Fourier-Poisson Duality

• Fourier-Poisson duality is a framework that exchanges spatial functions and their frequency counterparts through lattice summation and dual measures.
• It extends classical Poisson summation to discrete-support and nonperiodic settings, offering new insights in spectral analysis, tiling, and diffraction.
• The theory generalizes to weighted, geometric, and quantum formulations, providing computational tools for solving PDEs and exploring quantum group dynamics.

Fourier-Poisson duality is a family of correspondences in which Fourier transform exchanges a spatial object and a frequency-side object linked by lattice summation, periodicity, or a dual measure. Its classical prototype is the Poisson Summation Formula (PSF), which in one dimension may be written as

$$\sum_{n\in\mathbb Z} f(n)=\sum_{n\in\mathbb Z} \widehat f(n),\qquad \widehat f(\xi)=\int_{\mathbb R} e^{-2\pi i \xi x}f(x)\,dx,$$

or distributionally as

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$

In the literature considered here, that template reappears in several distinct but related forms: discrete-support Fourier pairs on $\mathbb R$, formal duality for periodic configurations, diffraction of model sets, spectral-pair measures for unbounded tilings, weighted and geometric Fourier transforms, symbol-level formulations of the Poisson equation, and local Fourier dualities in quantum and Poisson-Lie settings (Kolountzakis, 2015, Cohn et al., 2013, Chakraborty et al., 27 Oct 2025, Tudoran, 2024, Luu et al., 2015, Massar, 1 Dec 2025).

1. Classical lattice duality and the Poisson-summation template

The classical PSF states that for a lattice $\Lambda\subset \mathbb R^n$,

$\sum_{x \in \Lambda} f(x) = \frac{1}{\covol(\Lambda)} \sum_{y \in \Lambda^*} \widehat f(y),$

where

$$\widehat f(y)=\int_{\mathbb{R}^n} f(x)e^{-2\pi i\langle x,y\rangle}\,dx$$

and $$\Lambda^*=\{y:\langle x,y\rangle\in\mathbb{Z}\text{ for all }x\in\Lambda\}$$. In one dimension, the identity

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n$$

gives a self-dual discrete measure, and by translating, modulating, and dilating this identity one obtains many other discrete-support Fourier pairs (Kolountzakis, 2015).

For radial potentials, the Gaussian is the standard analytic model. If

$G_c(r)=e^{-\pi cr^2},$

then

$\widehat{G_c}(r)=c^{-n/2}e^{-\pi r^2/c},$

so the parameter transformation $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$0 is built directly into Fourier transform. In this form, Poisson summation expresses a relation between a lattice and its dual lattice, and also between the energies of dual periodic configurations (Cohn et al., 2013).

The significance of this prototype is its rigidity. The PSF generates Fourier pairs whose supports inherit lattice or periodic structure, and much of the later literature can be read either as an extension of this pattern or as a demonstration that the PSF-generated class is not exhaustive.

2. Discrete-support Fourier pairs beyond finite PSF generation

A central nonclassical result is the existence of a translation-bounded measure

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$1

with $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$2 discrete, such that

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$3

with $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$4 discrete, while neither $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$5 nor $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$6 is contained in a finite union of arithmetic progressions. Here “translation bounded” means that the total variation in any unit interval is uniformly bounded, and “discrete” means that every bounded interval contains only finitely many support points. Since measures obtained from finitely many applications of the PSF, together with translation, modulation, and dilation, have support contained in a finite union of arithmetic progressions, this pair cannot be derived from finitely many applications of the PSF (Kolountzakis, 2015).

The construction proceeds in two stages. First, for any prescribed $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$7, one builds a nonzero Fourier pair $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$8 of discrete-support measures such that both $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$9 and $\mathbb R$0 vanish on $\mathbb R$1. The starting point is a finite cyclic group $\mathbb R$2, where one chooses a nonzero function $\mathbb R$3 vanishing on

$\mathbb R$4

with discrete Fourier transform $\mathbb R$5 also vanishing on $\mathbb R$6. From $\mathbb R$7, one forms an $\mathbb R$8-periodic measure

$\mathbb R$9

whose Fourier transform is supported on $\Lambda\subset \mathbb R^n$0: $\Lambda\subset \mathbb R^n$1 After scaling,

$\Lambda\subset \mathbb R^n$2

and with $\Lambda\subset \mathbb R^n$3, both measures vanish on $\Lambda\subset \mathbb R^n$4 (Kolountzakis, 2015).

