Poisson Summation with Signature

• Poisson summation with signature is a refined duality where extra data such as phases, weights, and character twists encode underlying geometric and arithmetic structures.
• Key methodologies include the integration of hyper-Kloosterman sums in Voronoi formulas on GL(n) and the incorporation of local factors like γ- and ε-factors in automorphic trace formulas.
• The approach unifies disparate treatments—from weighted Fourier-Poisson operators to quadratic forms—by modifying the classical summation-transform cycle to reveal detailed spectral and arithmetic signatures.

The Poisson summation formula with signature is a family of refinements of the classical identity

$$\sum_{n\in\mathbb{Z}} f(n)=\sum_{m\in\mathbb{Z}} \widehat f(m),$$

or, more generally,

$$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$

in which the passage from a sum over a lattice, quotient, or orbit to a dual sum is modified by additional structural data. In the current literature, “signature” is not a single standardized term; it is used interpretively for weights, parity, homogeneity, arithmetic phases, hyper-Kloosterman twists, local $\gamma$- and $\varepsilon$-factors, transfer factors, cone orientations, character data, or the local signature of a quadratic form. The common feature is that the basic Poisson architecture survives, but the transform carries extra signs, phases, or correction terms that encode geometry, representation theory, or arithmetic structure (Zhou, 2014, Faifman, 2011, Li, 2024, Getz, 2022).

1. Classical duality and the emergence of signature

In its classical form, Poisson summation expresses a symmetry between a lattice and its dual. For a Schwartz function $f:\mathbb{R}\to\mathbb{C}$ with Fourier transform

$$\widehat f(\xi)=\int_{\mathbb{R}} f(x)e(-x\xi)\,dx,\qquad e(t):=e^{2\pi i t},$$

the formula

$$\sum_{n\in\mathbb{Z}} f(n)=\sum_{m\in\mathbb{Z}} \widehat f(m)$$

is the simplest instance of the principle “sums over a lattice $\leftrightarrow$ sums over its dual lattice.” In higher dimension, the same duality is expressed through $\Lambda\subset\mathbb{R}^d$ and $\Lambda^*\subset\widehat{\mathbb{R}^d}$ (Zhou, 2014).

Already in elementary analytic number theory, additive twists introduce a first layer of signature-like structure. A sum

$$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$0

transforms into a dual sum with the additive character $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$1, where $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$2 is the inverse of $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$3. The duality remains Poissonian, but the summands now carry a phase pattern. This anticipates later settings in which the transform is weighted by automorphic coefficients, character families, or local factors (Zhou, 2014).

A useful interpretive point is that several papers explicitly treat “signature” as an explanatory label rather than a formal theorem name. In some works the term does not literally appear, even though signs, orientations, character twists, or spectral phases play exactly that role. This suggests that “Poisson summation with signature” denotes a class of structures rather than a single canonical identity (Vergne, 2013, Cohn et al., 2013).

2. Main types of signature data

Across the literature, the extra data attached to Poisson summation takes several distinct forms. In each case the original sum, the dual sum, and the transform are governed by structural invariants beyond bare Fourier duality.

Framework	Signature data	Source
Weighted Fourier–Poisson operators	Sequence $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$4, Dirichlet symbol $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$5, spectral phase $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$6	(Faifman, 2011)
Formal duality in finite abelian groups	Difference multiset statistics and normalized Fourier amplitudes $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$7	(Cohn et al., 2013)
Box splines and hyperplane poles	Cone orientation, signs $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$8, vertex characters $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$9, tangent-cone conditions	(Vergne, 2013)
Algebraic summation of homogeneous series	Homogeneity type $\gamma$0, parity $\gamma$1, regularized Poisson object $\gamma$2	(Nori, 2012)
Automorphic trace formulas	Local $\gamma$3- and $\gamma$4-factors, root numbers, irregular orbital terms	(Li, 2024)

These forms of signature are not equivalent. A weight sequence $\gamma$5 in operator theory, a difference multiset in formal duality, and a transfer factor $\gamma$6 in endoscopy encode different phenomena. What unifies them is that each modifies the summation-transform-summation cycle by structural data intrinsic to the problem.

3. Hyper-Kloosterman signatures in Voronoi summation on $\gamma$7

A central automorphic realization of Poisson summation with signature is Fan Zhou’s family of Voronoi summation formulae on $\gamma$8. For an automorphic form $\gamma$9 on $\varepsilon$0 with Fourier–Whittaker coefficients $\varepsilon$1, the standard $\varepsilon$2-function is

$\varepsilon$3

Zhou shows that for $\varepsilon$4 there are $\varepsilon$5 different Voronoi formulae on $\varepsilon$6, parametrized by $\varepsilon$7, and each may be viewed as a Poisson summation formula weighted by automorphic coefficients and twisted by hyper-Kloosterman sums (Zhou, 2014).

