Lecture 6 Affine Group: the Real and Complex Techniques in Harmonic Analysis

In this chapter we reviews complex and real techniques in harmonic analysis. We demonstrated that both, real and complex, techniques in harmonic analysis have the same group-theoretical origin: the covariant transform generated by the affine group. Moreover, they are complemented by the wavelet construction. Therefore, there is no any confrontation between these approaches and they can be lined up as in Table 6.1. In other words, the binary opposition of the real and complex methods resolves via Kant’s triad thesis-antithesis-synthesis: complex-real-covariant.

6.1  Introduction

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: among competing hypotheses, the one with the fewest assumptions should be selected);
2. allows a straightforward generalization to several real variables.

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, 68], happily combine the both techniques. However, the reasons for switching between two methds at particular places may look mysterious.

The purpose of the present paper is to revise the origins of the real and complex variable techniques. Thereafter, we describe the common group-theoretical root of both. Such a unification deepens our understanding of both methods and illuminates their interaction.

6.2  Two approaches to harmonic analysis

As a starting point of our discussion, we provide a schematic outline of complex and real variables techniques in the one-dimensional harmonic analysis. The application of complex analysis may be summarised in the following sequence of principal steps:

Integral transforms.

For a function f∈L_p(ℝ), we apply the Cauchy or Poisson integral transforms:

[C f] (x+i y)  =  1 2π i   ∫   ℝ f(t) t−(x+i y)   dt ,                 (1)   [Pf](x, y)  = 1 π ∫   ℝ   y (t−x)2 + y2    f(t)  dt  .                 (2)

An equivalent transformation on the unit circle replaces the Fourier series ∑_k c_k e^{i kt} by the Taylor series ∑_k=0^∞c_k z^k in the complex variable z=r e^{i t}, 0≤ r<1. It is used for the Abel summation of trigonometric series [344]*§ III.6. Some other summations methods are in use as well [245].

Domains.

Above integrals (1)–(2) map the domain of functions from the real line to the upper half-plane, which can be conveniently identified with the set of complex numbers having a positive imaginary part. The larger domain allows us to inspect functions in greater details.

Differential operators.

The image of integrals (1) and (2) consists of functions,belonging to the kernel of the Cauchy–Riemann operator ∂_z and Laplace operator Δ respectively, i.e.:

∂z=

∂ ∂ x

+i

∂ ∂ y

,   Δ=

∂2 ∂ x2

+

∂2 ∂ y2

. (3)

Such functions have numerous nice properties in the upper half-plane, e.g. they are infinitely differentiable, which make their study interesting and fruitful.

Boundary values and SIO.

To describe properties of the initial function f on the real line we consider the boundary values of [C f] (x+i y) or [Pf](x, y), i.e. their limits as y→ 0 in some sense. The Sokhotsky–Plemelj formula provides the boundary value of the Cauchy integral [257]*(2.6.6):

[Cf](x,0)=

1 2

f(x)+

1 2πi

∫

ℝ

f(t) t−x

dt. (4)

The last term is a singular integral operator defined through the principal value in the Cauchy sense:

1 2πi

∫

ℝ

f(t) t−x

dt=

lim ε→ 0

1 2πi

x−ε ∫ −∞

+

∞ ∫ x+ε

f(t) t−x

dt  . (5)

For the Abel summation the boundary values are replaced by the limit as r→ 1⁻ in the series ∑_k=0^∞c_k (r e^{i t})^k.

Hardy space.

Sokhotsky–Plemelj formula (4) shows, that the boundary value [Cf](x,0) may be different from f(x). The vector space of functions f(x) such that [Cf](x,0)=f(x) is called the Hardy space on the real line [264]*A.6.3.

Summing up this scheme: we replace a function (distribution) on the real line by a nicer (analytic or harmonic) function on a larger domain—the upper half-plane. Then, we trace down properties of the extensions to its boundary values and, eventually, to the initial function.

