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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.
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:
where
.
Definition[]
The Weil–Brezin map is the unitary transformation given by
for every Schwartz function , where convergence is pointwise.
The inverse of the Weil–Brezin map is given by
for every smooth function on the Heisenberg manifold that is in .
Fundamental unitary representation of the Heisenberg group[]
For each real number , the fundamental unitary representation of the Heisenberg group is an irreducibleunitary representation of on defined by
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.
Poisson summation formula[]
For any Schwartz function ,
.
This is just the Poisson summation formula.
Relation to the finite Fourier transform[]
For each , the subspace can further be decomposed into right-translation-invariant orthogonal subspaces
where
.
The left translation is well-defined on , and are its eigenspaces.
The left translation is well-defined on , and the map
is a unitary transformation.
For each , and , define the map by
for every Schwartz function , where convergence is pointwise.
The inverse map is given by
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
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
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 be the complexified Lie algebra of the Heisenberg group . A basis of is given by the left-invariant vector fields on :
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
the space of nil-theta functions of degree .
Algebra structure of nil-theta functions[]
The nil-theta functions with pointwise multiplication on form a graded algebra (here ).
Auslander and Tolimieri showed that this graded algebra is isomorphic to