Abstract
We examine the affine Wigner distribution from a quantization perspective with an emphasis on the underlying group structure. One of our main results expresses the scalogram as (affine) convolution of affine Wigner distributions. We strive to unite the literature on affine Wigner distributions and we provide the connection to the Mellin transform in a rigorous manner. Moreover, we present an affine ambiguity function and show how this can be used to illuminate properties of the affine Wigner distribution. In contrast with the usual Wigner distribution, we demonstrate that the affine Wigner distribution is never an analytic function. Our approach naturally leads to several applications, one of which is an approximation problem for the affine Wigner distribution. We show that the deviation for a symbol to be an affine Wigner distribution can be expressed purely in terms of intrinsic operator-related properties of the symbol. Finally, we present a positivity conjecture regarding the non-negativity of the affine Wigner distribution.
Highlights
The most studied quadratic time-frequency representation is the Wigner distribution defined by Wf (x, ω) := f Rd x + t 2 f x − t 2 e−2πiωt dt, (x, ω) ∈ R2d. (1.1)Originally invented by Wigner in [24] almost a century ago, the Wigner distribution is essential in quantum mechanics as it gives the expectation values for Weyl quantization of symbols [8]
Invented by Wigner in [24] almost a century ago, the Wigner distribution is essential in quantum mechanics as it gives the expectation values for Weyl quantization of symbols [8]
The authors showed that the affine Wigner distribution satisfies WAψff ∈ L2r(Aff) for every ψ ∈ L2(R+, a−1 da), where L2r(Aff) denotes all measurable functions on the upper half-plane R × R+ that are square integrable with respect to the measure a−1 da dx
Summary
The most studied quadratic time-frequency representation is the Wigner distribution defined by. The authors showed that the affine Wigner distribution satisfies WAψff ∈ L2r(Aff) for every ψ ∈ L2(R+, a−1 da), where L2r(Aff) denotes all measurable functions on the upper half-plane R × R+ that are square integrable with respect to the measure a−1 da dx. The first significant contribution is to develop a connection between the affine Wigner distribution and the scalogram defined by SCALgf (x, a) := |Wgf (x, a)|2, (x, a) ∈ R × R+, where Wgf denotes the continuous wavelet transform of f with respect to g defined precisely in (2.8). We will show that the affine Wigner distribution and the affine ambiguity function are related through the Mellin transform by. The authors are grateful for helpful suggestions from Eirik Skrettingland and Luıs Daniel Abreu
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