Affine Quantum Harmonic Analysis

We develop a quantum harmonic analysis framework for the affine group. This encapsulates several examples in the literature such as affine localization operators, covariant integral quantizations, and affine quadratic time-frequency representations. In the process, we develop a notion of admissibility for operators and extend well known results to the operator setting. A major theme of the paper is the interaction between operator convolutions, affine Weyl quantization, and admissibility.


Introduction
The affine group and the Heisenberg group play prominent roles in wavelet theory and Gabor analysis, respectively. As is well-known, the representation theory of the Heisenberg group is intrinsically linked to quantization on phase space R 2n . 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.
As has been shown by two of the authors in [36], the theory of quantum harmonic analysis on phase space introduced by Werner [46] provides a coherent framework for many aspects of quantization and Gabor analysis associated with the Heisenberg group. Based on this connection, advances in the understanding of time-frequency analysis have been made [37,38,39]. In this paper we aim to develop a variant of Werner's quantum harmonic analysis in [46] for time-scale analysis. This is based on unitary representations of the affine group in a similar way to the Schrödinger representation of the Heisenberg group being used in Werner's framework. We will refer to this theory on the affine group as affine quantum harmonic analysis.

Affine Operator Convolutions
In Werner's quantum harmonic analysis on phase space, a crucial component is extending convolutions to operators. Recall that the affine group Aff has the underlying set R × R + and group operation modeling composition of affine transformations. A key feature of this group is that the left Haar measure a −2 dx da and the right Haar measure a −1 dx da are not equal, making the group non-unimodular. Both measures play a role in affine quantum harmonic analysis, making the theory more involved than the case of the Heisenberg group. In addition to the standard function (right-)convolution on the affine group f * Aff g(x, a) := Aff f (y, b)g((x, a) · (y, b) −1 ) dy db b , • Let f ∈ L 1 r (Aff) := L 1 (Aff, a −1 dx da) and let S be a trace-class operator on L 2 (R + ). We define the convolution f Aff S between f and S to be the operator on L 2 (R + ) given by where U is the unitary representation of Aff on L 2 (R + ) given by U (x, a)ψ(r) := e 2πixr ψ(ar).
• Let S be a trace-class operator and let T be a bounded operator on L 2 (R + ). Then we define the convolution S Aff T between S and T to be the function on Aff given by The three convolutions are compatible in the following sense: Let f, g ∈ L 1 r (Aff) and denote by S a trace-class operator and by T a bounded operator, both on L 2 (R + ). Then

Interplay Between Affine Weyl Quantization and Convolutions
Integral to the theory in this paper is the affine Wigner distribution and the associated affine Weyl quantization. The affine (cross-)Wigner distribution W ψ,φ Aff of φ, ψ ∈ L 2 (R + ) is the function on Aff given by Although at first glance the definition (1.1) might look unnatural, it can be motivated through the representation theory of the affine group as illustrated in [3]. We will elaborate on this viewpoint in Section 5. One defines the affine Weyl quantization of f ∈ L 2 r (Aff) := L 2 (Aff, a −1 dx da) as the operator A f given by , for all φ, ψ ∈ L 2 (R + ).
We will explore the intimate relation between the convolutions and the affine Weyl quantization. The following theorem, being a combination of Proposition 3.6 and Proposition 3.7, highlights this relation.
Theorem A. Let f, g ∈ L 2 r (Aff), where g is additionally in L 1 r (Aff) and square integrable with respect to the left Haar measure. Then whereǧ(x, a) := g((x, a) −1 ).
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. Let f x (x, a) := x and f a (x, a) := a be the coordinate functions on Aff. The affine Weyl quantizations A fx and A fa satisfy the commutation relation This is, up to re-normalization, precisely the infinitesimal structure of the affine group.
We define affine parity operator P Aff as where δ (0,1) denotes the Dirac distribution at the identity element (0, 1) ∈ Aff. The following result, which will be rigorously stated in Section 3.5, builds on these definitions.
Theorem C. The affine Weyl quantization A g of g ∈ S (Aff) can be written as Moreover, for φ, ψ such that φ(e x ), ψ(e x ) ∈ S (R), the affine Weyl symbol W ψ,φ Aff of the rankone operator ψ ⊗ φ can be written as

Operator Admissibility
One of the key features of representations of non-unimodular groups is the concept of admissibility. Recall that the Duflo-Moore operator D −1 corresponding to the representation U is the densely defined positive operator on L 2 (R + ) given by D −1 ψ(r) = r −1/2 ψ(r). We will often use that D −1 has a densely defined inverse given by Dψ(r) = r 1/2 ψ(r). A function ψ is said to be an admissible wavelet if ψ ∈ dom(D −1 ). It is well known [13] that admissible wavelets satisfy the orthogonality relation We extend the definition of admissibility to operators as follows: Definition. Let S be a non-zero bounded operator on L 2 (R + ) that maps dom(D) into dom(D −1 ). We say that S is admissible if the composition D −1 SD −1 is bounded on dom(D −1 ) and extends to a trace-class operator D −1 SD −1 on L 2 (R + ).
Note that the rank-one operator S = ψ ⊗ ψ for ψ ∈ L 2 (R + ) is admissible precisely when ψ is an admissible wavelet. In Section 4.2 we show that a large class of admissible operators can be constructed from Laguerre bases. The following result, which we prove in Corollary 4.5, is motivated by [46,Lemma 3.1] and extends (1.2) to the operator setting.
Theorem D. Let S be an admissible operator on L 2 (R + ). For any trace-class operator T on L 2 (R + ), we have that T Aff S ∈ L 1 r (Aff) with Aff T Aff S(x, a) dx da a = tr(T ) tr(D −1 SD −1 ).
Determining whether an operator is admissible or not can be a daunting task. We managed in Corollary 4.9 to find an elegant characterization in terms of operator convolutions of admissible operators that are additionally positive trace-class operators.
Theorem E. Let S be a non-zero, positive trace-class operator. Then S is admissible if and only if S Aff S ∈ L 1 r (Aff).
The following result is derived in Section 4.4 and uses the affine Weyl quantization to show that admissibility is an operator manifestation of the non-unimodularity of the affine group.
Theorem F.
• Let f ∈ L 1 r (Aff) be such that A f is a trace-class operator on L 2 (R + ). Then • Let g ∈ L 1 l (Aff) := L 1 (Aff, a −2 dx da) be such that A g is an admissible Hilbert-Schmidt operator. Then dx da a 2 .

Relationship with Fourier Transforms
For completeness, we will also investigate how notions of Fourier transforms on the affine group fit into the theory, and use known results from abstract harmonic analysis to explore the relationship between affine Weyl quantization and affine Fourier transforms. Recall that the integrated representation U (f ) of f ∈ L 1 l (Aff) is the operator on L 2 (R + ) given by U (f )ψ := Aff f (x, a)U (x, a)ψ dx da a 2 , ψ ∈ L 2 (R + ).
We define the following operator Fourier transform in the affine setting.
Definition. The affine Fourier-Wigner transform is the isometry F W sending a Hilbert-Schmidt operator on L 2 (R + ) to a function in L 2 r (Aff) such that The following result is proved in Proposition 5.7 and provides a connection between the affine Fourier-Wigner transform and admissibility.
Theorem G. Let A be a trace-class operator on L 2 (R + ). The following are equivalent: 2) AD −1 extends from dom(D −1 ) to a Hilbert-Schmidt operator on L 2 (R + ).

3) A * A is admissible.
Another Fourier transform of interest is the (modified) Fourier-Kirillov transform on the affine group F KO given by As in quantum harmonic analysis on phase space, we have that the affine Weyl quantization is the composition of these Fourier transforms, see Proposition 5.8. In the affine setting we have in general that This contrasts the analogous result in Werner's original quantum harmonic analysis, see (5.6).
In spite of this, not all properties typically associated with the Fourier transform are lost: In Section 5.2 we prove a quantum Bochner theorem in the affine setting.

