Lecture 4 Covariant Transform

A general group-theoretical construction [279, 95, 173, 5, 106, 63, 220] of wavelets (or coherent state) starts from an irreducible square integrable representation—in the proper sense or modulo a subgroup. Then a mother wavelet is chosen to be admissible. This leads to a wavelet transform which is an isometry to L₂ space with respect to the Haar measure on the group or (quasi)invariant measure on a homogeneous space.

The importance of the above situation shall not be diminished, however an exclusive restriction to such a setup is not necessary, in fact. Here is a classical example from complex analysis: the Hardy space H₂(T) on the unit circle and Bergman spaces B₂ⁿ(ⅅ), n≥ 2 in the unit disk produce wavelets associated with representations ρ₁ and ρ_n of the group SL₂(ℝ) respectively [170]. While representations ρ_n, n≥ 2 are from square integrable discrete series, the mock discrete series representation ρ₁ is not square integrable [240]*§ VI.5 [321]*§ 8.4. However it would be natural to treat the Hardy space in the same framework as Bergman ones. Some more examples will be presented below.

4.1 Extending Wavelet Transform

To make a sharp but still natural generalisation of wavelets we give the following definition.

We borrow the name for operator F from fiducial vectors of Klauder and Skagerstam [220].

As we will see below covariant transform is a close relative of wavelet transform. The name is chosen due to the following common property of both transformations.

Proof. We have a calculation similar to wavelet transform [173]*Prop. 2.6. Take u=(g) v and calculate its covariant transform:

The next result follows immediately:

4.2 Examples of Covariant Transform

In this Subsection we will provide several examples of covariant transforms. Some of them will be expanded in subsequent sections, however a detailed study of all aspects will not fit into the present work. We start from the classical example of the group-theoretical wavelet transform:

This scheme is typically carried out for a square integrable representation with v₀ being an admissible vector [279, 95, 5, 106, 63, 87]. In this case the wavelet (covariant) transform is a map into the square integrable functions [87] with respect to the left Haar measure on G. The map becomes an isometry if v₀ is properly scaled. Moreover, we are able to recover the input v from its wavelet transform through the reconstruction formula, which requires an admissible vector as well, see Example 28 below. The most popularised case of the above scheme is as follows.

However, square integrable representations and admissible vectors do not cover all interesting cases.

Since linearity has clear advantages, we may prefer to reformulate the last example through linear covariant transforms. The idea is similar to the representation of a convex function as an envelope of linear ones, cf. [108]*Ch. I, Lem. 6.1. To this end, we take a collection F of linear fiducial functionals and, for a given function f, consider the set of all covariant transforms W_F f, F∈F.

4.3 Symbolic Calculi

There is a very important class of the covariant transforms which maps operators to functions. Among numerous sources we wish to single out works of Berezin [34, 35]. We start from the Berezin covariant symbol.

There are several variants of the last Example which are of a separate interest.

The intertwining property in the previous examples was obtained as a consequence of the general Prop. 6 about the covariant transform. However it may be worth to select it as a separate definition:

There is a dual class of covariant transforms acting in the opposite direction: from functions to operators. The prominent examples are the Berezin contravariant symbol [34, 173] and symbols of a pseudodifferential operators (PDO) [139, 173].

In line with the Defn. 23 we can directly define the corresponding calculus through the intertwining property [168, 182]:

The duality between co- and contravariant calculi is the particular case of the duality between covariant transform and the contravariant transform defined in the next Subsection. In many cases a proper choice of spaces makes covariant and/or contravariant calculus a bijection between functions and operators. Subsequently only one form of calculus, either co- or contravariant, is defined explicitly, although both of them are there in fact.

4.4 Contravariant Transform

An object invariant under the left action Λ (2) is called left invariant. For example, let L and L′ be two left invariant spaces of functions on G. We say that a pairing ⟨ ·,· ⟩: L× L′ → ℂ is left invariant if

For a representation of G in V and w₀∈ V, we construct a function w(g)=(g)w₀. We assume that the pairing can be extended in its second component to this V-valued functions. For example, such an extension can be defined in the weak sense.

We can drop out sup/subscripts in M_w0 as we are doing for W_F.

If the invariant pairing is defined by integration over the Haar measure, cf. Example 28, then we can show an intertwining property for the contravariant transform as well.

Let

• be not a square integrable representation (even modulo a subgroup); or
• w₀ be an inadmissible vector of a square integrable representation .

A suitable invariant pairing in this case is not associated with integration over the Haar measure on G. In this case we speak about a Hardy pairing. The following example explains the name.

Similar pairings can be defined for other semi-direct products of two groups. We can also extend a Hardy pairing to a group, which has a subgroup with such a pairing.

Here is an important example of non-linear pairing.

There is the obvious inequality ⟨ f₁,f₂ ⟩_∞ ≥ ⟨ f₁,f₂ ⟩_∞^H between pairings (23) and (27), which produces the corresponding relation between respective contravariant transforms.