For a prime $\varepsilon$8, the $\varepsilon$9-dimensional hyper-Kloosterman sum is

$f:\mathbb{R}\to\mathbb{C}$0

Two limiting cases are structurally important. For $f:\mathbb{R}\to\mathbb{C}$1,

$f:\mathbb{R}\to\mathbb{C}$2

so the twist is just an additive character. For $f:\mathbb{R}\to\mathbb{C}$3, one recovers the classical one-dimensional Kloosterman sum. This is the point at which the passage from ordinary Poisson summation to higher-rank Voronoi summation becomes visible as an increase in phase complexity (Zhou, 2014).

The main formula relates a sum of $f:\mathbb{R}\to\mathbb{C}$4 twisted by a $f:\mathbb{R}\to\mathbb{C}$5-dimensional hyper-Kloosterman sum to a dual sum of $f:\mathbb{R}\to\mathbb{C}$6 twisted by an $f:\mathbb{R}\to\mathbb{C}$7-dimensional hyper-Kloosterman sum, together with correction sums coming from Hecke relations and coefficients with a prime $f:\mathbb{R}\to\mathbb{C}$8 inserted in intermediate positions. For an even Maass cusp form $f:\mathbb{R}\to\mathbb{C}$9 for $$\widehat f(\xi)=\int_{\mathbb{R}} f(x)e(-x\xi)\,dx,\qquad e(t):=e^{2\pi i t},$$0, one side has the schematic form

$$\widehat f(\xi)=\int_{\mathbb{R}} f(x)e(-x\xi)\,dx,\qquad e(t):=e^{2\pi i t},$$1

and the dual side has

$$\widehat f(\xi)=\int_{\mathbb{R}} f(x)e(-x\xi)\,dx,\qquad e(t):=e^{2\pi i t},$$2

where $$\widehat f(\xi)=\int_{\mathbb{R}} f(x)e(-x\xi)\,dx,\qquad e(t):=e^{2\pi i t},$$3 is a Mellin-type transform built from the $$\widehat f(\xi)=\int_{\mathbb{R}} f(x)e(-x\xi)\,dx,\qquad e(t):=e^{2\pi i t},$$4 gamma factors

$$\widehat f(\xi)=\int_{\mathbb{R}} f(x)e(-x\xi)\,dx,\qquad e(t):=e^{2\pi i t},$$5

The powers of $$\widehat f(\xi)=\int_{\mathbb{R}} f(x)e(-x\xi)\,dx,\qquad e(t):=e^{2\pi i t},$$6 and the signs $$\widehat f(\xi)=\int_{\mathbb{R}} f(x)e(-x\xi)\,dx,\qquad e(t):=e^{2\pi i t},$$7, $$\widehat f(\xi)=\int_{\mathbb{R}} f(x)e(-x\xi)\,dx,\qquad e(t):=e^{2\pi i t},$$8 in the correction terms record the arithmetic contribution of the $$\widehat f(\xi)=\int_{\mathbb{R}} f(x)e(-x\xi)\,dx,\qquad e(t):=e^{2\pi i t},$$9-twisted functional equations and Hecke relations (Zhou, 2014).

The phrase “with signature” is especially apt here because the formulas separate even and odd components through

$$\sum_{n\in\mathbb{Z}} f(n)=\sum_{m\in\mathbb{Z}} \widehat f(m)$$0

so the transform distinguishes symmetric and antisymmetric phase patterns under $$\sum_{n\in\mathbb{Z}} f(n)=\sum_{m\in\mathbb{Z}} \widehat f(m)$$1. The original side carries hyper-Kloosterman signature of dimension $$\sum_{n\in\mathbb{Z}} f(n)=\sum_{m\in\mathbb{Z}} \widehat f(m)$$2, the dual side signature of dimension $$\sum_{n\in\mathbb{Z}} f(n)=\sum_{m\in\mathbb{Z}} \widehat f(m)$$3, and the family $$\sum_{n\in\mathbb{Z}} f(n)=\sum_{m\in\mathbb{Z}} \widehat f(m)$$4 records the different ways in which this signature can be distributed across the transform. Zhou’s paper also identifies the previously known $$\sum_{n\in\mathbb{Z}} f(n)=\sum_{m\in\mathbb{Z}} \widehat f(m)$$5 formula as one endpoint and the Li–Miller $$\sum_{n\in\mathbb{Z}} f(n)=\sum_{m\in\mathbb{Z}} \widehat f(m)$$6 formula as the case $$\sum_{n\in\mathbb{Z}} f(n)=\sum_{m\in\mathbb{Z}} \widehat f(m)$$7, $$\sum_{n\in\mathbb{Z}} f(n)=\sum_{m\in\mathbb{Z}} \widehat f(m)$$8 (Zhou, 2014).

