Lecture 5 Wavelet Transform, Uncertainty Relation and Analyticity

There are two and a half main examples of reproducing kernel spaces of analytic function. One is the Fock–Segal–Bargmann (FSB) space and others (one and a half) – the Bergman and Hardy spaces on the upper half-plane. The first space is generated by the Heisenberg group [197]*§ 7.3 [104]*§ 1.6, two others – by the group SL₂(ℝ) [197]*§ 4.2 (this explains our way of counting).

Those spaces have the following properties, which make their study particularly pleasant and fruitful:

1. There is a group, which acts transitively on functions’ domain.
2. There is a reproducing kernel.
3. The space consists of holomorphic functions.

Furthermore, for FSB space there is the following property:

1. The reproducing kernel is generated by a function, which minimises the uncertainty for coordinate and momentum observables.

It is known, that a transformation group is responsible for the appearance of the reproducing kernel [5]*Thm. 8.1.3. This paper shows that the last two properties are equivalent and connected to the group as well.

5.1 Induced Wavelet (Covariant) Transform

The following object is common in quantum mechanics [181], signal processing, harmonic analysis [205], operator theory [201, 204] and many other areas [197]. Therefore, it has various names [5]: coherent states, wavelets, matrix coefficients, etc. In the most fundamental situation [5]*Ch. 8, we start from an irreducible unitary representation of a Lie group G in a Hilbert space H. For a vector f∈ H (called mother wavelet, vacuum state, etc.), we define the map W_f from H to a space of functions on G by:

Under the above assumptions, v′(g) is a bounded continuous function on G. The map W_f intertwines (g) with the left shifts on G:

Thus, the image W_f H is invariant under the left shifts on G. If is square integrable and f is admissible [5]*§ 8.1, then v′(g) is square-integrable with respect to the Haar measure on G. Moreover, it is a reproducing kernel Hilbert space and the kernel is k(g)= [W_f f](g). At this point, none of admissible vectors has an advantage over others.

It is common [197]*§ 5.1, that there exists a closed subgroup H⊂ G and a respective f∈ H such that (h) f=χ(h) f for some character χ of H. In this case, it is enough to know values of v′(s(x)), for any continuous section s from the homogeneous space X=G/H to G. The map v ↦ v′(x)=v′(s(x)) intertwines with the representation χ in a certain function space on X induced by the character χ of H [159]*§ 13.2. We call the map

the induced wavelet transform [197]*§ 5.1.

For example, if G=ℍ, H={(s,0,0)∈ℍ: s∈ℝ} and its character χ_ℏ(s,0,0)= e^{2πiℏ s}, then any vector f∈L₂(ℝ) satisfies ℏ(s,0,0) f=χ_ℏ(s) f for the representation (16). Thus, we still do not have a reason to prefer any admissible vector to others.

5.2 The Uncertainty Relation

In quantum mechanics [104]*§ 1.1, an observable (that is, a self-adjoint operator on a Hilbert space H) A produces the expectation value Ā on a pure state (that is, a unit vector) φ∈ H by Ā=⟨ Aφ,φ ⟩. Then, the dispersion is evaluated as follow:

The next theorem links obstructions of exact simultaneous measurements with non-commutativity of observables.

Proof. The proof is well-known [104]*§ 1.3, but it is short, instructive and relevant for the following discussion, thus we include it in full. We start from simple algebraic transformations:

Then by the Cauchy–Schwartz inequality:

The equality holds if and only if (B−b)u and (A−a)u are proportional by a purely imaginary scalar.

The famous application of the above theorem is the following fundamental relation in quantum mechanics. We use [199]*(3.5) the Schrödinger representation (16) of the Heisenberg group (16):

Elements of the Lie algebra h, corresponding to the infinitesimal generators X and Y of one-parameters subgroups (0,t/(2π),0) and (0,0,t) in ℍ, are represented in (7) by the (unbounded) operators M′ and D′ on L₂(ℝ):

In the Schrödinger model of quantum mechanics, f(q)∈L₂(ℝ) is interpreted as a wave function (a state) of a particle, with M=i M′ and 1/i D′ are the observables of its coordinate and momentum.

Proof. The relation follows from the commutator [M,D]=iℏ I, which, in turn, is the representation of the Lie algebra h of the Heisenberg group. By Thm. 1, the minimal uncertainty state in the Schrödinger representation is a solution of the differential equation: (M−i r D)φ=0 for some r∈ℝ, or, explicitly:

The solution is the Gaussian φ(q)= e^{−c q2}, c=1/2rℏ. For c>0, this function is in the state space L₂(ℝ).

5.3 The Gaussian

The Gaussian is the crucial element in the theory of the Heisenberg group and will repeatedly use its remarkable properties. For this reasons we devote a separate section to their description.

