Weil–Brezin Map

In mathematics, the Weil–Brezin map, named after André Weil[1] and Jonathan Brezin,[2] 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.[3][4][5] 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,[6] 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 $N$ is the 3-dimensional Lie group that can be represented by triples of real numbers with multiplication rule

\langle x,y,t\rangle \langle a,b,c\rangle =\langle x+a,y+b,t+c+xb\rangle .

The discrete Heisenberg group $\Gamma$ is the discrete subgroup of $N$ whose elements are represented by the triples of integers. Considering $\Gamma$ acts on $N$ on the left, the quotient manifold $\Gamma \backslash N$ is called the Heisenberg manifold. The Heisenberg group acts on the Heisenberg manifold on the right. The Haar measure $\mu =dx\wedge dy\wedge dt$ 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:

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

where

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 $W:L^{2}(\mathbb {R} )\to H_{1}$ is the unitary transformation given by

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 $\psi$ , where convergence is pointwise.

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

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

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

Fundamental unitary representation of the Heisenberg group

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

(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

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

WU(\langle a,b,c\rangle )=R(\langle a,b,c\rangle )W

.

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

Relation to Fourier transform

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

J(\langle x,y,t\rangle )=\langle y,-x,t-xy\rangle

.

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

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

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

Plancherel theorem

The norm-preserving property of $W$ and $J^{*}$ , 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 $\psi$ ,

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

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

where

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

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

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

is a unitary transformation.

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

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 $\psi$ , where convergence is pointwise.

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

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

(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 $f$ on the Heisenberg manifold that is in $H_{n,m}$ .

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

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

.

For any $m,m'$ ,

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

.

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

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

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

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

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

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

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

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

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

We call

\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 $n$ .

Algebra structure of nil-theta functions

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

Auslander and Tolimieri showed that this graded algebra is isomorphic to

\mathbb {C} [x_{1},x_{2}^{2},x_{3}^{3}]/(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 $\vartheta (z;\tau )=\sum _{l=-\infty }^{\infty }\exp(\pi il^{2}\tau +2\pi ilz)$ be the Jacobi theta function. Then

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

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

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

\dim \Theta _{n}[_{b}^{a}](\tau ,A)=n

.

A basis of $\Theta _{n}[_{0}^{0}](i,A)$ is

\theta _{n,m}(z)=\sum _{l\in \mathbb {Z} }\exp[-\pi n(l+{m \over n})^{2}+2\pi i(ln+m)z)]

.

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

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

.

References

Weil, André. "Sur certains groupes d'opérateurs unitaires." Acta mathematica 111.1 (1964): 143-211.
Brezin, Jonathan. "Harmonic analysis on nilmanifolds." Transactions of the American Mathematical Society 150.2 (1970): 611-618.
Auslander, Louis, and Richard Tolimieri. Abelian harmonic analysis, theta functions and function algebras on a nilmanifold. Springer, 1975.
Auslander, Louis. "Lecture notes on nil-theta functions." Conference Board of the Mathematical Sciences, 1977.
Zhang, D. "Integer Linear Canonical Transforms, Their Discretization, and Poisson Summation Formulae"
http://mathworld.wolfram.com/ZakTransform.html
Auslander, L., and R. Tolimieri. "Algebraic structures for⨁∑ _ {𝑛≥ 1} 𝐿² (𝑍/𝑛) compatible with the finite Fourier transform." Transactions of the American Mathematical Society 244 (1978): 263-272.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[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] ttp://mathworld.wolfram.com/ZakTransform.html

[7] Auslander, L., and R. Tolimieri. "Algebraic structures for⨁∑ _ {𝑛≥ 1} 𝐿² (𝑍/𝑛) compatible with the finite Fourier transform." Transactions of the American Mathematical Society 244 (1978): 263-272.