The real variable approach does not have a clearly designated path in the above sense. Rather, it looks like a collection of interrelated tools, which are efficient for various purposes. To highlight similarity and differences between real and complex analysis, we line up the elements of the real variable technique in the following way:

Hardy–Littlewood maximal function

is, probably, the most important component [231]*§ VIII.B.1 [314]*Ch. 2 [108]*§ I.4 [54] of this technique. The maximal function f^M is defined on the real line by the identity:

fM(t)=

sup a>0

⎧ ⎪ ⎨ ⎪ ⎩

1 2a

t+a ∫ t−a

⎪ ⎪

f

⎛ ⎝

x

⎞ ⎠

⎪ ⎪

dx

⎫ ⎪ ⎬ ⎪ ⎭

. (6)

Domain

is not apparently changed, the maximal function f^M is again defined on the real line. However, an efficient treatment of the maximal functions requires consideration of tents [314]*§ II.2, which are parametrised by their vertices, i.e. points (a,b), a>0, of the upper half-plane. In other words, we repeatedly need values of all integrals 1/2a ∫_t−a^t+a | f(x) | dx, rather than the single value of the supremum over a.

Littlewood–Paley theory

 [68]*§ 3 and associated dyadic squares technique [108]*Ch. VII, Thm. 1.1 [314]*§ IV.3 as well as stopping time argument [108]*Ch. VI, Lem. 2.2 are based on bisection of a function’s domain into two equal parts.

SIO

is a natural class of bounded linear operators in L_p(ℝ). Moreover, maximal operator M: f→ f^M (32) and singular integrals are intimately related [314]*Ch. I.

Hardy space

can be defined in several equivalent ways from previous notions. For example, it is the class of such functions that their image under maximal operator (32) or singular integral (5) belongs to L_p(ℝ) [314]*Ch. III.

The following discussion will line up real variable objects along the same axis as complex variables. We will summarize this in Table 6.1.

6.3  Affine group and its representations

It is hard to present harmonic analysis and wavelets without touching the affine group one way or another, e.g. through the doubling condition on the measure, cf. [340]. Unfortunately, many sources only mention the group and do not use it explicitly. On the other hand, it is equally difficult to speak about the affine group without a reference to results in harmonic analysis: two theories are intimately intertwined. In this section we collect fundamentals of the affine group and its representations, which are not yet a standard background of an analyst.

Let G=Aff be the ax+b (or the affine) group [5]*§ 8.2, which is represented (as a topological set) by the upper half-plane {(a,b)  ∣  a∈ℝ₊, b∈ℝ}. The group law is:

As any other group, Aff has the left regular representation by shifts on functions Aff→ ℂ:

A left invariant measure on Aff is dg=a⁻² da db, g=(a,b). By the definition, the left regular representation (8) acts by unitary operators on L₂(Aff,dg). The group is not unimodular and a right invariant measure is a⁻¹ da db.

There are two important subgroups of the ax+b group:

An isometric representation of Aff on L_p(ℝ) is given by the formula:

Here, we identify the real line with the subgroup N or, even more accurately, with the homogeneous space Aff/N [92]*§ 2. This representation is known as quasi-regular for its similarity with (8). The action of the subgroup N in (10) reduces to shifts, the subgroup A acts by dilations.

The representation (10) in L₂(ℝ) is reducible and the space can be split into irreducible subspaces. Following the philosophy presented in the Introduction to the paper [203]*§ 1 we give the following

Some ambiguity in picking the Hardy space out of all (well, two, as we will see below) irreducible components is resolved by the traditional preference.

To clarify a decomposition of L_p(ℝ) into irreducible subspaces of representation (10) we need another realization of this representation. It is called co-adjoint and is related to the orbit method of Kirillov  [162]*§ 4.1.4 [105]*§ 6.7.1. Again, this isometric representation can be defined on L_p(ℝ) by the formula:

Since a>0, there is an obvious decomposition into invariant subspaces of [3]p:

It is possible to demonstrate, that these components are irreducible. This decomposition has a spatial nature, i.e., the subspaces have disjoint supports. Each half-line can be identified with the subgroup A or with the homogeneous space Aff/N.

The restrictions [3]+p and [3]−p of the co-adjoint representation [3]p to invariant subspaces (13) for p=2 are not unitary equivalent. Any irreducible unitary representation of Aff is unitary equivalent either to [3]+2 or [3]−2. Although there is no intertwining operator between [3]+p and [3]−p, the map:

has the property

which corresponds to the outer automorphism (a,b)↦ (a,−b) of Aff.