Cor. 2 identifies the Gaussian φ(q)= e^{−c q2} as the vector with minimal uncertainty. It is also common to say that the Gaussian φ(q)= e^{−c q2} represents the ground state, which minimises the uncertainty of coordinate and momentum.

5.4 Right Shifts and Analyticity

To discover some preferable mother wavelets, we use the following general result from [197]*§ 5. Let G be a locally compact group and be its representation in a Hilbert space H. Let [W_f v](g)=⟨ v,(g)f ⟩ be the wavelet transform defined by a vacuum state f∈ H. Then, the right shift R(g): [W_f v](g′)↦ [W_f v](g′g) for g∈ G coincides with the wavelet transform [W_fg v](g′)=⟨ v,(g′)f_g ⟩ defined by the vacuum state f_g=(g) f. In other words, the covariant transform intertwines right shifts on the group G with the associated action on vacuum states, cf. (2):

Although, the above observation is almost trivial, applications of the following corollary are not.

Some applications (including discrete one) produced by the ax+b group can be found in [205]*§ 6. We turn to the Heisenberg group now.

This example shows, that the Gaussian is a preferred vacuum state (as producing analytic functions through FSB transform) exactly for the same reason as being the minimal uncertainty state: the both are derived from the identity (ℏ c M+i D) e^{− c q2/2}=0.

5.5 Uncertainty and Analyticity

The main result of this paper is a generalisation of the previous observation, which bridges together Cor. 11 and Thm. 1. Let G, H, and H be as before. Assume, that the homogeneous space X=G/H has a (quasi-)invariant measure dµ(x) [159]*§ 13.2. Then, for a function (or a suitable distribution) k on X we can define the integrated representation:

which is (possibly, unbounded) operators on (possibly, dense subspace of) H. It is a homomorphism of the convolution algebra L₁(G,dg) to an algebra of bounded operators on H. In particular, R(k) denotes the integrated right shifts, for H={e}.

Proof. This is an immediate consequence of a combination of Thm. 1 and Cor. 11.

Example 4 is a particular case of this theorem with k₁(x,y)=δ′_x(x,y) and k₂(x,y)=δ′_y(x,y) (partial derivatives of the delta function), which represent vectors X and Y from the Lie algebra h. The next example will be of this type as well.

5.6 Hardy Space on the Real Line

We consider the induced representation 1 (8) for k=1 of the group SL₂(ℝ). A SL₂(ℝ)-quasi-invariant measure on the real line is | cx+d |⁻² d x. Thus, the following form of the representation (8)

is unitary in L₂(ℝ) with the Lebesgue measure dx.

We can calculate the derived representations for the basis of sl₂ (13):

The linear combination of the above vector fields producing ladder operators L^±=±i A+B are, cf. (17):

Obviously, the function f₊(x)=(x+i)⁻¹ satisfies dL⁺1 f₊=0. Recalling the commutator [A,B]=−1/2Z we note that dZ1f₊=−i f₊. Therefore, there is the following identity for dispersions on this state:

with the minimal value of uncertainty among all eigenvectors of the operator dZ1.

Furthermore, the vacuum state f₊ generates the induced wavelet transform for the subgroup K={ e^tZ  ∣  t∈ℝ}. We identify SL₂(ℝ)/K with the upper half-plane ℂ₊={z∈ℂ  ∣  ℑ z > 0} [197]*§ 5.5 [204]. The map s: SL₂(ℝ)/K → SL₂(ℝ) is defined as s(x+i y)=1/√y (

) (8). Then, the induced wavelet transform (3) is:

Clearly, this is the Cauchy integral up to the factor √y, which is related to the conformal metric on the upper half-plane. Similarly, we can consider the operator B−i A1=1/2 ( (x±i)· I +(x± i)²·∂ _x ) and the function f₋(z)=1/x−i simultaneously solving the equations B−i A1 f₋=0 and dZ1f₋=i f₋. It produces the integral with the conjugated Cauchy kernel.

Finally, we can calculate the operator (12) annihilating the image of the wavelet transform. In the coordinates (x+i y,t)∈ (SL₂(ℝ)/K)× K, the restriction to the induced subrepresentation is, cf. [240]*§ IX.5:

Then, the left-invariant vector field corresponding to the ladder operator contains the Cauchu–Riemann operator as the main ingredient:

Furthermore, if L^{−i A+B}v′(x+i y)=0, then (∂_x+i∂_y)(v′(x+i y)/√y)=0. That is, V(x+i y)=v′(x+i y)/√y is a holomorphic function on the upper half-plane.

Similarly, we can treat representations of SL₂(ℝ) in the space of square integrable functions on the upper half-plane. The irreducible components of this representation are isometrically isomorphic [197]*§ 4–5 to the weighted Bergman spaces of (purely poly-)analytic functions on the unit disk, cf. [329]. Further connections between analytic function theory and group representations can be found in [170, 197].