There is an explicit duality between the covariant transform and the contravariant transform. Discussion of the grand maximal function in the Rem. 17 shows usefulness of the covariant transform over the a family of fiducial functional. Thus, we shall not be surprised by the contravariant transform over a family of reconstructing vectors as well.

Note, that (30) reduces to the contravariant transform (19) if we start from the constant function w(g)=w₀.

Clearly, for a nucleus r, the function s=1(a,b) r has properties:

1. s is supported in a ball centred at b and radius a,
2. | s |<1/2a almost everywhere, and
3. ∫_ℝ s(x) dx=0.

In other words, s=1(a,b) r is an atom, cf. [314]*§ III.2.2 and any atom may be obtained in this way from some nucleus and certain (a,b)∈Aff.

4.5 Composing the Co- and Contravariant Transforms

In the case of classical wavelets, the relation between the wavelet transform (3) and the inverse wavelet transform (19) is suggested by their names.

It is of interest, that two different vectors can be used as analysing vector in (3) and for the reconstructing formula (19). Even a greater variety can be achieved if we use additional fiducial operators and invariant pairings.

It is the classical question of harmonic analysis to identify a class of functions on the real line such that Mv0*H W becomes the identity operator on it. Combining intertwining properties of the covariant and contravariant transforms (Props. 6, 30 and Rem. 32) we conclude that Mv0*H W will intertwine the representation with itself. If we restrict our attention to -irreducible subspace, then a sort of Schur’s lemma suggests that such an operator is a (possible zero) multiple of the identity operator. This motivates the following template definition, cf. [203]*§ 1.

We illustrate the group-theoretical technique by the following statement.

Proof. First of all we note that Mv0+H WF is a bounded operator on B. Let v(a,b)*=∞(a,b) v*. Obviously, v(a,b)*(0)=v*(−b/a) is an eigenfunction for operators Λ(a′,0), a′∈ℝ+ of the left regular representation of Aff:

This and the left invariance of the pairing (30) imply Mv0*H∘ Λ (1/a,0)=Mv0*H for any (a,0)∈Aff. Then, applying the intertwining properties (44) we obtain:

Using the limit a→ 0 (33) and the continuity of F∘ ∞(1,b) we conclude that the linear functional l:f↦ [Mv0*H∘ WF f](0) vanishes for any f∈B such that f(0)=0. Take a function f1∈B such that f1(0)=1 and define c=l(f1). From linearity of l, for any f∈ B we have:

Furthermore, using the intertwining properties (44) and (20):

This finishes the proof.

To get the classical statement we need the following lemma.

Proof. Let f be a continuous bounded function such that f(0)=0. For ε>0 chose

• δ>0 such that | f(t) |<ε for all | t |<δ;
• M>0 such that ∫_{| t |>M}| w(t) | dt<ε.

Then, for a<δ/M, we have the estimation:

Finally, for a uniformly continuous function g: for ε>0 there is δ>0 such that | g(t+b)−g(t) |<ε for all b<δ and t∈ℝ. Then:

That demonstrates the continuity of F(∞(1,b) g ) at b=0 and, by the group property, at any other point as well.

The table integral ∫_ℝ dx/x²+1=π tells that the “wavelet” p(t)=1/π1/1+t² is in L₁(ℝ) with c=1, the corresponding wavelet transform is the Poisson integral. Its boundary behaviour from Prop. 46 is the classical result, cf. [108]*Ch. I, Cor. 3.2.

The comparison between our demonstrations and the traditional proofs, e.g. in [108], does not reveal any significant distinctions. We simply made an explicit usage of the relevant group structure, which is implicitly employed in traditional texts anyway.

4.5.1 Real and Complex Technique in Harmonic Analysis

There are two main approaches in harmonic analysis on the real line. The real variables technique uses various maximal functions, dyadic cubes and, occasionally, the Poisson integral [314]. The complex variable technique is based on the Cauchy integral and fine properties of analytic functions [264, 265].

Both methods seem to have clear advantages. The real variable technique:

1. does not require an introduction of the imaginary unit for a study of real-valued harmonic functions of a real variable (Occam’s Razor);
2. allows a straightforward generalisation to several dimensions.

By contrast, access to the beauty and power of analytic functions (e.g., Möbius transformations, factorisation of zeroes, etc. [231]) is the main reason to use the complex variable technique. A posteriori, a multidimensional analytic version was also discovered [252], it is based on the monogenic Clifford-valued functions [49].

Therefore, propensity for either techniques becomes a personal choice of a researcher. Some of them prefer the real variable method, explicitly cleaning out any reference to analytic or harmonic functions [314]*Ch. III, p. 88. Others, e.g. [233], happily combine the both techniques. However, the reasons for switching beetween two methds at particular places may look mysterious.

We demonstrated above that both—real and complex—techniques in harmonic analysis have the same group-theoretical origin. Moreover, they are complemented by the wavelet construction. Therefore, there is no any confrontation between these approaches. In other words, the binary opposition of the real and complex methods resolves into Kant’s triad thesis-antithesis-synthesis: complex-real-covariant.