The affine Wigner distribution

Affine Quantum Harmonic Analysis

“…We will exploit the previous theorem to define the affine Weyl quantization of tempered distributions in Section 3.3. To do this rigorously, we will utilize a Schwartz space S (Aff) on the affine group introduced in [5]. An important example we prove in Theorem 3.11 is the affine Weyl quantization of the coordinate functions: Theorem B.…”

“…We will later also need the space S (R + ) of bounded, anti-linear functionals on S (R + ) called the tempered distributions on R + . For more information regarding the affine Wigner distribution the reader is referred to [5].…”

“…Define the functions Ψ and Φ to be Ψ(x) := ψ(e x ) and Φ(x) := φ(e x ) for ψ, φ ∈ L 2 (R + ). A reformulation of [5,Lem. 6.4] shows that we have the relation…”

“…Notice that we used a straightforward generalization of the marginal property of the affine Wigner distribution given in (2.15), see the proof of [5,Prop. 3.4] for details.…”

“…Similarly, the relation between quantization schemes on the affine group and its representation theory has received some attention and several schemes have been proposed, e.g. [18,5,21]. However, there are still many open questions awaiting a definite answer in the case of the affine group.…”