Main Applications
In Section 6 we show that affine quantum harmonic analysis provides a conceptual framework for the study of covariant integral quantizations and a version of the Cohen class for the affine group. In addition, we show in Section 6.1 that if S is a rank-one operator, then the study of operators f Aff S for functions f on Aff reduces to the study of time-scale localization operators [12]. We have seen that affine Weyl quantization is given by f → f Aff P Aff for f ∈ S (Aff). Inspired by this, we consider a whole class of quantization procedures: For any suitably nice operator S on L 2 (R + ) we define a quantization procedure Γ S for functions f on Aff by This class of quantization procedures coincides with the covariant integral quantizations studied by Gazeau and his collaborators motivated by applications in physics, see e.g. [21,20,19]. Our results on affine quantum harmonic analysis are therefore also results on covariant integral quantizations. In particular, the abstract notion of admissibility of an operator S implies that Γ S satisfies the simple property where c is some constant, I L 2 (R + ) is the identity operator on L 2 (R + ), and 1(x, a) = 1 for all (x, a) ∈ Aff.
As the name suggests, covariant integral quantizations Γ S satisfy a covariance property, namely where R denotes right translations of functions on Aff. In Theorem 6.5 we point out that, by a known result on covariant positive operator valued measures [34,9], this covariance assumption together with other mild assumptions completely characterize the covariant integral quantizations. We have also seen that the affine cross-Wigner distribution is given for sufficiently nice ψ, φ by W ψ,φ Aff = (ψ ⊗ φ) Aff P Aff . Inspired by this and the description in [37] of the Cohen class of time-frequency distributions on R 2n , we make the following definition.
We will show how properties of S (such as admissibility) influence properties of Q S , and obtain an abstract characterization of the affine Cohen class. Readers familiar with the Cohen class on R 2n [11] will know that it is defined in terms of convolutions with the Wigner function. In the affine setting, we have the analogous result As we explain in Proposition 6.14, the affine class of quadratic time-frequency representations from [41] may be identified with a subclass of the affine Cohen class.

Structure of the Paper
In Section 2 we recall necessary background material for completeness. In particular, Section 2.2 should serve as a brief reference for quantum harmonic analysis on phase space. We define affine operator convolution in Section 3.1 and show the relationship with the affine Weyl quantization in Section 3.2. The affine parity operator will be introduced in Section 3.4, and its relationship to affine Weyl quantization will be explored in Section 3.5. We have dedicated the entirety of Section 4 to operator admissibility. Section 5 discusses affine Weyl quantization from the viewpoint of representation theory. In particular, in Section 5.2 we derive a Bochner type theorem for our setting. In Section 6.1 and Section 6.2 we relate our work to time-scale localization operators and covariant integral quantizations, respectively. Finally, in Section 6.3 we define the affine Cohen class and derive some basic properties.

Preliminaries
Notation: Given a Hilbert space H we let L(H) denote the bounded operators on H. The notation S p (H) for 1 ≤ p < ∞ will be used for the Schatten-p class operators on H. We remark that S 1 (H)and S 2 (H) are respectively the trace-class operators and the Hilbert-Schmidt operators on H. The space S ∞ (H) is by definition L(H) for duality reasons. When the Hilbert space in question is H = L 2 (R + ) := L 2 (R + , r −1 dr), we will simplify the notation to S p := S p (L 2 (R + )) for readability. We will denote by S (R n ) the space of Schwartz functions on R n . For a function f on a group G, the functionf is defined byf (g) = f (g −1 ) for all g ∈ G.

Basic Constructions on the Affine Group
We begin by giving a brief introduction to the affine group and relevant constructions on it. The (reduced) affine group (Aff, · Aff ) is the Lie group whose underlying set is the upper half plane Aff := R × R + := R × (0, ∞), while the group operation is given by (x, a) · Aff (y, b) := (ay + x, ab), (x, a), (y, b) ∈ Aff.
We will often neglect the subscript in the group operation to improve readability. Moreover, we use the notation L (x,a) and R (x,a) to denote respectively the left-translation and righttranslation by (x, a) ∈ Aff, acting on a function f : Aff → C by Recall that the translation operator T x and the dilation operator D a are respectively given by The following computation motivates the group operation on the affine group: We can represent the affine group Aff and its Lie algebra aff in matrix form The Lie algebra structure of aff is completely determined by An important feature of the affine group is that it is non-unimodular; the left and right Haar measures are respectively given by As such, the modular function on the affine group is given by ∆(x, a) = a −1 . The affine group is exponential, meaning that the exponential map exp : aff → Aff given by is a global diffeomorphism. Hence we can write the left and right Haar measures in exponential coordinates by the formulas Throughout the paper, we will heavily use the spaces L p l (Aff) := L p (Aff, µ L ) and L p r (Aff) := L p (Aff, µ R ) for 1 ≤ p ≤ ∞.

Quantum Harmonic Analysis on the Heisenberg Group
Before delving into quantum harmonic analysis on the affine group, it is advantageous to review the Heisenberg setting, originally introduced by Werner [46]. There are three primary constructions that appear: (a) A quantization scheme, (b) an integrated representation, and (c) a way to define convolution that incorporates operators. We give a brief overview of these three constructions and refer the reader to [46,23,36] for more details.

Weyl Quantization
The cross-Wigner distribution of φ, ψ ∈ L 2 (R n ) is given by When φ = ψ we refer to W φ := W (φ, φ) as the Wigner distribution of φ ∈ L 2 (R n ). The cross-Wigner distribution satisfies the orthogonality relation Moreover, the Wigner distribution satisfies the marginal properties for φ ∈ S (R n ). Our primary interest in the cross-Wigner distribution stems from the following connection: For each f ∈ L 2 (R 2n ) we define the operator L f : L 2 (R n ) → L 2 (R n ) by the formula Then L f is the Weyl quantization of f , see [23,Ch. 14] for details. It is a non-trivial fact, see [42], that the Weyl quantization gives a well-defined isomorphism between L 2 (R 2n ) and S 2 (L 2 (R n )), the space of Hilbert-Schmidt operators on L 2 (R n ).

Integrated Schrödinger Representation
Recall that the Heisenberg group H n is the Lie group with underlying manifold R n × R n × R and with the group multiplication (x, ω, t) · (x , ω , t ) := x + x , ω + ω , t + t + 1 2 x ω − xω .
The Heisenberg group is omnipresent in modern mathematics and theoretical physics, see [27]. For a Hilbert space H we let U(H) denote the unitary operators on H. The most important representation of the Heisenberg group is the Schrödinger representation ρ : H n → U(L 2 (R n )) given by where T x is the n-dimensional analogue of the translation operator defined in (2.1) and M ω is the modulation operator given by The Schrödinger representation is both irreducible and unitary. Let us use the abbreviated notation z := (x, ω) ∈ R 2n and π(z) = M ω T x . Ignoring the central variable t, we can consider the integrated Schrödinger representation ρ : L 1 (R 2n ) → L(L 2 (R n )) given by where L(L 2 (R n )) denotes the bounded linear operators on L 2 (R n ). We remark that the integral in (2.4) is defined weakly. It turns out, see [15,Thm. 1.30], that the integrated representation ρ extends from L 1 (R 2n )∩L 2 (R 2n ) to a unitary map ρ : L 2 (R 2n ) → S 2 (L 2 (R n )).

Operator Convolution
Given a function f ∈ L 1 (R 2n ) and a trace-class operator S ∈ S 1 (L 2 (R n )), their convolution is the trace-class operator on L 2 (R n ) defined by The convolution f S satisfies the estimate f S S 1 ≤ f L 1 S S 1 . One can also define the convolution between two operators: For two trace-class operators S, T ∈ S 1 (L 2 (R n )) we define their convolution to be the function on R 2n given by  To see the connection with the Wigner distribution, we note that the cross-Wigner distribution of ψ, φ ∈ L 2 (R n ) can be written as where ψ ⊗ φ denotes the rank-one operator on L 2 (R n ) given by Similarly, the Weyl quantization of f ∈ L 1 (R 2n ) may be expressed in terms of operator convolutions: Hence convolution with the parity operator P gives a convenient way to represent the Wigner distribution and the Weyl quantization.
Finally, there is a Fourier transform for operators: Given a trace-class operator S ∈ S 1 (L 2 (R n )) we define the Fourier-Wigner transform F W (S) of S to be the function on R 2n given by F W (S)(z) := e iπxω tr(Sπ(z) * ), z ∈ R 2n . (2.8) The Fourier-Wigner transform extends to a unitary map F W : S 2 (L 2 (R n )) → L 2 (R 2n ), where it turns out the to be inverse of the integrated Schrödinger representation given in (2.4). By [15,Prop. 2.5] it is related to the Weyl transform by the elegant formula where F σ denotes the symplectic Fourier transform.