As was already mentioned, for the Hilbert space L₂(ℝ), representations (10) and (12) are unitary equivalent, i.e., there is a unitary intertwining operator between them. We may guess its nature as follows. The eigenfunctions of the operators 2(1,b) are e^{2πi ω x} and the eigenfunctions of [3]2(1,b) are δ(λ −ω ). Both sets form “continuous bases” of L₂(ℝ) and the unitary operator which maps one to another is the Fourier transform:

Although, the above arguments were informal, the intertwining property F2(a,b)=[3]2(1,b)F can be directly verified by the appropriate change of variables in the Fourier transform. Thus, cf. [264]*Lem. A.6.2.2:

Reflection J (14) anticommutes with the Fourier transform: F J=−J F. Thus, J also interchange the irreducible components +p and −p of quasi-regular representation (10) according to (15).

Summing up, the unique rôle of the Fourier transform in harmonic analysis is based on the following facts from the representation theory. The Fourier transform

• intertwines shifts in quasi-regular representation (10) to operators of multiplication in co-adjoint representation (12);
• intertwines dilations in (10) to dilations in (12);
• maps the decomposition L₂(ℝ)=H₂⊕ H₂^⊥ into spatially separated spaces with disjoint supports;
• anticommutes with J, which interchanges +2 and −2.

Armed with this knowledge we are ready to proceed to harmonic analysis.

6.4  Covariant transform

We make an extension of the wavelet construction defined in terms of group representations. See [159] for a background in the representation theory, however, the only treated case in this paper is the ax+b group.

We may drop the sup/subscripts from W_F if the functional F and/or the representation are clear from the context.

We start the list of examples with the classical case 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], i.e. satisfying the condition:

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 popularized case of the above scheme is provided by the affine group.

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 using 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.

6.5  The contravariant transform

Define the left action Λ of a group G on a space of functions over G by:

For example, in the case of the affine group it is (8). An object invariant under the left action Λ is called left invariant. In particular, 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 on G. 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 did for W_F.

Let either

• 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.

Here is an important example of non-linear pairing.

Note 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 a family of fiducial functionals. 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 the following 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.

6.6  Intertwining properties of covariant transforms

The covariant transform has obtained its name because of the following property.

The covariant transform is also a natural source of relative convolutions [171, 204], which are operators A_k=∫_G k(g)(g) dg obtained by integration a representation of a group G with a suitable kernel k on G. In particular, inverse wavelet transform M_w0 f (19) can be defined from the relative convolution A_f as well: M_w0 f= A_f w₀.

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

There is also a simple connection between a covariant transform and right shifts.

Although the above result is obvious, its infinitesimal version has interesting consequences. Let G be a Lie group with a Lie algebra g and be a smooth representation of G. We denote by dB the derived representation of the associated representation B (46) on fiducial operators.

The moral of the above Examples 7–10 is: there is a significant freedom in choice of covariant transforms. However, some fiducial functionals have special properties, which suggest the suitable technique (e.g., analytic, harmonic, dyadic, etc.) following from this choice.

6.7  Composing the covariant and the contravariant transforms

From Props. 6, 30 and Rem. 32 we deduce the following

Since the image space of M_w ∘ W_F is an Aff-invariant space, we shall be interested in the smallest building blocks with the same property. For the Hilbert spaces, any group invariant subspace V can be decomposed into a direct integral V=⊕∫V_µ dµ of irreducible subspaces V_µ, i.e. V_µ does not have any non-trivial invariant subspace [159]*§ 8.4. For representations in Banach spaces complete reducibility may not occur and we shall look for primary subspace, i.e. space which is not a direct sum of two invariant subspaces [159]*§ 8.3. We already identified such subspaces as generalized Hardy spaces in Defn. 3. They are also related to covariant functional calculus [182] [197]*§ 6.

For irreducible Hardy spaces, we can use the following general principle, which has several different formulations, cf. [159]*Thm. 8.2.1:

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

It is of interest here, 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.

