), providing harmonic analytic proofs of meromorphic continuation and functional equations for standard $L$-functions of $GL(2)$ (Li, 21 Oct 2024), and is fundamental in the paper of Eisenstein series on Schubert varieties (Choie et al., 2021).

5. Analytical, Numerical, and Physical Applications

The variety and flexibility of modified Poisson summation formulas are highlighted in their applications to diverse analytical and physical problems. For example, in statistical mechanics, the modified Euler–Maclaurin formula interpolates between the classical Euler–Maclaurin and Poisson summation formulas,

$$\sum_{i=a}^b f(i) = \sum_{i=a}^{a+m-1} f(i) + \int_{a+m}^b \frac{\sin((2p+1)\pi x)}{\sin(\pi x)} f(x) dx + \frac{f(a+m) + f(b)}{2} + \sum_{r=1}^\infty (-1)^r T_{2r,p} M_{2r-1}(a+m, b) + R_{mnp},$$

with explicit error control and tunable parameters making it suitable for precise partition-function calculations in low-temperature regimes (Guo et al., 2020).

Similarly, band-limited or compactly supported kernels (e.g., products of sinc functions) enable exact evaluation of integrals and sums using Poisson summation, with modifications for modulation (e.g., by cosines), leading to sum-form Siegel-type lower bounds and novel identities in harmonic analysis (Almkvist et al., 2014).

Finally, for transforms such as the special affine wavelet transform (SAWT), analogues of Poisson summation formula are derived that incorporate the complex phase, dilation, and translation symmetries inherent in advanced time-frequency analysis: $$\sum_{k \in \mathbb{Z}} f(t + kT) = \frac{1}{a} \sum_{k \in \mathbb{Z}} e^{iA k^2 T^2 + 2iA kT t + 2ip kT - i\frac{DB kT}{2B} - i\frac{kT(Dp - Bq)}{2B}} W_M(b, \xi),$$ with $W_M$ the SAWT coefficient, and $(A, B, D, p, q)$ the unimodular matrix parameters (Shah et al., 2020).

6. Connections to Discrete, Finite, and Combinatorial Frameworks

Formal duality and the combinatorial recasting of the Poisson summation formula find application in the analysis of point configurations, periodic sets, and potential energy minimization in finite abelian groups. Poisson summation-like identities become combinatorial equalities,

$$\left| \frac{1}{|S|} \sum_{v \in S} \langle v, y \rangle \right|^2 = \frac{1}{|T|} \#\{(w, w') \in T \times T : y = w-w'\}$$

for subsets $S, T$ of dual finite abelian groups, encoding the energy duality of periodic configurations. Such formal duality is critical for classifying energy-minimizing lattices and for constructing non-lattice “formal duals,” with deep ties to Gauss sums and higher reciprocity (Cohn et al., 2013).

7. Contemporary Impact and Outlook

The modified Poisson summation formula continues to be a central analytic tool across modern mathematics and physics. Its flexibility is evidenced by its role in analytic number theory (e.g., functional equations for zeta and $L$-functions), spectral theory (e.g., trace formulas, automorphic representations), harmonic analysis (wavelets, index transforms), mathematical physics (partition functions, diffraction), and the combinatorics and symmetry of discrete structures.

The common thread in these generalizations is the systematic adaptation of the analytic duality principle—summing a function over a discrete set and expressing this sum via an appropriate (generalized or modified) transform or dual object—under increasingly broad hypotheses (allowing singularities, modular corrections, arithmetic or spectral weights, generalized domains, and relaxations of smoothness or decay conditions).

Advances in the theory of modified Poisson summation formulas continue to deepen the connections between analysis, number theory, geometry, combinatorics, and mathematical physics, as well as to yield new tools for computation and problem-solving in these fields.