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In mathematics, the Weil–Brezin map, named after André Weil and Jonathan Brezin, is a unitary transformation that maps a Schwartz function on the real line to a smooth function on the Heisenberg manifold. The Weil–Brezin map gives a geometric interpretation of the Fourier transform, the Plancherel theorem and the Poisson summation formula. The image of Gaussian functions under the Weil–Brezin map are nil-theta functions, which are related to theta functions. The Weil–Brezin map is sometimes referred to as the Zak transform, which is widely applied in the field of physics and signal processing; however, the Weil–Brezin Map is defined via Heisenberg group geometrically, whereas there is no direct geometric or group theoretic interpretation from the Zak transform.

Heisenberg manifold

The (continuous) Heisenberg group ${\displaystyle N}$ is the 3-dimensional Lie group that can be represented by triples of real numbers with multiplication rule

${\displaystyle \langle x,y,t\rangle \langle a,b,c\rangle =\langle x+a,y+b,t+c+xb\rangle .}$

The discrete Heisenberg group ${\displaystyle \Gamma }$ is the discrete subgroup of ${\displaystyle N}$ whose elements are represented by the triples of integers. Considering ${\displaystyle \Gamma }$ acts on ${\displaystyle N}$ on the left, the quotient manifold ${\displaystyle \Gamma \backslash N}$ is called the Heisenberg manifold. The Heisenberg group acts on the Heisenberg manifold on the right. The Haar measure ${\displaystyle \mu =dx\wedge dy\wedge dt}$ on the Heisenberg group induces a right-translation-invariant measure on the Heisenberg manifold. The space of complex-valued square-integrable functions on the Heisenberg manifold has a right-translation-invariant orthogonal decomposition:

${\displaystyle L^{2}(\Gamma \backslash N)=\oplus _{n\in \mathbb {Z} }H_{n}}$

where

${\displaystyle H_{n}=\{f\in L^{2}(\Gamma \backslash N)\mid f(\Gamma \langle x,y,t+s\rangle )=\exp(2\pi ins)f(\Gamma \langle x,y,t\rangle )\}}$.

Definition

The Weil–Brezin map ${\displaystyle W:L^{2}(\mathbb {R} )\to H_{1}}$ is the unitary transformation given by

${\displaystyle W(\psi )(\Gamma \langle x,y,t\rangle )=\sum _{l\in \mathbb {Z} }\psi (x+l)e^{2\pi ily}e^{2\pi it}}$

for every Schwartz function ${\displaystyle \psi }$, where convergence is pointwise.

The inverse of the Weil–Brezin map ${\displaystyle W^{-1}:H_{1}\to L^{2}(\mathbb {R} )}$ is given by

${\displaystyle (W^{-1}f)(x)=\int _{0}^{1}f(\Gamma \langle x,y,0\rangle )dy}$

for every smooth function ${\displaystyle f}$ on the Heisenberg manifold that is in ${\displaystyle H_{1}}$.

Fundamental unitary representation of the Heisenberg group

For each real number ${\displaystyle \lambda \neq 0}$, the fundamental unitary representation ${\displaystyle U_{\lambda }}$ of the Heisenberg group is an irreducible unitary representation of ${\displaystyle N}$ on ${\displaystyle L^{2}(\mathbb {R} )}$ defined by

${\displaystyle (U_{\lambda }(\langle a,b,c\rangle )\psi )(x)=e^{2\pi i\lambda (c+bx)}\psi (x+a)}$.

By Stone–von Neumann theorem, this is the unique irreducible representation up to unitary equivalence satisfying the canonical commutation relation

${\displaystyle U_{\lambda }(\langle a,0,0\rangle )U_{\lambda }(\langle 0,b,0\rangle )=e^{2\pi i\lambda ab}U_{\lambda }(\langle 0,b,0\rangle )U_{\lambda }(\langle a,0,0\rangle )}$.

The fundamental representation ${\displaystyle U=U_{1}}$ of ${\displaystyle N}$ on ${\displaystyle L^{2}(\mathbb {R} )}$ and the right translation ${\displaystyle R}$ of ${\displaystyle N}$ on ${\displaystyle H_{1}\subset L^{2}(\Gamma \backslash N)}$ are intertwined by the Weil–Brezin map

${\displaystyle WU(\langle a,b,c\rangle )=R(\langle a,b,c\rangle )W}$.

In other words, the fundamental representation ${\displaystyle U}$ on ${\displaystyle L^{2}(\mathbb {R} )}$ is unitarily equivalent to the right translation ${\displaystyle R}$ on ${\displaystyle H_{1}}$ through the Weil-Brezin map.

Relation to Fourier transform

Let ${\displaystyle J:N\to N}$ be the automorphism on the Heisenberg group given by

${\displaystyle J(\langle x,y,t\rangle )=\langle y,-x,t-xy\rangle }$.

It naturally induces a unitary operator ${\displaystyle J^{*}:H_{1}\to H_{1}}$, then the Fourier transform

${\displaystyle {\mathcal {F}}=W^{-1}J^{*}W}$

as a unitary operator on ${\displaystyle L^{2}(\mathbb {R} )}$.