Second, one combines infinitely many such periodic pairs with increasing gaps. Choosing $\Lambda\subset \mathbb R^n$5, a sequence $\Lambda\subset \mathbb R^n$6 that is linearly independent over $\Lambda\subset \mathbb R^n$7, and normalizing factors $\Lambda\subset \mathbb R^n$8, one defines

$\Lambda\subset \mathbb R^n$9

The weights ensure convergence and translation boundedness, the expanding zero neighborhoods ensure discreteness of the supports, and the $\sum_{x \in \Lambda} f(x) = \frac{1}{\covol(\Lambda)} \sum_{y \in \Lambda^*} \widehat f(y),$0-linearly independent shifts obstruct containment in any finite union of arithmetic progressions (Kolountzakis, 2015).

This result broadens Fourier-Poisson duality decisively. It shows that a measure and its Fourier transform can both be purely atomic without arising from any finite PSF construction and without possessing global periodic structure.

3. Formal duality, model sets, and diffraction

For periodic configurations, a direct identity

$\sum_{x \in \Lambda} f(x) = \frac{1}{\covol(\Lambda)} \sum_{y \in \Lambda^*} \widehat f(y),$1

is too strong and forces lattices. A weaker invariant is the average pair sum. If

$\sum_{x \in \Lambda} f(x) = \frac{1}{\covol(\Lambda)} \sum_{y \in \Lambda^*} \widehat f(y),$2

then

$\sum_{x \in \Lambda} f(x) = \frac{1}{\covol(\Lambda)} \sum_{y \in \Lambda^*} \widehat f(y),$3

Periodic configurations $\sum_{x \in \Lambda} f(x) = \frac{1}{\covol(\Lambda)} \sum_{y \in \Lambda^*} \widehat f(y),$4 are formally dual if

$\sum_{x \in \Lambda} f(x) = \frac{1}{\covol(\Lambda)} \sum_{y \in \Lambda^*} \widehat f(y),$5

for every Schwartz function $\sum_{x \in \Lambda} f(x) = \frac{1}{\covol(\Lambda)} \sum_{y \in \Lambda^*} \widehat f(y),$6. In the finite-group reformulation, with $\sum_{x \in \Lambda} f(x) = \frac{1}{\covol(\Lambda)} \sum_{y \in \Lambda^*} \widehat f(y),$7 for a finite abelian group $\sum_{x \in \Lambda} f(x) = \frac{1}{\covol(\Lambda)} \sum_{y \in \Lambda^*} \widehat f(y),$8, formal duality is equivalent to

$\sum_{x \in \Lambda} f(x) = \frac{1}{\covol(\Lambda)} \sum_{y \in \Lambda^*} \widehat f(y),$9

or equivalently

$$\widehat f(y)=\int_{\mathbb{R}^n} f(x)e^{-2\pi i\langle x,y\rangle}\,dx$$0

This makes formal duality a combinatorial statement about difference sets and character sums. The paper gives concrete examples such as TITO, with $$\widehat f(y)=\int_{\mathbb{R}^n} f(x)e^{-2\pi i\langle x,y\rangle}\,dx$$1 and $$\widehat f(y)=\int_{\mathbb{R}^n} f(x)e^{-2\pi i\langle x,y\rangle}\,dx$$2, and a family arising from quadratic Gauss sums in $$\widehat f(y)=\int_{\mathbb{R}^n} f(x)e^{-2\pi i\langle x,y\rangle}\,dx$$3 (Cohn et al., 2013).

A different but related realization occurs in cut-and-project theory. For a lattice $$\widehat f(y)=\int_{\mathbb{R}^n} f(x)e^{-2\pi i\langle x,y\rangle}\,dx$$4 in a cut-and-project scheme and a weight $$\widehat f(y)=\int_{\mathbb{R}^n} f(x)e^{-2\pi i\langle x,y\rangle}\,dx$$5, the weighted Dirac comb is

$$\widehat f(y)=\int_{\mathbb{R}^n} f(x)e^{-2\pi i\langle x,y\rangle}\,dx$$6

The generalized PSF states that, under the transformability hypotheses of the paper,

$$\widehat f(y)=\int_{\mathbb{R}^n} f(x)e^{-2\pi i\langle x,y\rangle}\,dx$$7

For Riemann integrable $$\widehat f(y)=\int_{\mathbb{R}^n} f(x)e^{-2\pi i\langle x,y\rangle}\,dx$$8, the autocorrelation is

$$\widehat f(y)=\int_{\mathbb{R}^n} f(x)e^{-2\pi i\langle x,y\rangle}\,dx$$9

and the diffraction measure is

$$\Lambda^*=\{y:\langle x,y\rangle\in\mathbb{Z}\text{ for all }x\in\Lambda\}$$0

For regular model sets, this yields pure point diffraction. The paper proves that the diffraction formula for regular model sets is equivalent to the PSF for the underlying lattice (Richard et al., 2015).