Affine Weyl Quantization
We briefly describe affine Weyl quantization and how this gives rise to the affine Wigner distribution. There is a unitary representation π of the affine group Aff on L 2 (R + , r −1 dr) given by Since r −1 dr is the Haar measure on R + we will write L 2 (R + ) := L 2 (R + , r −1 dr). Later we also consider another measure on R + and will be more explicit when the situation requires it.
To define the quantization scheme we will utilize the Stratonovich-Weyl operator on L 2 (R + ) given by , e u ψ(r) du dv. (2.11) The following result was shown in [18] and provides us with an affine analogue of Weyl quantization.
Proposition 2.1 ( [18]). There is a norm-preserving isomorphism between L 2 r (Aff) and the space of Hilbert-Schmidt operators on L 2 (R + ). The isomorphism sends f ∈ L 2 r (Aff) to the operator A f on L 2 (R + ) defined weakly by We will refer to the association f → A f as affine Weyl quantization, while f is called the affine (Weyl) symbol of A f . To emphasize the correspondence between a Hilbert-Schmidt operator A and its affine symbol f we use the notation f A := f . The affine Weyl symbol of an operator A is explicitly given by where A K : R + × R + → C is the integral kernel of A defined by By taking the affine Weyl symbol of the rank-one operator ψ ⊗ φ on L 2 (R + ) given by for ψ, φ, ξ ∈ L 2 (R + ), we obtain the following definition.
Definition 2.2. For φ, ψ ∈ L 2 (R + ) we define the affine (cross-)Wigner distribution W ψ,φ Aff to be the function on Aff given for (x, a) ∈ Aff by When φ = ψ we refer to W ψ Aff := W ψ,ψ Aff as the affine Wigner distribution of ψ. The weak interpretation of the integral defining A f means that we have the relation for f ∈ L 2 r (Aff) and φ, ψ ∈ L 2 (R + ). The affine Wigner distribution satisfies the orthogonality relation for ψ 1 , ψ 2 , φ 1 , φ 2 ∈ L 2 (R + ). Moreover, the affine Wigner distribution also satisfies the marginal property for all rapidly decaying smooth functions ψ on R + . We remark that a rapidly decaying smooth function (also called a Schwartz function) ψ : R + → C is by definition a smooth function such that x → ψ(e x ) is a rapidly decaying function on R. The space of all rapidly decaying smooth functions on R + will be denoted by S (R + ). 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].

Affine Operator Convolutions
In this part we introduce operator convolutions in the affine setting. We show that this notion is intimately related to affine Weyl quantization in Section 3.2. In Section 3.4 we will introduce the affine Grossmann-Royer operator, which will be essential in Section 3.5 where we prove the main connection between the affine Weyl quantization and the operator convolutions in Theorem 3.21.

Definitions and Basic Properties
We begin by defining operator convolutions in the affine setting and derive basic properties.
Recall that the usual convolution on the affine group with respect to the right Haar measure is given by Remark. Other sources, e.g. [16], use the left Haar measure and define the convolution to be . We will mainly work with the right Haar measure, and our definition ensures that r (Aff) and let S be a trace-class operator on L 2 (R + ). We define the convolution f Aff S between f and S to be the operator on L 2 (R + ) given by where U is the unitary representation given in (2.10). The integral is a convergent Bochner integral in the space of trace-class operators. Remark.
1. As we will see later, using U (−x, a) instead of U (x, a) in Definition 3.1 ensures that the convolution is compatible with the following covariance property of the affine Wigner distribution: W (3.1) 2. The notation has a different meaning in [18], where it is used to denote the so-called Moyal product of two functions defined on Aff.
Definition 3.2. Let S be a trace-class operator and let T be a bounded operator on L 2 (R + ). Then we define the convolution S Aff T between S and T to be the function on Aff given by Remark. Recently, [10] defined another notion of convolution of trace-class operators. Unlike our definition, this convolution produces a new trace-class operator, with the aim of interpreting the trace-class operators as an analogue of the Fourier algebra. It is straightforward to check that if f is a positive function and S, T are positive operators, then f Aff S is a positive operator and S Aff T is a positive function. Moreover, we have the elementary estimate and The following result is proved by a simple computation.
In particular, for η, ξ ∈ L 2 (R + ) we have A natural question to ask is whether the three different notions of convolution we have introduced are compatible. The following proposition gives an affirmative answer to this question.
r (Aff), S ∈ S 1 , and let T be a bounded operator on L 2 (R + ). Then we have the compatibility equations Proof. The first equality follows from the computation We are allowed to take the trace outside the integral since the second to last line is essentially the duality action of the bounded operator U (−x, a) * T U (−x, a) on a convergent Bochner integral in the space of trace-class operators.
For the second equality, we use change of variables and obtain Changing the order of integration above is allowed by Fubini's theorem for Bochner integrals dz dc c is bounded from above by

Relationship With Affine Weyl Quantization
The goal of this section is to connect the affine Weyl quantization described in Section 2.3 with the convolutions defined in Section 3.1. We first establish a preliminary result describing how right multiplication on the affine group affects the affine Weyl quantization.
Proof. The result follows from (2.13) and the computation We are now ready to prove the first result showing the connection between convolution and affine Weyl quantization.
Proof. The operator g Aff A f is defined as the S 2 -convergent Bochner integral By Proposition 2.1, the map W : S 2 → L 2 r (Aff) given by W(A f ) = f is unitary. Since bounded operators commute with convergent Bochner integrals, we have using Lemma 3.5 that We can also express the convolution of two operators in terms of their affine Weyl symbols.
Proof. Using Proposition 2.1 and Lemma 3.5 we compute that The result follows asǧ ∈ L 2 r (Aff) if and only if g ∈ L 2 l (Aff).

Affine Weyl Quantization of Coordinate Functions
Of particular interest is the affine Weyl quantization of the coordinate functions f x (x, a) := x and f a (x, a) := a for (x, a) ∈ Aff. Due to the fact that the coordinate functions are not in L 2 r (Aff), we first need to interpret the quantizations A fx and A fa in a rigorous manner. We begin this task by defining rapidly decaying smooth function and tempered distributions on the affine group.
We refer to S (Aff) as the space of rapidly decaying smooth functions (or Schwartz functions) on the affine group.
There is a natural topology on S (Aff) induced by the semi-norms for α = (α 1 , α 2 ) and β = (β 1 , β 2 ) in N 0 × N 0 . With these semi-norms, the space S (Aff) becomes a Fréchet space. The space of bounded, anti-linear functionals on S (Aff) is denoted by S (Aff) and called the space of tempered distributions on Aff.
Additionally, the map f → A f is injective.
For the injectivity it suffices to show that A f = 0 implies that f = 0. Let us first reformulate this slightly: If A f = 0, then we have that We could conclude that f = 0 if we knew that any g ∈ S (Aff) could be approximated (in the Fréchet topology) by linear combinations of elements on the form W φ,ψ Aff for ψ, φ ∈ S (R + ). To see that this is the case, we translate the problem to the Heisenberg setting.
The Mellin transform M is given by 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 where W is the cross-Wigner distribution. The correspondence preserves Schwartz functions, due to the term being smooth with polynomially bounded derivatives. This gives a bijective correspondence between W ψ,φ Aff ∈ S (Aff) and W (Ψ, Φ) ∈ S (R 2 ). As such, the injectivity question is reduced to asking whether the linear span of elements on the form W (f, g) for f, g ∈ S (R) is dense in S (R 2 ). One way to verify this well-known fact is to note that the map f ⊗ g → W (f, g), where f ⊗g(x, y) = f (x)g(y), extends to a topological isomorphism on S (R 2 ), see for instance [23, (14.21)] for the formula of this isomorphism. The density of elements on the form W (f, g) for f, g ∈ S (R) therefore follows as the functions h m ⊗ h n , where {h n } ∞ n=0 are the Hermite functions, span a dense subspace of S (R 2 ) by [43,Thm. V.13].
Example 3.10. Consider the constant function on the affine group given by 1(x, a) = 1 for all (x, a) ∈ Aff. Then the quantization A 1 is the identity operator since for ψ, φ ∈ S (R + ) 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.
To motivate the next result, consider the coordinate functions σ x (x, ω) := x and σ ω (x, ω) := ω for (x, ω) ∈ R 2n . The Weyl quantizations L σx and L σω are the well-known position operator and momentum operator in quantum mechanics. In particular, the commutator is a constant times the identity by [26,Prop. 3.8]. This is precisely the relation for the Lie algebra of the Heisenberg group. In light of this, the following proposition shows that the affine Weyl quantization has the expected expression for the coordinate functions.
Theorem 3.11. Let f x and f a be the coordinate functions on the affine group. The affine Weyl quantizations A fx and A fa are well-defined as maps from S (R + ) to S (R + ) and are explicitly given by In particular, we have the commutation relation This is, up to re-normalization, precisely the Lie algebra structure of aff given in (2.2).
Proof. Let us begin by computing A fx . We can change the order of integrating by Fubini's theorem and obtain for ψ, φ ∈ S (R + ) that Notice that the inner integral is equal to Hence we have the relation By using the formulas λ(0) = 1 and λ (0) = 1/2 we can simplify and obtain Using integration by parts we obtain the claim since For A fa we have by similar calculations as above that The commutation relation follows from straightforward computation.