These formulas are not merely formal analogues. The paper states that Voronoi formulas of this type are central tools for bounding exponential sums, studying moments of $$\sum_{n\in\mathbb{Z}} f(n)=\sum_{m\in\mathbb{Z}} \widehat f(m)$$9-functions, subconvexity, equidistribution, and even conjectural functorial lifts when $\leftrightarrow$0-twisted functional equations are available (Zhou, 2014).

4. Spectral and trace-formula signatures in automorphic harmonic analysis

In the representation-theoretic literature, the signature may be local, spectral, or endoscopic. On the Whittaker space of $\leftrightarrow$1, a local Poisson summation formula is realized through a Jacquet-type Hankel transform rather than an ordinary Fourier transform. The local orbital integral space on $\leftrightarrow$2 is acted on by a transform

$\leftrightarrow$3

built from multiplicative Fourier operators along cocharacters and a phase factor $\leftrightarrow$4. The global formula

$\leftrightarrow$5

shows that the Poisson summation formula on the Whittaker space yields the functional equation for the standard $\leftrightarrow$6-functions of $\leftrightarrow$7. Here the signature is carried by local $\leftrightarrow$8- and $\leftrightarrow$9-factors, root numbers, the phase $\Lambda\subset\mathbb{R}^d$0, and the irregular boundary distributions that appear in the Kuznetsov quotient (Li, 2024).

For $\Lambda\subset\mathbb{R}^d$1, the corresponding phenomenon is endoscopic rather than Whittaker-theoretic. The trace of $\Lambda\subset\mathbb{R}^d$2 on $\Lambda\subset\mathbb{R}^d$3 is rewritten as a sum over orbital integrals on the geometric side and as a discrete-series expansion on the spectral side. Endoscopy introduces a sign character $\Lambda\subset\mathbb{R}^d$4 and transfer factors $\Lambda\subset\mathbb{R}^d$5, so the stable trace formula takes the form

$\Lambda\subset\mathbb{R}^d$6

The paper explicitly interprets this as a noncommutative Poisson summation formula in which the underlying group signature $\Lambda\subset\mathbb{R}^d$7, the Galois action, Weyl-group parity, and endoscopic transfer signs are inseparable parts of the transform (Diep et al., 2014).

In the Braverman–Kazhdan program, signature appears through normalized intertwining operators and residue terms. Getz and Liu construct a Schwartz space on

$\Lambda\subset\mathbb{R}^d$8

define a generalized Fourier transform by

$\Lambda\subset\mathbb{R}^d$9

and prove a refined Poisson summation formula in which the raw sums are accompanied by explicit Eisenstein-residue corrections. Those residue terms occur for the trivial character and for quadratic characters $\Lambda^*\subset\widehat{\mathbb{R}^d}$0, and the normalized intertwiner is built from products of $\Lambda^*\subset\widehat{\mathbb{R}^d}$1-factors. In this setting the signature consists of character parity, gamma factors, and the pole structure of degenerate Eisenstein series (Getz et al., 2017).

Cheng’s ramified beyond-endoscopy formula for $\Lambda^*\subset\widehat{\mathbb{R}^d}$2 pushes this further into a semilocal setting. The Poisson summation formula is carried out on

$\Lambda^*\subset\widehat{\mathbb{R}^d}$3

with lattice $\Lambda^*\subset\widehat{\mathbb{R}^d}$4, ramification set $\Lambda^*\subset\widehat{\mathbb{R}^d}$5, local orbital-integral factors $\Lambda^*\subset\widehat{\mathbb{R}^d}$6 and $\Lambda^*\subset\widehat{\mathbb{R}^d}$7, modified norms $\Lambda^*\subset\widehat{\mathbb{R}^d}$8, and generalized Kloosterman weights $\Lambda^*\subset\widehat{\mathbb{R}^d}$9. The approximate functional equation is used to validate Poisson summation and then residues are computed to isolate one-dimensional and Eisenstein contributions. In this case the signature is precisely the ramification profile and the local orbital-integral data that survive the semilocal transform (Cheng, 25 May 2025).

