Weil–Brezin Map
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 is the 3-dimensional Lie group that can be represented by triples of real numbers with multiplication rule 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 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. Relation to Fourier transform Let be the automorphism on the Heisenberg group given by . It naturally induces a unitary operator , then the Fourier transform as a unitary operator on . Plancherel theorem 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 .