Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions

Abstract

We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. We first prove that a Plancherel inversion formula, well known for Bruhat functions on the group, holds for a much larger class of functions. This result allows us to view the wavelet transform as essentially the inverse Plancherel transform. The wavelet transform of a signal is an L 2 -function on an appropriately chosen group while the Wigner function is defined on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps L 2 -functions on a group unitarily to fields of Hilbert-Schmidt operators, indexed by unitary irreducible representations of the group. The wavelet transform can essentially be looked upon as a restricted inverse Plancherel transform, while Wigner functions are modified Fourier transforms of inverse Plancherel transforms, usually restricted to a subset of the unitary dual of the group. Some known results on both Wigner functions and wavelet transforms, appearing in the literature from very different perspectives, are naturally unified within our approach. Explicit computations on a number of groups illustrate the theory.

Wigner functions for gauge equivalence classes of unitary irreducible representations of noncommutative quantum mechanics

“…In this section, we will be dealing with he construction of Wigner function associated with a 2-dimensional system of NCQM using the general method developed in [1]. The UIRs of G NC that we will be focussing on for this purpose are given by (3.1).…”

“…Now the Plancherel measure forĜ NC can be computed using a general orthogonality relation (see, for example, [1]) given by…”

“…It has been argued in [1,15] that for nilpotent Lie group things become more tractable in the sense that the underlying Wigner functions are decomposable, meaning that the Wigner function only picks up the contribution of the Hilbert space of Hilbert-Schmidt operators associated with the underlying coadjoint orbit for the measurable operator fields (ρ, σ, τ ) −→ A(ρ, σ, τ ) ∈ B 2 (ρ, σ, τ ), so that…”

Admissible vectors for the regular representation

“…Hence it is necessary to have a pointwise Fourier inversion formula. The following theorem is proven in [2]. It can be seen as a generalization of [ …”

Square-integrability modulo a subgroup

“…In particular, let G be the Poincaré group R 4 × SO (3,1), H = R 4 × SO(3) and σ the one-dimensional representation of H given by…”