5. Quadrics and probabilistic signatures

A different line of development replaces the lattice by the zero locus of a quadratic form. For an even-dimensional quadratic space $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$00 over a number field, one defines

$$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$01

Using the Weil representation and coinvariants, one obtains Schwartz spaces $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$02, a Fourier transform $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$03, and dimension-lowering maps $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$04. The Poisson summation formula for quadrics is not a two-term identity; its novelty is that all boundary terms are given either by constants or by sums over smaller quadrics related to the original quadric. In its explicit form,

$$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$05

equals the same expression with $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$06 replaced by $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$07. The local signature of the quadratic form governs the archimedean behavior of these terms, and the formula is linked to the counting problem for rational or integral solutions of $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$08 in even dimension (Getz, 2022).

The later geometrization of this formula for split quadratic forms makes explicit the relation between the Braverman–Kazhdan and theta-lift definitions of Schwartz spaces on quadrics. The summation domain is

$$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$09

and the paper states that because these quadrics are defined by a quadratic form, the local behavior of that form—including its signature at archimedean places—is reflected in the analytic and geometric structure of the Poisson summation formula. The geometrization shows that the two constructions of the Fourier transform on $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$10 coincide (Hsu, 31 May 2026).

A probabilistic version of signature arises on locally compact abelian groups. Let $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$11 be LCA, $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$12 discrete, and $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$13 compact. For a convolution semigroup $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$14 with continuous densities $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$15, periodization gives

$$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$16

and under Poisson summation one has

$$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$17

If $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$18, then

$$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$19

This probabilistic trace formula identifies the diagonal value of the wrapped density with the spectral signature of the Markov semigroup on $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$20. The Gaussian satisfies the full Poisson summation formula, rotationally invariant $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$21-stable densities satisfy the trace formula but the paper cannot verify Poisson summation for them, and certain adelic semistable constructions yield densities that fail even the probabilistic trace formula (Applebaum, 2016).

6. Applications, extensions, and terminological issues

The range of applications is broad precisely because the notion of signature is broad. In analytic number theory, hyper-Kloosterman signatures in Voronoi summation are used to transform additively twisted sums, to study moments of $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$22-functions, subconvexity, and equidistribution questions (Zhou, 2014). In discrete and Euclidean energy minimization, formal duality turns Poisson summation into a statement about pair sums and difference multiplicities, explaining self-duality phenomena for Gaussian potentials and giving nonexistence results in settings such as $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$23, periodic Barlow packings, and the Best packing in $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$24 (Cohn et al., 2013).

In signal analysis, the special affine wavelet transform provides a distinctly different signature model. The special affine Fourier transform is determined by a six-parameter unimodular matrix

$$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$25

and the corresponding Poisson summation formula relates time-domain samples weighted by SAFT chirps to discrete samples of $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$26. The paper explicitly treats $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$27 together with the wavelet parameters $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$28 as the signature of the transform, and its constant $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$29-property is independent of $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$30 (Shah et al., 2020).

Two interpretive cautions are important. First, “signature” does not always mean the signature of a quadratic form. Depending on the setting, it may mean parity $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$31, a weight sequence $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$32, a spectral phase $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$33, a difference multiset, a cone orientation, a local root number, a transfer factor, or the actual archimedean signature of $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$34 (Faifman, 2011, Nori, 2012, Getz, 2022). Second, a trace formula is not automatically equivalent to Poisson summation: the Gaussian case exhibits both, but the $$\sum_{\lambda\in\Lambda} f(\lambda)=\frac{1}{\operatorname{vol}(\mathbb{R}^d/\Lambda)}\sum_{\lambda^*\in\Lambda^*}\widehat f(\lambda^*),$$35-stable and adelic semistable examples show that a spectral trace may exist when a genuine Poisson summation identity is unavailable, and may fail altogether in more singular adelic situations (Applebaum, 2016).

Taken together, these developments show that Poisson summation with signature is best understood as a structural paradigm. The invariant core is always a duality between an original summation problem and a transformed one. What changes from paper to paper is the data that must accompany the transform: automorphic coefficients, hyper-Kloosterman phases, Dirichlet-series symbols, formal-duality spectra, cone signs, endoscopic transfer factors, gamma and epsilon factors, ramification sets, or boundary terms from smaller quadrics. The modern theory therefore extends classical Poisson summation not by abandoning duality, but by refining it until the relevant arithmetic, geometric, spectral, or probabilistic signature is visible on both sides of the formula.