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.
Contents
- Heisenberg manifold
- Definition
- Fundamental unitary representation of the Heisenberg group
- Relation to Fourier transform
- Plancherel theorem
- Poisson summation formula
- Relation to the finite Fourier transform
- Nil theta functions
- Definition of nil theta functions
- Algebra structure of nil theta functions
- Relation to Jacobi theta functions
- Higher order theta functions with characteristics
- References
Heisenberg manifold
The (continuous) Heisenberg group
The discrete Heisenberg group
where
Definition
The Weil–Brezin map
for every Schwartz function
The inverse of the Weil–Brezin map
for every smooth function
Fundamental unitary representation of the Heisenberg group
For each real number
By Stone-von Neumann theorem, this is the unique irreducible representation up to unitary equivalence satisfying the canonical commutation relation
The fundamental representation
In other words, the fundamental representation
Relation to Fourier transform
Let
It naturally induces a unitary operator
as a unitary operator on
Plancherel theorem
The norm-preserving property of
Poisson summation formula
For any Schwartz function
This is just the Poisson summation formula.
Relation to the finite Fourier transform
For each
where
The left translation
The left translation
is a unitary transformation.
For each
for every Schwartz function
The inverse map
for every smooth function
Similarly, the fundamental unitary representation
For any
For each
Then the left translations
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
These vector fields are well-defined on the Heisenberg manifold
Introduce the notation
We call
the space of nil-theta functions of degree
Algebra structure of nil-theta functions
The nil-theta functions with pointwise multiplication on
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
Let
Higher order theta functions with characteristics
An entire function
-
f ( z + 1 ) = exp ( π i a ) f ( z ) , -
f ( z + τ ) = exp ( π i b ) exp ( − π i n ( 2 z + τ ) ) f ( z ) .
The space of theta functions of order
A basis of
These higher order theta functions are related to the nil-theta functions by