The Affine Grossmann-Royer Operator
In this section we introduce the affine Grossmann-Royer operator with the aim of obtaining an affine parity operator analogous to the (Heisenberg) parity operator P in Section 2.2.3. The main reason for this is to obtain affine version of the formulas (2.6) and (2.7) so that we can describe the affine Weyl quantization through convolution. Recall that the (Heisenberg) Grossmann-Royer operator R(x, ω) for (x, ω) ∈ R 2n is defined by the relation Analogously, we have the following definition.
Definition 3.12. We define the affine Grossmann-Royer operator R Aff (x, a) for (x, a) ∈ Aff by the relation We restrict our attention to Schwartz functions for convenience since then W ψ,φ Aff ∈ S (Aff) by [5,Cor. 6.6], and hence have well-defined point values. The Grossmann-Royer operator R Aff (x, a) is precisely the affine Weyl quantization of the point mass δ Aff (x, a) ∈ S (Aff) for (x, a) ∈ Aff defined by Since this is also true for the Stratonovich-Weyl operator Ω(x, a) given in (2.11), it follows that R Aff (x, a) = Ω(x, a) for all (x, a) ∈ Aff. From [18, p. 12] it follows that we have the affine covariance relation The following result, which is a straightforward computation, shows that R Aff (x, a) is an unbounded and densely defined operator on L 2 (R + ).
Lemma 3.13. Fix ψ ∈ S (R + ) and (x, a) ∈ Aff. The affine Grossmann-Royer operator R Aff (x, a) has the explicit form where λ is the function given in (2.3).
We will be particularly interested in the affine parity operator P Aff given by the affine Grossmann-Royer operator at the identity element, that is, for ψ ∈ S (R + ). The affine parity operator P Aff is symmetric as an unbounded operator on L 2 (R + ). Moreover, we see from the relation An important commutation relation for the (Heisenberg) Grossman-Royer operator R(x, ω) for (x, ω) ∈ R 2n is given by (3.7) The following proposition shows that the analogue of (3.7) breaks down in the affine setting due to Aff being non-unimodular. As the proof is a straightforward computation, we leave the details to the reader.
Proposition 3.14. The commutation relation We will now show that both the function λ in (2.3) and the affine parity operator P Aff are related to the Lambert W function. Recall that the (real) Lambert W function is the multivalued function defined to be the inverse relation of the function f (x) = xe x for x ∈ R. The function f (x) for x < 0 is not injective. There exist for each y ∈ (−1/e, 0) precisely two values x 1 , x 2 ∈ (−∞, 0) such that As the solutions appear in pairs, we can define σ to be the function that permutes these solutions, that is, σ(x 1 ) = x 2 and σ(x 2 ) = x 1 . For y = −1/e there is only one solution to the equation xe x = y, namely x = −1. Hence we define σ(−1) = −1. We can represent the function σ as where W 0 , W −1 are the two branches of the Lambert W function satisfying Lemma 3.15. The inverse of λ is given by Proof. To find the inverse of λ we solve the equation A simple computation shows that −r = −u − re −u . Making the substitution v = e −u together with straightforward manipulations shows that The trivial solution to (3. −re −r = σ(−r) + r.

Remark.
A minor variation of the function σ appeared in [18,Section 3] where it was defined by the relation in Lemma 3.15. The advantage of understanding the connection to the Lambert W function is that properties such as σ(σ(x)) = x for every x < 0 become trivial in this description.
Proof. The formula for P Aff (ψ) is obtained from Lemma 3.15 together with (3.6). To find the value P Aff (ψ)(1), we use (3.6) and the fact that Hence the claim follows from L'Hopital's rule since

Operator Convolution for Tempered Distributions
This section is all about expressing the affine Weyl quantization of a function f ∈ S (Aff) by using affine convolution. To be able to do this, we will first define what it means for A f to be a Schwartz operator.
Definition 3.17. We say that a Hilbert-Schmidt operator A : Proof. Assume that A is a Schwartz operator. In [18,Equation (4.8)] it is shown that the integral kernel A K of A is related to the affine Weyl symbol f A of A by the formula Since the inverse-Fourier transform preserves Schwartz functions, together with the definition of S (R + × R + ), we have that By performing the change of variable x = log(r/s) and s = e ω for ω ∈ R we obtain Finally, by letting u = log((e x − 1)/x) + ω we see that due to the fact that x → log((e x − 1)/x) has polynomial growth. Conversely, assume that A = A f for f ∈ S (Aff). The integral kernel A K is then given by r − s log(r/s) .
By using that the inverse-Fourier transform F −1 1 in the first component preserves S (Aff) together with similar substitutions as previously, we have that A K ∈ S (R + × R + ). We will use the notation S (L 2 (R + )) for all Schwartz operators on L 2 (R + ). There is a natural topology on S (L 2 (R + )) induced by the semi-norms A f α,β := f α,β where · α,β are the semi-norms on S (Aff) given in (3.4).
Moreover, for fixed A ∈ S (L 2 (R + )) the map is continuous.
Proof. Let f ∈ S (Aff) and A ∈ S (L 2 (R + )). Then A = A g for some g ∈ S (Aff) and we have by Proposition 3.6 that (3.9) Hence the first statement reduces to showing that the usual affine group convolution is a well-defined map After a change of variables, the question becomes whether the map is an element in S (R 2 ). It is straightforward to check that (3.10) is a smooth function.
Moreover, since f and g are both in S (Aff), it suffices to show that (3.10) decays faster than any polynomial towards infinity; we can then iterate the argument to obtain the required decay statements for the derivatives. We claim that where A g k,l is a constant that depends only on the indices k, l ∈ N 0 and g ∈ S (Aff). To show this, we need to individually consider three cases: • Assume that we only take the supremum over x and u satisfying 2|z| ≥ |u| and 2|y| ≥ |x|.
Then clearly (3.11) is satisfied with A g k,l = 2 k+l max |g|. • Assume that we only take the supremum over u satisfying 2|z| ≤ |u| and let x ∈ R be arbitrary. Then e u−z is outside the interval [e −|u|/2 , e |u|/2 ]. Since g ∈ S (Aff) the left-hand side of (3.11) will eventually decrease when increasing u. When y ≤ 0 the left hand-side of (3.11) will also obviously eventually decrease by increasing x. When y > 0 then any increase of x would necessitate an increase of u on the scale of u ∼ ln(x) to compensate so that the first coordinate in g does not blow up. However, this again forces the second coordinate to grow on the scale of x and we would again, due to g ∈ S (Aff), have that the left hand-side of (3.11) would eventually decrease.
• Finally, we can consider taking the supremum over x and u satisfying 2|z| ≥ |u| and 2|y| ≤ |x|. As this case uses similar arguments as above, we leave the straightforward verification to the reader.
where the last inequality follows from that f ∈ S (Aff). Finally, the continuity of the map f → f Aff A follows from (3.9) and (3.12).
Remark. Notice that the proof of Proposition 3.19 shows that affine convolution between f, g ∈ S (Aff) satisfies f * Aff g ∈ S (Aff). This fact, together with Proposition 3.18, strengthens the claim that S (Aff) is the correct definition for Schwartz functions on the group Aff.
The main result in this section is Theorem 3.21 presented below. To state the result rigorously, we first need to make sense of the convolution between Schwartz functions g ∈ S (Aff) and the affine parity operator P Aff . As motivation for our definition we will use the following computation: Let S, T ∈ S 2 with affine Weyl symbols f S , f T ∈ L 2 r (Aff). Fix g ∈ S (Aff) and consider the affine Weyl symbol f g Aff S corresponding to the convolution g Aff S. Then .
With this motivation in mind we get the following definition.
Definition 3.20. Let S : S (R + ) → S (R + ) be the operator with affine Weyl symbol f S ∈ S (Aff) and let g ∈ S (Aff). Then g Aff S is defined by its Weyl symbol f g Aff S ∈ S (Aff) satisfying , for all h ∈ S (Aff).
Recall that the injectivity in Lemma 3.9 ensures that the operator S in Definition 3.20 is well-defined. The argument to show f g Aff S ∈ S (Aff) is similar to the one presented in Proposition 3.19. Hence g Aff S is well-defined.
Remark. We could similarly have defined S Aff A f for S ∈ S (L 2 (R + )) and f ∈ S (Aff) by using Proposition 3.7. For brevity, we restrict ourselves in the next theorem to the case where S = φ ⊗ ψ for ψ, φ ∈ S (Aff). In this case, we can extend Lemma 3.3 and define We can now finally state the main theorem in this section.
Theorem 3.21. The affine Weyl quantization A g of g ∈ S (Aff) can be written as where P Aff is the affine parity operator. Moreover, for ψ, φ ∈ S (R + ) we have that the affine Weyl symbol W ψ,φ Aff of the rank-one operator ψ ⊗ φ can be written as Proof. Recall that the affine parity operator P Aff is the affine Weyl quantization of the point measure δ (0,1) ∈ S (Aff). As such, the convolution g Aff P Aff is well-defined with the interpretation given in Definition 3.20. The affine Weyl symbol f g Aff P Aff of g Aff P Aff is acting on h ∈ S (Aff) by Since S (Aff) ⊂ L 2 r (Aff) is dense, we can conclude that f g Aff P Aff = g and thus A g = g Aff P Aff . For the second statement, we get that Aff (x, a).