Taken together, these results shift Fourier-Poisson duality away from the single question of “lattice versus dual lattice.” In formal duality it becomes a relation between pair correlations and Fourier magnitudes; in model-set diffraction it becomes a statement that Bragg structure is precisely the PSF of a hidden higher-dimensional lattice.

4. Spectral pairs and unbounded tiling sets

In the spectral-pair framework, a measurable set $$\Lambda^*=\{y:\langle x,y\rangle\in\mathbb{Z}\text{ for all }x\in\Lambda\}$$1 and a positive Radon measure $$\Lambda^*=\{y:\langle x,y\rangle\in\mathbb{Z}\text{ for all }x\in\Lambda\}$$2 form a spectral pair if the Fourier transform gives an isometric isomorphism

$$\Lambda^*=\{y:\langle x,y\rangle\in\mathbb{Z}\text{ for all }x\in\Lambda\}$$3

When $$\Lambda^*=\{y:\langle x,y\rangle\in\mathbb{Z}\text{ for all }x\in\Lambda\}$$4 has finite measure, this reduces to the classical notion of a spectrum $$\Lambda^*=\{y:\langle x,y\rangle\in\mathbb{Z}\text{ for all }x\in\Lambda\}$$5, where $$\Lambda^*=\{y:\langle x,y\rangle\in\mathbb{Z}\text{ for all }x\in\Lambda\}$$6 is counting measure on $$\Lambda^*=\{y:\langle x,y\rangle\in\mathbb{Z}\text{ for all }x\in\Lambda\}$$7 and $$\Lambda^*=\{y:\langle x,y\rangle\in\mathbb{Z}\text{ for all }x\in\Lambda\}$$8 is an orthogonal basis for $$\Lambda^*=\{y:\langle x,y\rangle\in\mathbb{Z}\text{ for all }x\in\Lambda\}$$9, with

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n$$0

For unbounded $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n$$1, the spectrum is replaced by a measure (Chakraborty et al., 27 Oct 2025).

A concrete one-dimensional theorem identifies a tiling condition with an exact dual measure. If $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n$$2 is open and $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n$$3, then

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n$$4

if and only if $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n$$5 is spectral with pair measure

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n$$6

The corresponding Plancherel identity is

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n$$7

for $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n$$8, and the Fourier transform is onto: $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n$$9 The proof constructs finite approximants

$G_c(r)=e^{-\pi cr^2},$0

whose discrete Parseval identities converge to the limiting periodic dual measure (Chakraborty et al., 27 Oct 2025).

The reverse implication derives structural constraints from the spectral side. For $G_c(r)=e^{-\pi cr^2},$1, the periodization

$G_c(r)=e^{-\pi cr^2},$2

has period $G_c(r)=e^{-\pi cr^2},$3, which implies

$G_c(r)=e^{-\pi cr^2},$4

and therefore

$G_c(r)=e^{-\pi cr^2},$5

In this formulation, Fourier-Poisson duality is not between two discrete sets. It is a duality between an unbounded tiling set and a periodic Lebesgue measure, with Fourier transform acting as an onto isometry between the two Hilbert spaces (Chakraborty et al., 27 Oct 2025).

5. Weighted, geometric, and algebraic generalizations

One line of generalization replaces the classical Fourier transform by a weighted operator built from dilation sums. For a sequence $G_c(r)=e^{-\pi cr^2},$6,

$G_c(r)=e^{-\pi cr^2},$7

and with convolution inverse $G_c(r)=e^{-\pi cr^2},$8, the Fourier-Poisson operator is defined by

$G_c(r)=e^{-\pi cr^2},$9

Under the decay hypothesis

$\widehat{G_c}(r)=c^{-n/2}e^{-\pi r^2/c},$0

for some $\widehat{G_c}(r)=c^{-n/2}e^{-\pi r^2/c},$1, and the same condition for $\widehat{G_c}(r)=c^{-n/2}e^{-\pi r^2/c},$2, the paper proves that $\widehat{G_c}(r)=c^{-n/2}e^{-\pi r^2/c},$3 is unitary and is uniquely determined by

$\widehat{G_c}(r)=c^{-n/2}e^{-\pi r^2/c},$4

After logarithmic-Fourier conjugation it becomes multiplication by

$\widehat{G_c}(r)=c^{-n/2}e^{-\pi r^2/c},$5

so the operator is diagonalized by a ratio of Dirichlet series on the critical line (Faifman, 2011).

A second line starts from a nondegenerate bilinear form $\widehat{G_c}(r)=c^{-n/2}e^{-\pi r^2/c},$6 on $\widehat{G_c}(r)=c^{-n/2}e^{-\pi r^2/c},$7. The associated left and right geometric Fourier transforms are

$\widehat{G_c}(r)=c^{-n/2}e^{-\pi r^2/c},$8

For the geometric pair $\widehat{G_c}(r)=c^{-n/2}e^{-\pi r^2/c},$9, they satisfy

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$00

and they are exactly the transforms that place Poisson summation for the full lattices $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$01 and $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$02 into the canonical forms

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$03

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$04

When $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$05 is positive definite, the same framework yields a geometric fractional Laplacian defined by the multiplier $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$06 (Tudoran, 2024).