Plancherel theorem

The norm-preserving property of ${\displaystyle W}$ and ${\displaystyle J^{*}}$, which is easily seen, yields the norm-preserving property of the Fourier transform, which is referred to as the Plancherel theorem.

Poisson summation formula

For any Schwartz function ${\displaystyle \psi }$,

${\displaystyle \sum _{l}\psi (l)=W(\psi )(\Gamma \langle 0,0,0)\rangle )=(J^{*}W(\psi ))(\Gamma \langle 0,0,0)\rangle )=W({\hat {\psi }})(\Gamma \langle 0,0,0)\rangle )=\sum _{l}{\hat {\psi }}(l)}$.

This is just the Poisson summation formula.

Relation to the finite Fourier transform

For each ${\displaystyle n\neq 0}$, the subspace ${\displaystyle H_{n}\subset L^{2}(\Gamma \backslash N)}$ can further be decomposed into right-translation-invariant orthogonal subspaces

${\displaystyle H_{n}=\oplus _{m=0}^{|n|-1}H_{n,m}}$

where

${\displaystyle H_{n,m}=\{f\in H_{n}\mid f(\Gamma \langle x,y+{1 \over n},t\rangle )=e^{2\pi im/n}f(\Gamma \langle x,y,t\rangle )\}}$.

The left translation ${\displaystyle L(\langle 0,1/n,0\rangle )}$ is well-defined on ${\displaystyle H_{n}}$, and ${\displaystyle H_{n,0},...,H_{n,|n|-1}}$ are its eigenspaces.

The left translation ${\displaystyle L(\langle m/n,0,0\rangle )}$ is well-defined on ${\displaystyle H_{n}}$, and the map

${\displaystyle L(\langle m/n,0,0\rangle ):H_{n,0}\to H_{n,m}}$

is a unitary transformation.

For each ${\displaystyle n\neq 0}$, and ${\displaystyle m=0,...,|n|-1}$, define the map ${\displaystyle W_{n,m}:L^{2}(\mathbb {R} )\to H_{n,m}}$ by

${\displaystyle W_{n,m}(\psi )(\Gamma \langle x,y,t\rangle )=\sum _{l\in \mathbb {Z} }\psi (x+l+{m \over n})e^{2\pi i(nl+m)y}e^{2\pi int}}$

for every Schwartz function ${\displaystyle \psi }$, where convergence is pointwise.

${\displaystyle W_{n,m}=L(\langle m/n,0,0\rangle )\circ W_{n,0}.}$

The inverse map ${\displaystyle W_{n,m}^{-1}:H_{n,m}\to L^{2}(\mathbb {R} )}$ is given by

${\displaystyle (W_{n,m}^{-1}f)(x)=\int _{0}^{1}e^{-2\pi imy}f(\Gamma \langle x-{m \over n},y,0\rangle )dy}$

for every smooth function ${\displaystyle f}$ on the Heisenberg manifold that is in ${\displaystyle H_{n,m}}$.

Similarly, the fundamental unitary representation ${\displaystyle U_{n}}$ of the Heisenberg group is unitarily equivalent to the right translation on ${\displaystyle H_{n,m}}$ through ${\displaystyle W_{n,m}}$:

${\displaystyle W_{n,m}U_{n}(\langle a,b,c\rangle )=R(\langle a,b,c\rangle )W_{n,m}}$.

For any ${\displaystyle m,m'}$,

${\displaystyle (W_{n,m'}^{-1}J^{*}W_{n,m}\psi )(x)=e^{2\pi im'm/n}{\hat {\psi }}(nx)}$.

For each ${\displaystyle n>0}$, let ${\displaystyle \phi _{n}(x)=(2n)^{1/4}e^{-\pi nx^{2}}}$. Consider the finite dimensional subspace ${\displaystyle K_{n}}$ of ${\displaystyle H_{n}}$ generated by ${\displaystyle \{{\boldsymbol {e}}_{n,0},...,{\boldsymbol {e}}_{n,n-1}\}}$ where

${\displaystyle {\boldsymbol {e}}_{n,m}=W_{n,m}(\phi _{n})\in H_{n,m}.}$

Then the left translations ${\displaystyle L(\langle 1/n,0,0\rangle )}$ and ${\displaystyle L(\langle 0,1/n,0\rangle )}$ act on ${\displaystyle K_{n}}$ and give rise to the irreducible representation of the finite Heisenberg group. The map ${\displaystyle J^{*}}$ acts on ${\displaystyle K_{n}}$ and gives rise to the finite Fourier transform

${\displaystyle J^{*}{\boldsymbol {e}}_{n,m}={1 \over {\sqrt {n}}}\sum _{m'}e^{2\pi im'm/n}{\boldsymbol {e}}_{n,m'}.}$

Nil-theta functions

Nil-theta functions are functions on the Heisenberg manifold that are analogous to the theta functions on the complex plane. The image of Gaussian functions under the Weil–Brezin Map are nil-theta functions. There is a model of the finite Fourier transform defined with nil-theta functions, and the nice property of the model is that the finite Fourier transform is compatible with the algebra structure of the space of nil-theta functions.