Operator Admissibility
For operator convolutions on the Heisenberg group, we have from (2.5) the important integral relation R 2n

S T (z) dz = tr(S) tr(T ).
A similar formula for the integral of operator convolutions will not hold generally in the affine setting. We therefore search for a class of operators where such a relation does hold: the admissible operators. As a first step, we recall the notion of admissible functions.
This definition of admissibility is motivated by the theorem of Duflo and Moore [13], see also [24]. The Duflo-Moore operator D −1 in our setting is formally given by It is clear that the Duflo-Moore operator D −1 is a densely defined, self-adjoint positive operator on L 2 (R + ) with a densely defined inverse, namely Dψ(r) := √ rψ(r).
Hence a function ψ ∈ L 2 (R + ) is admissible if and only if D −1 ψ ∈ L 2 (R + ). We will on several occasions use the commutation relations The following orthogonality relation is a trivial reformulation of the classic orthogonality relations for wavelets, see for instance [25].

Admissibility for Operators
Our goal is now to extend the notion of admissibility to bounded operators on L 2 (R + ), with the aim of obtaining a class of operators where a formula for the integral of operator convolutions similar to (2.5) holds. We will often use that any compact operator S on L 2 (R + ) has a singular value decomposition where {ξ n } N n=1 and {η n } N n=1 are orthonormal sets in L 2 (R + ). The singular values {s n } N n=1 with s n > 0 will converge to zero when N = ∞. If S is a trace-class operator we have {s n } N n=1 ∈ 1 (N) with S S 1 = s n 1 . Since the admissible functions in L 2 (R + ) form a dense subspace, we can always find an orthonormal basis consisting of admissible functions.
The next result concerns bounded operators DSD for a trace-class operator S. To be precise, this means that we assume that S maps dom(D −1 ) into dom(D), and that the operator DSD defined on dom(D) extends to a bounded operator. Theorem 4.3. Let S ∈ S 1 satisfy that DSD ∈ L(L 2 (R + )). For any T ∈ S 1 we have that Proof. We divide the proof into three steps.
Step 1: We first assume that T = ψ ⊗ φ for ψ, φ ∈ dom(D). Recall that S can be written in the form (4.2). From Lemma 3.3 and (4.1) we find that Integrating with respect to the right Haar measure and using that (x, a) → (x, a) −1 interchanges left and right Haar measure, we get where the last line uses Proposition 4.2. It follows that the sum in the expression for T Aff DSD(x, a) converges absolutely in L 1 r (Aff) with

Equation (4.3) follows in a similar way by integrating the sum expressing T Aff DSD and using Proposition 4.2.
Step 2: We now assume that T = ψ ⊗ φ for arbitrary ψ, φ ∈ L 2 (R + ). Pick sequences {ψ n } ∞ n=1 , {φ n } ∞ n=1 in dom(D) converging to ψ and φ, respectively, and let T n = ψ n ⊗ φ n . It is straightforward to check that T n converges to T in S 1 . By (3.3) this implies that T n Aff DSD converges uniformly to T Aff DSD. On the other hand, T n Aff DSD is a Cauchy sequence in L 1 r (Aff): for m, n ∈ N we find by Step 1 that which clearly goes to zero as m, n → ∞. This means that T n Aff DSD converges in L 1 r (Aff), and the limit must be T Aff DSD as we already know that T n Aff DSD converges uniformly to this function. In particular, this implies Equation (4.3) also follows by taking the limit of Aff T n Aff DSD(x, a) dx da a .
Step 3: We now assume that T ∈ S 1 . Consider the singular value decomposition of T given by By (3.3) we have, with uniform convergence of the sum, that Notice that Step 2 implies that the convergence is also in L 1 In particular, T Aff DSD ∈ L 1 r (Aff). Finally, (4.3) follows by integrating (4.4) and using that the sum converges in L 1 r (Aff) and Step 2.
The integral relation (4.3) is somewhat artificial in the sense that it introduces D in the integrand. We will typically be interested in the integral of T Aff S, not of T Aff DSD. This motivates the following definition.
Definition 4.4. Let S be a non-zero bounded operator on L 2 (R + ) that maps dom(D) into dom(D −1 ). We say that S is admissible if the composition D −1 SD −1 is bounded on dom(D −1 ) and extends to a trace-class operator D −1 SD −1 ∈ S 1 .
Assume now that S is admissible, and define R := D −1 SD −1 . Clearly R maps dom(D −1 ) into dom(D) as we assume that S maps dom(D) into dom(D −1 ). The following corollary is therefore immediate from Theorem 4.3. We also note that it extends [34, Cor. 1] to nonpositive, non-compact operators.
Corollary 4.5. Let S ∈ L(L 2 (R + )) be an admissible operator. For any T ∈ S 1 we have that and Aff T Aff S(x, a) dx da a = tr(T ) tr(D −1 SD −1 ).
Example 4.6. A rank-one operator S = η⊗ξ for non-zero η, ξ is an admissible operator if and only if η, ξ ∈ L 2 (R + ) are admissible functions. Requiring that S maps dom(D) into dom(D −1 ) clearly implies that η ∈ dom(D −1 ), i.e. η is admissible. For D −1 SD −1 to be trace-class, the map must at least be bounded for ψ L 2 (R + ) ≤ 1. This is bounded if and only if is bounded, which is precisely the condition that ξ ∈ dom D −1 * = dom(D −1 ). Hence our notion of admissibility for operators naturally extends the classical function admissibility. In the case of rank-one operators, it follows from Lemma 3.3 and the computation that Corollary 4.5 reduces to Proposition 4.2.
When both S and T are admissible trace-class operators, their convolution T Aff S behaves well with respect to both the left and right Haar measures. Proof. The first equation and the claim that T Aff S ∈ L 1 r (Aff) is Corollary 4.5. The second equation and the claim that T Aff S ∈ L 1 l (Aff) follows since We now turn to the case where S is a positive compact operator. We first note that admissibility in this case becomes a statement about the eigenvectors and eigenvalues of S. Proof. We first assume that S is admissible. By linearity and Lemma 3.3 we get for ξ ∈ L 2 (R + ) with ξ L 2 (R + ) = 1 that (4.5) Integrating (4.5) using the monotone convergence theorem and Proposition 4.2, we obtain The claim now follows from Corollary 4.5.
For the converse, it is clear by the assumption that the operator is a trace-class operator. It only remains to show that S maps dom(D) into dom(D −1 ) and that D −1 SD −1 is given by (4.6). This is easily shown when N is finite, so we do the proof for N = ∞. The partial sums for ψ ∈ L 2 (R + ) are denoted by The sequence of partial sums D −1 (Sψ) M also converges in L 2 (R + ), since by using Hölder's inequality and Bessel's inequality we obtain Since D −1 is a closed operator, we get that Sψ belongs to the domain of D −1 and s n ψ, ξ n L 2 (R + ) D −1 ξ n .
For any φ ∈ dom(D −1 ), we have that so D −1 SD −1 agrees with (4.6) on this dense subspace. In fact, they agree on all of L 2 (R + ) since shows that D −1 SD −1 extends to a bounded operator.
As a consequence of Proposition 4.8, we obtain a compact reformulation of admissibility for positive trace-class operators.
In particular, if S is a non-zero, positive trace-class operator, then S is admissible if and only if S Aff S ∈ L 1 r (Aff). The claims now follow immediately from Proposition 4.8.  for L 2 (R + ) by

