Weil–Brezin Map Explained

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

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) = ⊕ _{n} H _{n} where H _{n} = \{f\in L ^{2(\Gamma\backslash} N) \mid f(\Gamma \langle x, y, t + s\rangle) = \exp( 2 \pi i ns) f(\Gamma \langle x,y,t\rangle)\} .

Definition

The Weil–Brezin map

W: L ^{2(R)\to} H _{1} is the unitary transformation given by W(\psi) (\Gamma \langle x, y, t \rangle) = \sum _{l\in} \psi(x + l) e ^{2} e ^{2\pi} for every Schwartz function \psi , where convergence is pointwise.

The inverse of the Weil–Brezin map

W ^{-1} : H _{1} \to L ^{2(R)} is given by(W ^{-1} f) (x) = \int ^{1} _{0} 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

λ \ne 0 , the fundamental unitary representation U _{λ} of the Heisenberg group is an irreducible unitary representation of N on L ^{2(R)} defined by(U _{λ} (\langle a, b, c \rangle) \psi) (x) = e ^{2} \psi(x + a) . By Stone–von Neumann theorem, this is the unique irreducible representation up to unitary equivalence satisfying the canonical commutation relationU _{λ} (\langle a, 0 , 0 \rangle) U _{λ} (\langle 0 , b, 0 \rangle) = e ^{2\pi} U _{λ} (\langle 0 , b, 0 \rangle) U _{λ} (\langle a, 0 , 0 \rangle) .The fundamental representation U = U _{1} of N on L ^{2(R)} and the right translation R of N on H _{1} \subset L ^{2(\Gamma} \backslash N) are intertwined by the Weil–Brezin mapW U(\langle a, b, c \rangle) = R(\langle a, b, c \rangle) W .In other words, the fundamental representation U on L ^{2(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 byl F = W ^{-1} J ^{*} W as a unitary operator on L ^{2(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\ne 0 , the subspace H _{n\subset} L ^{2(\Gamma\backslash} N) can further be decomposed into right-translation-invariant orthogonal subspacesH _{n} = ⊕ ^{|n| -1} _{m =0} 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} 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,} are its eigenspaces.

The left translation

L(\langle m/n, 0 , 0 \rangle) is well-defined on H _{n} , and the mapL(\langle m/n, 0 , 0 \rangle) : H _{n,0} \to H _{n,m} is a unitary transformation.

For each

n\ne 0 , and m = 0 , ..., |n| -1 , define the map W _{n,} : L ^{2(R)\to} H _{n,} by W _{n,m} (\psi) (\Gamma \langle x, y, t \rangle) = \sum _{l\in} \psi(x + l + {m \over n}) e ^{2} e ^{2\pi} 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 ^{-1} _{n,m} : H _{n,m} \to L ^{2(R)} is given by (W ^{-1} _{n,m} f) (x) = \int ^{1} _{0} e ^{-2\pi} 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 ^{-1} _{n,m'} J ^{*} W _{n,m} \psi ) (x) = e ^{2\pi} \hat{\psi}(nx) .

For each

n> 0 , let \phi _{n(x)} = ( 2 n) ^{1/4} e ^{- \pi n x ^{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 transformJ ^{*} \boldsymbol{e} _{n,m} = { 1 \over \sqrt{n}}\sum _{m'} e ^{2\pi} \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

ak{n} be the complexified Lie algebra of the Heisenberg group N . A basis of ak{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 - i Y . 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 \ 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 ⊕ _{n\ge} K _{n} (here K _{0} = C ).

Auslander and Tolimieri showed that this graded algebra is isomorphic to

C [x _{1,} x ^{2 ,} _{2} x ^{6} _{3} + x ^{4} _{1} x ^{2} _{2} + x ^{6 )} _{2} ,and that the finite Fourier transform (see the preceding section

1. Relation to the finite Fourier transform

) is an automorphism of the graded algebra.

Relation to Jacobi theta functions

Let

\vartheta(z; \tau) = \sum ^{inf ty} _{l =-inf ty} \exp (\pi i l ^{2} \tau + 2 \pi i l z) be the Jacobi theta function. Then\vartheta(n(x + iy); ni) = ( 2 n) ^{-1/4} e ^{\pi n y ^{2}} \boldsymbol{e} _{n,0} (\Gamma\langle y, x, 0 \rangle) .

Higher order theta functions with characteristics

f on C is called a theta function of order n , period \tau (Im (\tau)> 0 ) and characteristic [ ^{a} _{b]} if it satisfies the following equations:f(z +1 ) = \exp(\pi i a ) f(z) ,f(z + \tau) = \exp(\pi i b) \exp( - \pi i n ( 2 z + \tau)) f(z) .The space of theta functions of order n , period \tau and characteristic [ ^{a} _{b]} is denoted by \Theta ^{a} _{b](\tau,} A) . \dim \Theta ^{a} _{b](\tau,} A) = n . A basis of \Theta ^{0} _{0 ](i,} A) is\theta _{n,m} (z) = \sum _{l\in} \exp [ - \pi n (l + {m \over n} ) ^{2} + 2 \pi i (l n + m) z ) ] .These higher order theta functions are related to the nil-theta functions by\theta _{n,m} (x + iy) = ( 2 n) ^{-1/4} e ^{\pi n y ^{2}} \boldsymbol{e} _{n,m} (\Gamma \langle y, x, 0 \rangle ) .

See also

• Nilmanifold
• Nilpotent group
• Nilpotent Lie algebra
• Weil representation
• Theta representation
• Oscillator representation

Notes and References

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. Web site: Zak Transform.
7. Auslander, L., and R. Tolimieri. "Algebraic structures for⨁Σ _ ² (/) compatible with the finite Fourier transform." Transactions of the American Mathematical Society 244 (1978): 263-272.