Other papers isolate algebraic aspects of the same theme. One interprets the continuous Fourier transform as a limiting CRT decomposition: sampled values of $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$07 are packaged into a polynomial $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$08, sampled values of $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$09 appear as residues modulo the factors of $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$10, and the Poisson summation formula

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$11

emerges from the finite cyclic structure in the limit $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$12 (Xu, 2018). Another shows that for the Gaussian $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$13, Poisson summation and cosine Fourier expansion on $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$14 are equivalent once the image terms are negligible for sufficiently large $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$15, specifically $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$16 (Abrarov et al., 2012).

These generalizations preserve the defining feature of Fourier-Poisson duality: an operator or summation identity exchanges two structures that are not identical on the nose but are tied by a transform law, often with an explicit reconstruction mechanism.

6. Poisson equations, symbols, and computational formulations

In numerical PDE analysis, the phrase denotes the diagonalization of the Poisson operator in frequency space. For

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$17

the Fourier differentiation rule gives

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$18

hence

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$19

and therefore

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$20

The singularity at $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$21 requires the solvability correction

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$22

so that $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$23. In the discrete setting, the DFT

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$24

and its FFT implementation reduce the cost from $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$25 to $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$26. The method is naturally suited to periodic grids and is constrained by the Nyquist frequency

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$27

and aliasing (Rodriguez-Lara et al., 2024).

A more recent symbol-level formulation removes the need for an explicit Green’s function. For the scalar Poisson equation, the inverse symbol is

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$28

and the key identity is

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$29

Splitting the time integral at $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$30 gives

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$31

so the solution decomposes into a localized part $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$32 and a smooth nonlocal history part $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$33. The localized component admits asymptotic expansions in powers of $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$34 for volume, single-layer, and double-layer potentials, with coefficients expressed in terms of local geometry and derivatives of the source data. The same Fourier-domain construction extends to coupled strongly elliptic systems by replacing $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$35 with a matrix symbol $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$36 and using $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$37 (Fryklund, 13 Apr 2026).

In this computational setting, Fourier-Poisson duality is no longer primarily a statement about dual supports or lattices. It is the statement that the inverse Poisson operator becomes an algebraic multiplier in Fourier space, and that this multiplier can be regularized, discretized, or asymptotically decomposed while remaining entirely at the level of symbols.

7. Local Fourier duality in quantum curves and Poisson-Lie quantum groups

For quantum curves, the basic object is a pair of ordinary differential operators

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$38

with bi-degree $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$39. There are two natural KP deformations: one obtained by normalizing $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$40, the other by normalizing $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$41. The paper associates to the quantum curve a companion-matrix connection $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$42 on the formal punctured disc and proves that the two KP orbits are related by local Fourier transform. If $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$43 are coprime and both $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$44 are normalized, then

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$45

where the new KP times are determined by compositional inversion relations. In 2D gravity, this yields a conceptual proof of the $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$46 duality: the local Fourier transform exchanges the connections attached to the two models, and therefore exchanges the corresponding KP data (Luu et al., 2015). A closely related treatment realizes the $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$47 duality of 2D gravity as a local Fourier duality of irregular $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$48-modules attached to Kac-Schwarz operators (Luu, 2014).

In Poisson-Lie and quantum-group theory, the duality takes yet another form. For semidirect-product Poisson-Lie groups

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$49

with $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$50 abelian, suitable abelian approximations

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$51

allow the construction of an explicit unitary operator

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$52

the Poisson-Fourier transform. It implements the quantum duality principle in the form

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$53

by inducing an isomorphism of locally compact quantum groups (Massar, 1 Dec 2025). In a related QP-manifold formulation of Poisson-Lie T-duality, canonical transformations on a doubled correspondence space lead to a proposed Fourier-Mukai-type integral transform with kernel $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$54, where

$$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$55

so that the kernel intertwines the twisted differential complexes on the dual sides (Arvanitakis et al., 2021).

These quantum-geometric formulations retain the recognizable Fourier-Poisson pattern—an explicit transform exchanges dual data—but the objects are now connections, KP orbits, bicrossed products, or twisted flux complexes rather than functions on $$\left(\sum_{n\in\mathbb Z}\delta_n\right)^\wedge=\sum_{n\in\mathbb Z}\delta_n.$$56. The common structure is that Fourier transform, or a Fourier-type operator, is the mechanism that converts one side of the duality into the other.