Admissible Operators from Laguerre Functions
where Γ denotes the gamma function and L (α) n denotes the generalized Laguerre polynomials given by The classical orthogonality relation for the generalized Laguerre polynomials ensures that the Laguerre bases are orthonormal bases for L 2 (R + ) for any fixed α ∈ R + . The following result shows that the Laguerre basis is especially compatible with the Duflo-Moore operator D −1 .
Proposition 4.11. For any α ∈ R + and n ∈ N 0 we have Proof. The first equality in (4.8) follows from unwinding the definitions. For the second equality in (4.8), we will use the well-known identity together with the orthogonality relation (4.7). This gives where the last equality follows from a straightforward induction argument.
The following consequence from Proposition 4.8 shows that we can explicitly construct admissible operators by using the Laguerre basis.
Corollary 4.12. Let {s n } ∞ n=0 ∈ 1 (N) be a sequence of non-negative numbers and let α ∈ R + . Then n is an admissible operator with Remark. The corollary may be considered a reformulation with slightly different proof of the calculations in [21,Section 3.3], where a resolution of the identity operator is constructed from thermal states that are diagonal in the Laguerre basis. We will return to resolutions of the identity operator and the relation to admissibility in Section 6.2.

Connection with Convolutions and Quantizations
We will now see how admissibility relates to the convolution of a function with an operator.
The following result shows that we can use convolutions to generate new admissible operators from a given admissible operator. Proof. It is clear from (3.2) that f Aff S is a trace-class operator, and positivity follows from the definition of the convolution f Aff S. Let T be a non-zero positive trace-class operator on L 2 (R + ). It suffices by Corollary 4.9 to show that We have that We may then use Fubini's theorem, which applies by our assumptions on f and S, to show that where we used the admissibility of S and Theorem 4.5 in the last line.
Remark. We can give a simple heuristic argument for Proposition 4.13 by ignoring that D −1 is unbounded as follows: We have by using (4.1) that Since D −1 SD −1 is a trace-class operator, the integral above is a convergent Bochner integral and we obtain the desired equality.

Admissibility as a Measure of Non-Unimodularity
In this section we will delve more into how the non-unimodularity of the affine group affects the affine Weyl quantization. As we will see, both the left and right Haar measures take on an active role in this picture.
Proposition 4.14. Let S be an admissible Hilbert-Schmidt operator on L 2 (R + ) such that its affine Weyl symbol f S satisfies f S ∈ L 1 l (Aff). Then Proof. Let T = ϕ ⊗ ϕ for some non-zero ϕ ∈ S (R + ). Then the affine Weyl symbol of T is f T = W ϕ Aff ∈ S (Aff). We know by Corollary 4.5 that On the other hand, Fubini's theorem together with Proposition 3.7 allows us to calculate that The marginal properties of the affine Wigner distribution (2.15) show that The claim now follows from combining the calculations we have done.
Remark. Assuming that T is a trace-class operator we have that which follows from a similar proof to the one in Proposition 4.14. This gives the interesting heuristic interpretation that taking D −1 T D −1 of an operator T coincides with multiplying f T by 1 a .
The following result shows that the affine Wigner distribution satisfies both left and right integrability when more is assumed of the input. This should be compared with the Heisenberg case where the Heisenberg group H n is unimodular.
Theorem 4.15. Assume that φ, ψ, Dφ, Dψ ∈ L 2 (R + ). Then the affine Wigner distribution satisfies Proof. We already know that W φ,ψ Aff is in L 2 r (Aff) by the orthogonality relations (2.14). Using the definition of the affine Wigner distribution and Plancherel's theorem, we have that where we used the change of variables v = aλ(x) and w = aλ(−x) in the last line. By our assumptions on φ and ψ, it will suffice to show that for all v, w ∈ R + we have the upper It will be enough by symmetry to consider Λ = {(v, w) ∈ R + × R + : v > w}. We have the decomposition Λ = C 1 ∪ C 2 ∪ C 3 , where where σ is the function appearing in Lemma 3.15. • The level surface g(v, w) = (v − w)/ log(v/w) = C for C > 0 is given by the equation On C 1 we are below the level surface (4.9) with C = 2. Notice that (1, 0.5) ∈ C 1 with g(1, 0.5) = log( √ 2) < 2. The continuity of g forces the inequality g(v, w) ≤ 2 for all Hence the case of C 2 follows from the previous the argument for C 1 by considering the level surface of g(1/v, 1/w) = 1.
• It is straightforward to verify that v > 2 and w < 1 when (v, w) ∈ C 3 . Hence we obtain for any Remark. The connection from this result to admissibility is that the assumptions boil down to S = Dψ ⊗ Dφ being an admissible operator.
Remark. Let A be a Hilbert-Schmidt operator on L 2 (R + ) with integral kernel A K . Then one can gauge from the proof of Theorem 4.15 that the affine Weyl symbol f A satisfies f A ∈ L 2 r (Aff) ∩ L 2 l (Aff) if and only if the integral kernel A K satisfies

Extending the Setting
Proof. By using (4.1) we get that Clearly Dψ ⊗ Dφ is an admissible operator with By Corollary 4.5 we therefore get The density of dom(D) implies that f Aff DSD extends to a bounded operator on L 2 (R + ).
Armed with Lemma 4.17 and Corollary 4.5, we prove the following result describing L p and S p properties of convolutions with admissible operators. The proof is essentially an application of complex interpolation: we refer to [45, Thm. 2.10] and [8, Thm. 5.1.1] for the interpolation theory of S p and L p r (Aff).
If R ∈ S p , g ∈ L p r (Aff), and S is an admissible trace-class operator, then: Proof. For g ∈ L 1 r (Aff) ∩ L ∞ (Aff), we have for p = ∞ that Lemma 4.17 gives Since we also have g Aff S S 1 ≤ g L 1 r (Aff) S S 1 , the first result follows by complex interpolation. For the second claim, if R ∈ S 1 we know from Corollary 4.5 that The result follows by complex interpolation since

From the Viewpoint of Representation Theory
We will for completeness investigate how various notions of affine Fourier transforms fit into our framework. As we will see, known results from abstract wavelet analysis give connections between affine Weyl quantization, affine Fourier transforms, and admissibility for operators.

Affine Fourier Transforms
Definition 5.1. For f ∈ L 1 l (Aff) we define the (left) integrated representation U (f ) to be the operator on L 2 (R + ) given by The inverse affine Fourier-Wigner transform F −1 W (f ) of f ∈ L 1 r (Aff) is given by The inverse affine Fourier-Wigner transform F −1 W (f ) of f ∈ L 1 r (Aff) is explicitly given by where F 1 denotes the Fourier transform in the first coordinate and ψ ∈ L 2 (R + ). Hence the integral kernel of F −1 W (f ) is given by It is straightforward to verify that we have the estimate for every f ∈ L 1 r (Aff) ∩ L 2 r (Aff). Hence we can extend F −1 W to be defined on L 2 r (Aff) and we have that F −1 W (f ) ∈ S 2 for any f ∈ L 2 r (Aff).