5.7 Contravariant Transform and Relative Convolutions

For a square integrable unitary irreducible representation and a fixed admissible vector ψ∈ V, the integrated representation (15) produces the contravariant transform M_ψ: L₁(G) → V, cf. [173, 204]:

The contravariant transform M_ψ intertwines the left regular representation Λ on L₂(G) and :

Combining with (2), we see that the composition M_ψ ∘ W_φ of the covariant and contravariant transform intertwines with itself. For an irreducible square integrable and suitably normalised admissible φ and ψ, we use the Schur’s lemma [5, Lem. 4.3.1], [159, Thm. 8.2.1] to conclude that:

Similarly to induced wavelet transform (3), we may define integrated representation and contravariant transform for a homogeneous space. Let H be a subgroup of G and X=G/H be the respective homogeneous space with a (quasi-)­invariant measure dx [159, § 9.1]. For the natural projection p: G → X we fix a continuous section s: X → G [159, § 13.2], which is a right inverse to p. Then, we define an operator of relative convolution on V [171, 204], cf. (15):

with a kernel k defined on X=G/H. There are many important classes of operators described by (24), notably pseudodifferential operators (PDO) and Toeplitz operators [139, 171, 173, 204]. Thus, it is important to have various norm estimations of (k). We already mentioned a straightforward inequality ||(k)||≤ C ||k||₁ for k∈L₁(G,dg), however, other classes are of interest as well.

5.8 Norm Estimations of Relative Convolutions

If G is the Heisenberg group and is its Schrödinger representation, then (â) (24) is a PDO a(X,D) with the symbol a [139, 104, 204], which is the Weyl quantization (6) of a classical observable a defined on phase space ℝ². Here, â is the Fourier transform of a, as usual. The Calderón–Vaillancourt theorem [320, Ch. XIII] estimates ||a(X,D)|| by L_∞-norm of a finite number of partial derivatives of a.

In this section we revise the method used in [139, § 3.1] to prove the Calderón–Vaillancourt estimations. It was described as “rather magical” in [104, § 2.5]. We hope, that a usage of the covariant transform dispel the mystery without undermining the power of the method. 0.2cm

We start from the following lemma, which has a transparent proof in terms of covariant transform, cf.  [139, § 3.1] and [104, (2.75)]. For the rest of the section we assume that is an irreducible square integrable representation of an exponential Lie group G in V and mother wavelet φ,ψ ∈ V are admissible.

Proof. We know from (23) that M_φ∘ W_(g)φ = ⟨ φ,(g)φ ⟩ I on V, thus:

On the other hand, the intertwining properties (2) and (11) of the wavelet transform imply:

Integrating the identity Φ (g) (g)= M_φ∘ (Λ ⊗ R)(g) ∘W_φ with the function fΦ⁻¹ and use the partial isometries W_φ and M_φ we get the inequality.

The Lemma is most efficient if Λ ⊗ R acts in a simple way. Thus, we give he following

For a Lie group G with the Lie algebra g define the Lie algebra h=[g,g]. The subgroup H=exp(h) (as well as any larger subgroup) has the complemented commutator property (26). Of course, X=G/H is non-trivial if H≠ G and this happens, for example, for a nilpotent G. In particular, for the Heisenberg group, its centre has the complemented commutator property.

Note, that the complemented commutator property (26) implies:

For a character χ of the subgroup H, we introduce an integral transformation :L₁(X)→ C(G):

where h(x,g)=g⁻¹s(x)⁻¹g s(x) is in H due to the relations (26). This transformation generalises the isotropic symbol defined for the Heisenberg group in [139, § 2.1].

Proof. For an induced representation χ [159, § 13.2], the covariant transform W_φ maps V to a space L₂^χ(G) of functions having the property F(gh)=χ(h)F(g) [204, § 3.1]. From (27), the restriction of Λ⊗ R to the space L₂^χ(G) is:

In other words, Λ⊗ R acts by multiplication on L₂^χ(G). Then, integrating the representation Λ ⊗ R over X with a function k we get an operator (L⊗ R)(k), which reduces on the irreducible component to multiplication by the function k(g). Put k=fΦ⁻¹ for Φ=W_φφ. Then, from the inequality (25), the norm of operator χ(f) can be estimated by ||Λ ⊗ R(f Φ⁻¹)||=||fΦ⁻¹||_∞.

For a nilpotent step 2 Lie group, the transformation (28) is almost the Fourier transform, cf. the case of the Heisenberg group in [139, § 2.1]. This allows to estimate ||fΦ⁻¹||_∞ through ||f||_∞, where f is in the essence the symbol of the respective PDO. For other groups, the expression g⁻¹s(x)⁻¹g s(x) in (28) contains non-linear terms and its analysis is more difficult. In some circumstance the integral Fourier operators [320, Ch. VIII] may be useful for this purpose.