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 N {\displaystyle N} is the 3-dimensional Lie group that can be represented by triples of real numbers with multiplication rule
⟨ x , y , t ⟩ ⟨ a , b , c ⟩ = ⟨ x + a , y + b , t + c + x b ⟩ . {\displaystyle \langle x,y,t\rangle \langle a,b,c\rangle =\langle x+a,y+b,t+c+xb\rangle .}
The discrete Heisenberg group Γ {\displaystyle \Gamma } is the discrete subgroup of N {\displaystyle N} whose elements are represented by the triples of integers. Considering Γ {\displaystyle \Gamma } acts on N {\displaystyle N} on the left, the quotient manifold Γ ∖ N {\displaystyle \Gamma \backslash N} is called the Heisenberg manifold. The Heisenberg group acts on the Heisenberg manifold on the right. The Haar measure μ = d x ∧ d y ∧ d t {\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:
L 2 ( Γ ∖ N ) = ⊕ n ∈ Z H n {\displaystyle L^{2}(\Gamma \backslash N)=\oplus _{n\in \mathbb {Z} }H_{n}}
where
H n = { f ∈ L 2 ( Γ ∖ N ) ∣ f ( Γ ⟨ x , y , t + s ⟩ ) = exp ( 2 π i n s ) f ( Γ ⟨ x , y , t ⟩ ) } {\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 W : L 2 ( R ) → H 1 {\displaystyle W:L^{2}(\mathbb {R} )\to H_{1}} is the unitary transformation given by
W ( ψ ) ( Γ ⟨ x , y , t ⟩ ) = ∑ l ∈ Z ψ ( x + l ) e 2 π i l y e 2 π i t {\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 ψ {\displaystyle \psi } , where convergence is pointwise. The inverse of the Weil–Brezin map W − 1 : H 1 → L 2 ( R ) {\displaystyle W^{-1}:H_{1}\to L^{2}(\mathbb {R} )} is given by
( W − 1 f ) ( x ) = ∫ 0 1 f ( Γ ⟨ x , y , 0 ⟩ ) d y {\displaystyle (W^{-1}f)(x)=\int _{0}^{1}f(\Gamma \langle x,y,0\rangle )dy}
for every smooth function f {\displaystyle f} on the Heisenberg manifold that is in H 1 {\displaystyle H_{1}} .
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