Proposition 5.2. The inverse affine Fourier-Wigner transform is a unitary transformation
Proof. Any function K ∈ L 2 (R + × R + ) can be written uniquely on the form K f in (5.1) for some f ∈ Q 1 . Moreover, we have Since there is a norm-preserving correspondence between integral kernels in L 2 (R + × R + ) and Hilbert-Schmidt operators on L 2 (R + ), the claim follows.
It is straightforward to check that the inverse affine Fourier-Wigner transform F −1 W satisfies for f, g ∈ Q 1 the properties Definition 5.3. The affine Fourier-Wigner transform F W : S 2 → Q 1 is defined to be the inverse of F −1 W | Q 1 . Remark.
• To avoid overly cluttered notation, we have used the symbol F W for both the classical Fourier-Wigner transform in Section 2.2.3, and the affine Fourier-Wigner transform. It should be clear from the context which operator we are referring to.
• Recall that the right multiplication R acts on elements in L 2 r (Aff) by for (x, a), (y, b) ∈ Aff. For a closed subspace H ⊂ L 2 r (Aff) invariant under R, we write From [13,Lem. 3] we deduce that as both spaces are the image of the Hilbert-Schmidt operators under the Fourier-Wigner transform. Note that [13] uses left Haar measure, but translating to right Haar measure is an easy exercise using that f →f is a unitary equivalence from the left regular representation on L 2 l (Aff) to the right regular representation on L 2 r (Aff).
Example 5.4. Let φ, ψ ∈ L 2 (R + ) with ψ ∈ dom(D). If f (x, a) = φ, U (x, a) * Dψ L 2 (R + ) , one finds using Proposition 4.2 that f ∈ L 2 r (Aff) and for η ∈ L 2 (R + ) and ξ ∈ dom(D). This implies that F −1 For the Heisenberg group, the Fourier-Wigner transform has a very convenient expression for trace-class operators, see (2.8). The corresponding expression on the affine group is F W (A)(x, a) = tr(ADU (x, a)), and the next result shows that it holds as long as the objects in the formula are well-defined. The result is due to Führ in this generality [17,Thm. 4.15], and builds on an earlier result due to Duflo and Moore [13,Cor. 2].
Proposition 5.5 (Führ, Duflo, and Moore). Let A ∈ S 1 be such that AD −1 extends to a Hilbert-Schmidt operator. Then Proof. To see how the result follows from [17,Thm. 4.15], we need some terminology regarding direct integrals, see [17,Section 3.3]. Recall that the Plancherel theorem [17,Thm. 3.48] supplies a measurable field of Hilbert spaces indexed by the dual group {H π } [π]∈Ĝ . For the affine group G = Aff, the Plancherel measure is counting measure supported on the two irreducible representations π 1 (x, a) = U (x, a) on L 2 (R + ) and For f, g ∈ L 2 (R) we denote by SCAL g f the scalogram of f with respect to g given by The following result, which follows from Lemma 3.3 and Example 5.4, gives a connection between the affine Fourier-Wigner transform, affine convolutions, and the scalogram.
Corollary 5.6. Let f, g ∈ L 2 (R) such that ψ :=f and φ :=ĝ are supported in R + and are in L 2 (R + ). If ψ is admissible then Remark. The condition that ψ is admissible in Corollary 5.6 is only necessary for the first equality in (5.3). Recall that the affine Wigner distribution W ψ Aff is the affine Weyl symbol of the rank-one operator ψ ⊗ ψ. If we use Proposition 3.7 together with Corollary 5.6, then we recover [5,Thm. 5.1].
Corollary 5.6 shows that we have the simple relation for positive rank-one operators A. By Corollary 4.9, admissibility therefore means that F W (AD −1 ) ∈ L 2 r (Aff) in this case. For more general operators, (5.4) will no longer hold. However, we still obtain a result relating admissibility to the Fourier-Wigner transform. Note that in the first statement in Proposition 5.7 if A ∈ S 1 we interpret F W (AD −1 ) := tr(AU (x, a)) if we do not know that AD −1 extends to a Hilbert-Schmidt operator.
Proposition 5.7. Let A be a trace-class operator on L 2 (R + ). Then the following are equivalent: 2) AD −1 extends from dom(D −1 ) to a Hilbert-Schmidt operator on L 2 (R + ).
3) A * A is admissible.
Proof. The equivalence of 1) and 2) follows from [17,Thm. 4.15], by applying that theorem to the element {A [π] } [π]∈Ĝ of the direct integral (see proof of Proposition 5.5) given by choosing A [π 1 ] = A and A [π] = 0 for [π] = [π 1 ]. The equivalence of 2) and 3) is clear apart from technicalities resulting from the unboundedness of D −1 . If we assume 2), then [44,Thm. 13.2] gives that (AD −1 ) * = D −1 A * , where the equality includes equality of domains. As the domain of the left term is all of L 2 (R + ) by assumption, this means that the range of A * is contained in dom(D −1 ). In particular, A * A maps dom(D) into dom(D −1 ), and as we also have D −1 A * AD −1 = (AD −1 ) * AD −1 where AD −1 is Hilbert-Schmidt, A * A satisfies all requirements for being admissible.
Conversely, if A * A is admissible, then we have for ψ ∈ dom(D −1 ) So AD −1 extends to a bounded operator, and as this operator satisfies that is trace-class, AD −1 is a Hilbert-Schmidt operator.
Remark. Recall that we consider F W a Fourier transform of operators. The inequality There is a second Fourier transform related to the affine group that comes from representation theory. We define the affine Fourier-Kirillov transform as the map F KO : Q 1 → L 2 r (Aff) given by More information about the Fourier-Kirillov transform can be found in [33]. The following result, which is motivated by (2.9) and is a slight generalization of [3, Section VIII.6], shows that the affine Weyl quantization is intrinsically linked with the Fourier transforms on the affine group.
Proposition 5.8. Let A f be a Hilbert-Schmidt operator on L 2 (R + ) with affine symbol f ∈ L 2 r (Aff). Then the following diagram commutes: Proof. Recall from (5.1) that the integral kernel of F −1 W (g) for g ∈ Q 1 is given by K g (s, r) = √ r(F 1 g)(r, s/r), s, r ∈ R + .
Hence by using (2.12) and a change of variables, we see that the affine Weyl symbol of F −1 W (g) is given at the point (x, a) ∈ Aff by , e u e −2πi(xu+av) du dv λ(−u) Remark.
• In [40] the authors define an alternative quantization scheme on general type 1 groups. Their quantization scheme together with the affine Weyl quantization is used in [40] to define a quantization scheme on the cotangent bundle T * Aff.
• Consider A f for some f ∈ L 2 r (Aff). Inserting f = F KO F W (A f ) into Proposition 4.14 allows us to obtain a formal expression for tr(D −1 A f D −1 ) in terms of F W (A f ): a formal calculation gives that for sufficiently nice operators A f we have where F 1 is the Fourier transform in the first coordinate. This is similar to a condition in [21,Cor. 5.2], where finiteness of (5.5) is used as a necessary condition for 1 Aff A f = I L 2 (R + ) to hold, where 1(x, a) = 1 for all (x, a) ∈ Aff. We will see in Section 6.2 that this is closely related to admissibility of A f . Unfortunately, the formal calculation leading to (5.5) does not give clear conditions on A f for the equality to hold.

Affine Quantum Bochner Theorem
On the Heisenberg group, the Fourier-Wigner transform behaves in many ways like the Fourier transform on functions. In particular, for f ∈ L 1 (R 2n ) and S, T ∈ S 1 (R n ) we get the decoupling equations   a 1 ), . . . , (x n , a n )} ⊂ Aff the matrix A Ω with entries is positive semi-definite. Before stating the general result we consider an illuminating special case.
Example 5.9. Assume that A = φ ⊗ ψ is a rank-one operator where φ, ψ ∈ L 2 (R + ). We will show that is Hence U (x, a)φ, ψ L 2 (R + ) = U (x, a)ψ, φ L 2 (R + ) and it follows from [22,Thm. 4.2] that φ = c · ψ for some c ∈ C. We can conclude from [16,Cor. 3.22] that c ≥ 0 since We are now ready to state the main result regarding positivity. This result is actually, when interpreted correctly, a special case of the general result [17,Thm. 4.12]. is of positive type on Aff.
Proof. We use the same notation as in the proof of Proposition 5.5. For G = Aff, the abstract result in [17] says that if consists of trace-class operators, then A

Examples
In this section, we show how the theory developed in this paper provides a common framework for various operators and functions studied by other authors. We also introduce an analogue of the Cohen class of time-frequency distributions for the affine group, and deduce its relation to the previously studied affine quadratic time-frequency representations.

