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
[edit]
The (continuous) Heisenberg group
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 .}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CoNrAoDC0zNi3zqe4yje3ytvFaAo1ygiNzDiNoDdEnDhByqvCytdB)
The discrete Heisenberg group
is the discrete subgroup of
whose elements are represented by the triples of integers. Considering
acts on
on the left, the quotient manifold
is called the Heisenberg manifold.
The Heisenberg group acts on the Heisenberg manifold on the right. The Haar measure
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}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84ntw2njePoqa4aAi3aDC5ygo5nDoOyqw5nji2njlFzgs1ztK2oDBB)
where
.
The Weil–Brezin map
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}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Bnqi5z2a4oNoNoNoQoNhBytvAaqoNntwOaAvDzts4zDCQnqrDaqoQ)
for every Schwartz function
, where convergence is pointwise.
The inverse of the Weil–Brezin map
is given by
![{\displaystyle (W^{-1}f)(x)=\int _{0}^{1}f(\Gamma \langle x,y,0\rangle )dy}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Eaqa2aNmOzqs5yqrDzqzFyqoNaNw4ati2ytlAzqzBzqsQajiPyjlC)
for every smooth function
on the Heisenberg manifold that is in
.
Fundamental unitary representation of the Heisenberg group
[edit]
For each real number
, the fundamental unitary representation
of the Heisenberg group is an irreducible unitary representation of
on
defined by
.
By Stone–von Neumann theorem, this is the unique irreducible representation up to unitary equivalence satisfying the canonical commutation relation
.
The fundamental representation
of
on
and the right translation
of
on
are intertwined by the Weil–Brezin map
.
In other words, the fundamental representation
on
is unitarily equivalent to the right translation
on
through the Wei-Brezin map.
Let
be the automorphism on the Heisenberg group given by
.
It naturally induces a unitary operator
, then the Fourier transform
![{\displaystyle {\mathcal {F}}=W^{-1}J^{*}W}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zDa0ntFFatKOatlAatdDajJAngaQajG1yjFBaAhCnjaNzqwPz2sP)
as a unitary operator on
.
The norm-preserving property of
and
, which is easily seen, yields the norm-preserving property of the Fourier transform, which is referred to as the Plancherel theorem.
For any Schwartz function
,
.
This is just the Poisson summation formula.
For each
, the subspace
can further be decomposed into right-translation-invariant orthogonal subspaces
![{\displaystyle H_{n}=\oplus _{m=0}^{|n|-1}H_{n,m}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CoAoNzjzAz2a1oNG0zghBatw3zqs0zge2agwQzqsNntvAyqiOyjs5)
where
.
The left translation
is well-defined on
, and
are its eigenspaces.
The left translation
is well-defined on
, and the map
![{\displaystyle L(\langle m/n,0,0\rangle ):H_{n,0}\to H_{n,m}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Aaqa4ztGQoAsPata2oAvCoNFCo2e3otBDoNm2aNa1yjdFoqe1nAnD)
is a unitary transformation.
For each
, and
, define the map
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}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AnAwNntlBythAzNeQngiQaAvEzjGPoqe4otFEztG1aAaPzgs5ngo3)
for every Schwartz function
, where convergence is pointwise.
![{\displaystyle W_{n,m}=L(\langle m/n,0,0\rangle )\circ W_{n,0}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85ygs2yjoPzjFCnti1zthBnji1yjm2yqaQztoOo2vFotdAotw3nAa1)
The inverse map
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}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CnthFzNG1njiOzNoPoDFFa2dAzjGOoAnBntzDajCPyjs1zjhFaqi3)
for every smooth function
on the Heisenberg manifold that is in
.
Similarly, the fundamental unitary representation
of the Heisenberg group is unitarily equivalent to the right translation on
through
:
.
For any
,
.
For each
, let
. Consider the finite dimensional subspace
of
generated by
where
![{\displaystyle {\boldsymbol {e}}_{n,m}=W_{n,m}(\phi _{n})\in H_{n,m}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85ajwPyqnFnAvAnqoQzgo1zNo5yjdFaNiPyjK5oAvDngrBzDBDoAe0)
Then the left translations
and
act on
and give rise to the irreducible representation of the finite Heisenberg group. The map
acts on
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'}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CoNw5zteOo2eOzDrCa2a5zNe0otw0yga3zgw0aqs2ajwQo2wPnqdB)
Nil-theta functions
[edit]
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
[edit]
Let
be the complexified Lie algebra of the Heisenberg group
. A basis of
is given by the left-invariant vector fields
on
:
![{\displaystyle X(x,y,t)={\partial \over \partial x},}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DntwPytm2nqrFzjlFo2iOzDi1ngaPatnAyjsPyjhCnAo5ytvAztJD)
![{\displaystyle Y(x,y,t)={\partial \over \partial y}+x{\partial \over \partial t},}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Cata4oAzDzNo2agvCyjlFzNiPaDi1aNs0ygoQatK1ygzCatvBnjw5)
![{\displaystyle T(x,y,t)={\partial \over \partial t}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EztlDngwNathEygw5ztK1aDG2aqsPnAvAzqiQyja3njiNoAnDoDBC)
These vector fields are well-defined on the Heisenberg manifold
.
Introduce the notation
. For each
, the vector field
on the Heisenberg manifold can be thought of as a differential operator on
with the kernel generated by
.
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.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QaDoOyqoNnjdAoAzAyto0zjG2otzAzqs5njoQoNe2aNK1nArCzDeO)
the space of nil-theta functions of degree
.
Algebra structure of nil-theta functions
[edit]
The nil-theta functions with pointwise multiplication on
form a graded algebra
(here
).
Auslander and Tolimieri showed that this graded algebra is isomorphic to
,
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
[edit]
Let
be the Jacobi theta function. Then
.
Higher order theta functions with characteristics
[edit]
An entire function
on
is called a theta function of order
, period
(
) and characteristic
if it satisfies the following equations:
,
.
The space of theta functions of order
, period
and characteristic
is denoted by
.
.
A basis of
is
.
These higher order theta functions are related to the nil-theta functions by
.
- ^ 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"
- ^ "Zak Transform".
- ^ Auslander, L., and R. Tolimieri. "Algebraic structures for⨁Σ _ {𝑛≥ 1} 𝐿2 (𝑍/𝑛) compatible with the finite Fourier transform." Transactions of the American Mathematical Society 244 (1978): 263-272.