Definition of nil-theta functions

Let ${\displaystyle {\mathfrak {n}}}$ be the complexified Lie algebra of the Heisenberg group ${\displaystyle N}$. A basis of ${\displaystyle {\mathfrak {n}}}$ is given by the left-invariant vector fields ${\displaystyle X,Y,T}$ on ${\displaystyle N}$:

${\displaystyle X(x,y,t)={\partial \over \partial x},}$

${\displaystyle Y(x,y,t)={\partial \over \partial y}+x{\partial \over \partial t},}$

${\displaystyle T(x,y,t)={\partial \over \partial t}.}$

These vector fields are well-defined on the Heisenberg manifold ${\displaystyle \Gamma \backslash N}$.

Introduce the notation ${\displaystyle V_{-i}=X-iY}$. For each ${\displaystyle n>0}$, the vector field ${\displaystyle V_{-i}}$ on the Heisenberg manifold can be thought of as a differential operator on ${\displaystyle C^{\infty }(\Gamma \backslash N)\cap H_{n,m}}$ with the kernel generated by ${\displaystyle {\boldsymbol {e}}_{n,m}}$.

We call

${\displaystyle \ker(V_{-i}:C^{\infty }(\Gamma \backslash N)\cap H_{n}\to H_{n})=\left\{{\begin{array}{lr}K_{n},&n>0\\\mathbb {C} ,&n=0\end{array}}\right.}$

the space of nil-theta functions of degree ${\displaystyle n}$.

Algebra structure of nil-theta functions

The nil-theta functions with pointwise multiplication on ${\displaystyle \Gamma \backslash N}$ form a graded algebra ${\displaystyle \oplus _{n\geq 0}K_{n}}$ (here ${\displaystyle K_{0}=\mathbb {C} }$).

Auslander and Tolimieri showed that this graded algebra is isomorphic to

${\displaystyle \mathbb {C} /(x_{3}^{6}+x_{1}^{4}x_{2}^{2}+x_{2}^{6})}$,

and that the finite Fourier transform (see the preceding section #Relation to the finite Fourier transform) is an automorphism of the graded algebra.

Relation to Jacobi theta functions

Let ${\displaystyle \vartheta (z;\tau )=\sum _{l=-\infty }^{\infty }\exp(\pi il^{2}\tau +2\pi ilz)}$ be the Jacobi theta function. Then

${\displaystyle \vartheta (n(x+iy);ni)=(2n)^{-1/4}e^{\pi ny^{2}}{\boldsymbol {e}}_{n,0}(\Gamma \langle y,x,0\rangle )}$.

Higher order theta functions with characteristics

An entire function ${\displaystyle f}$ on ${\displaystyle \mathbb {C} }$ is called a theta function of order ${\displaystyle n}$, period ${\displaystyle \tau }$ (${\displaystyle \mathrm {Im} (\tau )>0}$) and characteristic ${\displaystyle }$ if it satisfies the following equations:

1. ${\displaystyle f(z+1)=\exp(\pi ia)f(z)}$,
2. ${\displaystyle f(z+\tau )=\exp(\pi ib)\exp(-\pi in(2z+\tau ))f(z)}$.

The space of theta functions of order ${\displaystyle n}$, period ${\displaystyle \tau }$ and characteristic ${\displaystyle }$ is denoted by ${\displaystyle \Theta _{n}(\tau ,A)}$.

${\displaystyle \dim \Theta _{n}(\tau ,A)=n}$.

A basis of ${\displaystyle \Theta _{n}(i,A)}$ is

${\displaystyle \theta _{n,m}(z)=\sum _{l\in \mathbb {Z} }\exp}$.

These higher order theta functions are related to the nil-theta functions by

${\displaystyle \theta _{n,m}(x+iy)=(2n)^{-1/4}e^{\pi ny^{2}}{\boldsymbol {e}}_{n,m}(\Gamma \langle y,x,0\rangle )}$.

See also

• Nilmanifold
• Nilpotent group
• Nilpotent Lie algebra
• Weil representation
• Theta representation
• Oscillator representation

References

1. Weil, André. "Sur certains groupes d'opérateurs unitaires." Acta mathematica 111.1 (1964): 143-211.
2. Brezin, Jonathan. "Harmonic analysis on nilmanifolds." Transactions of the American Mathematical Society 150.2 (1970): 611-618.
3. Auslander, Louis, and Richard Tolimieri. Abelian harmonic analysis, theta functions and function algebras on a nilmanifold. Springer, 1975.
4. Auslander, Louis. "Lecture notes on nil-theta functions." Conference Board of the Mathematical Sciences, 1977.
5. Zhang, D. "Integer Linear Canonical Transforms, Their Discretization, and Poisson Summation Formulae"
6. "Zak Transform".
7. Auslander, L., and R. Tolimieri. "Algebraic structures for⨁Σ _ {𝑛≥ 1} 𝐿 (𝑍/𝑛) compatible with the finite Fourier transform." Transactions of the American Mathematical Society 244 (1978): 263-272.

• Harmonic analysis
• Representation theory