Affine Localization Operators
There is no general consensus of a localization operator in the affine setting. We will use the following definition based on the convolution framework. Definition 6.1. Let f ∈ L 1 r (Aff) and ϕ ∈ L 2 (R + ). We say that is an affine localization operator on L 2 (R + ).
Inequality (3.2) shows that an affine localization operator A is a trace-class operator on . Moreover, Proposition 4.13 implies that A is admissible whenever ϕ is admissible and f ∈ L 1 l (Aff) ∩ L 1 r (Aff). We will now see that the affine localization operators are naturally unitarily equivalent to the more commonly defined localization operators on the Hardy space H 2 + (R). Recall that the space H 2 + (R) is the subspace of L 2 (R) consisting of elements ψ whose Fourier transform Fψ is supported on R + . Note that the composition DF is a unitary map from H 2 + (R) to L 2 (R + ). An admissible wavelet ξ ∈ H 2 + (R) satisfies by definition that In other words, DFξ ∈ L 2 (R + ) is an admissible function in the sense of Definition 4.1. In [47,Thm. 18.13] the localization operator A ξ f on H 2 + (R), given an admissible wavelet ξ ∈ H 2 + (R) and f ∈ L 1 l (Aff), is defined by where π acts on H 2 + (R) by The next proposition is straightforward and relates operators on the form A ξ f with affine localization operators.  2. In [12], Daubechies and Paul define localization operators in the same way as in [47], except that they use π(−x, a) instead of π(x, a) in (6.1) and consider symbols f on the full affine group Aff F = R × R * . The eigenfunctions and eigenvalues of the resulting localization operators acting on L 2 (R) are studied in detail in [12] when the window is related to the first Laguerre function, and f = χ Ω C where The corresponding inverse problem, i.e. conditions on the eigenfunctions of the localization operator that imply that Ω = Ω C , is studied in [1].
3. Localization operators with windows related to Laguerre functions have also been extensively studied by Hutník, see for instance [29,30,31], with particular emphasis on symbols f depending only on either x or a. When f (x, a) = f (a), it is shown that the resulting localization operator is unitarily equivalent to multiplication with some function γ f . This correspondence allows properties of the localization operator to be deduced from properties of γ f .

Covariant Integral Quantizations
Operators of the form f Aff S form the basis of the study of covariant integral quantizations by Gazeau and his collaborators in [2,6,7,19,20,21]. Apart from differing conventions that we clarify at the end of this section, covariant integral quantizations on Aff are maps Γ S sending functions on Aff to operators given by for some fixed operator S. By varying S we obtain several quantization maps Γ with properties depending on the properties of S. Examples of such quantization procedures with a different parametrization of Aff are studied in [21,7]. Their approach is to define S either by F W (S) or by its kernel as an integral operator, and deduce conditions on this function that ensures the condition 1 Aff S = I L 2 (R + ) .
Example 6.3. The affine Weyl quantization is an example of a covariant integral quantization Γ S , where S is not a bounded operator. It corresponds to choosing S = P Aff by Theorem 3.21.
Remark. The example above leads to a natural question: could there be other operators P such that f Aff P behaves as an affine analogue of Weyl quantization? Since Weyl quantization on R 2n is given by convolving with the parity operator, a natural guess is P ψ(r) = ψ(1/r), ψ ∈ L 2 (R + ).
The resulting quantization Γ P (f ) = f Aff P has been studied by Gazeau and Murenzi in [21,Sec. 7]. It has the advantage that P is a bounded operator, but unfortunately by [21,Prop. 7.5] it does not satisfy the natural dequantization rule f = Γ P (f ) Aff P.
We also mention that Gazeau and Bergeron have shown that this choice of P is merely a special case corresponding to ν = −1/2 of a class P ν of operators defining possible affine versions of the Weyl quantization [7,Sec. 4.5].
In quantization theory one typically wishes that the domain of Γ S contains L ∞ (Aff). This, by Lemma 4.17, leads us to chose S = DT D for some trace-class operator T . In particular, one requires that Γ S (1) = I L 2 (R + ) , which can be easily satisfied as the following proposition shows.  The following result, which is a modification of the remark given at the end of [34], shows a remarkable converse to these observations. Theorem 6.5. Let Γ : L ∞ (Aff) → L(L 2 (R + )) be a linear map satisfying
Given a functionf on Π + and an operator S on L 2 (R + , dr), Gazeau and Murenzi define (note that the adjoint is now with respect to L 2 (R + , dr), not L 2 (R + )) (q, p)U G (q, p)SU G (q, p) * dq dp, where we assume that S satisfies ∞ −∞ ∞ 0 U G (q, p)SU G (q, p) * dq dp = C S · I L 2 (R + ,dr) .
The next proposition is straightforward and shows that Gazeau and Murenzi's framework is easily related to our affine operator convolutions.
Proposition 6.7. Let S be an operator on L 2 (R + , dr). Then D −1 SD is an operator on L 2 (R + , r −1 dr) and where f (x, a) =f (a, 2πx a ) for (x, a) ∈ Aff.

Affine Cohen Class Distributions
The cross-Wigner distribution W (ψ, φ) of ψ, φ ∈ L 2 (R n ) is known to have certain undesirable properties. A typical example is that one would like to interpret W (ψ, φ) as a probability distribution, but W (ψ, φ) is seldom a non-negative function as shown by Hudson in [28]. To remedy this, Cohen introduced in [11] a new class of time-frequency distributions Q f given by where f is a tempered distribution on R 2n . In light of our setup, it is natural to investigate the affine analogue of the Cohen class.
By Proposition 3.7 we get for S = A f that Q S (ψ, φ) = W ψ,φ Aff * Afff , (6.3) which shows that our definition of the affine Cohen class is a natural analogue of (6.2). It is straightforward to verify that Q S (ψ, φ) is a continuous function on Aff for all ψ, φ ∈ L 2 (R + ) and S ∈ L(L 2 (R + )). Since the affine Cohen class is defined in terms of the operator convolutions, we get some simple properties: The statements 1 and 2 in Proposition 6.9 follow from Proposition 4.18 and Corollary 4.5. Statement 3 is a simple calculation and the last statement follows from a short polarization argument.

We have the covariance property
2. If we relax the requirement that S is bounded in Definition 6.8, then it follows from Theorem 3.21 that Q P Aff (ψ) = W ψ Aff for ψ ∈ S (R + ). Hence the affine Wigner distribution can be represented as a (generalized) affine Cohen class operator. If we define an alternative affine Weyl quantization using an operator P as in Section 6.2, then it is clear that Q P gives an alternative Wigner function. See [21,Sec. 7.2] for the case of P ψ(r) = ψ(1/r).
The covariance property (6.4) and some rather weak continuity conditions completely characterize the affine Cohen class, as is shown in the following result.

Relation to the Affine Quadratic Time-Frequency Representations
The signal processing literature contains a wealth of two-dimensional representations of signals. Among them we find the affine class of quadratic time-frequency representations, see [41]. A member of the affine class of quadratic time-frequency representations is a map sending functions ψ on R to a function Q A Φ (ψ) on R 2 given by Φ(t/a, s/a)e 2πix(t−s) ψ(t)ψ(s) dt ds for some kernel function Φ on R 2 . There are clearly a few differences between our setup and the affine class of quadratic time-frequency representations. The domain of the affine class consists of functions on R, whereas the affine Cohen class acts on functions on R + . For a function ψ on R + we therefore define ψ 0 (t) = ψ(t) t > 0 0 otherwise.
Finally, we recall that a function K S defined on R + × R + defines an integral operator S with respect to the measure dt t by The following formal result is straightforward to verify.
Proposition 6.14. Let S be an integral operator with kernel K S and define