For the affine group, recall the decomposition from Prop. 7 into invariant subspaces L₂(ℝ)=H₂⊕ H₂^⊥ and the fact, that the restrictions +2 and −2 of 2 (10) on H₂ and H₂^⊥ are not unitary equivalent. Then, Schur’s lemma implies:

Of course, Corollary 42 is applicable to the composition of covariant and contravariant transforms. In particular, the constants k1 and k2 may have zero values: for example, the zero value occurs for W (3) with an admissible vector v0 and non-tangential limit Mv0*H (29)—because a square integrable function f(a,b) on Aff vanishes for a→ 0.

In fact, the previous reasons imply the following

The maximal function and SIO are often treated as elementary building blocks of harmonic analysis. In particular, it is common to define the Hardy space as a closed subspace of L_p(ℝ) which is mapped to L_p(ℝ) by either the maximal operator (50) or by the SIO (51) [314]*§ III.1.2 and § III.4.3 [89]. From this perspective, the coincidence of both characterizations seems to be non-trivial. On the contrast, we presented both the maximal operator and SIO as compositions of certain co- and contravariant transforms. Thus, these operators act between certain Aff-invariant subspaces, which we associated with generalized Hardy spaces in Defn. 3. For the right choice of fiducial functionals, the coincidence of the respective invariant subspaces is quite natural.

The potential of the group-theoretical approach is not limited to the Hilbert space L2(ℝ). One of possibilities is to look for a suitable modification of Schur’s Lemma 39, say, to Banach spaces. However, we can proceed with the affine group without such a generalisation. Here is an illustration to a classical question of harmonic analysis: to identify the class of functions on the real line such that Mv0*H W becomes the identity operator on it.

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 pairing (30) imply that Mv0*H∘ Λ (1/a,0)=Mv0*H for any (a,0)∈Aff. Then, applying intertwining properties (44) we obtain that

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 intertwining properties (44) and (20):

This completes 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

This proves 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 (23) 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 of our arguments with 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, cf. [54]. Further demonstrations of this type can be found in [4, 92].

6.8  Transported norms

If the functional F and the representation in (1) are both linear, then the resulting covariant transform is a linear map. If W_F is injective, e.g. due to irreducibility of , then W_F transports a norm ||·|| defined on V to a norm ||·||_F defined on the image space W_F V by the simple rule:

By the very definition, we have the following

A touch of non-triviality occurs if the transported norm can be naturally expressed in the original terms of G.

A reader expects that there are other interesting examples of the transported norms, which are not connected to the Haar integration.

The second possibility to transport a norm from V to a function space on G uses an contravariant transform M_v:

The above result is well-known for traditional wavelets.

In the theory of Hardy spaces the result is also classical.

The co- and contravariant transforms can be used to transport norms in the opposite direction: from a classical space of functions on G to a representation space V.

It was already mentioned in Rem. 17 and Example 39 that we may be interested to mix several different covariant and contravariant transforms. This motivate the following statement.

Proof. Indeed:

by the definition of transported norm (58) and the assumptions M∘W_i=c_i I.

Although the above result is simple, it does have important consequences.

Proof. We can derive (60) from (59) as follows. Let M_f be the inverse wavelet transform (19) defined by the admissible vector f, then M_f∘ W_f=I on V providing the right scaling of f. Furthermore, M_f∘ W_f′=cI by (49) for some complex constant c. Thus, by (59):

Now, through the polarisation identity [164]*Problem 476 we get the equality (60) of inner products.

The above result is known as the orthogonality relation in the theory of wavelets, for some further properties of the constant c see [5]*Thm. 8.2.1.

Here is an application of Prop. 58 to harmonic analysis, cf. [114]*Thm. 4.1.7:

Proof. As we establish in Example 46 for contravariant transform Mv0+H (28), Mv0+H ∘ Wq=−i I and i I on H2 and H2⊥, respectively. Take the unique presentation f=u+u⊥, for u∈ H2 and u⊥∈ H2⊥. Then, by (59)

This completes the proof.

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90

Covariant scheme	Complex variable	Real variable
Covariant transform is WF: v↦ v(g) = F((g−1) v). In particular, the wavelet transform for the mother wavelet v0 is v′(g) = ⟨ v,(g)v0 ⟩.	The Cauchy integral is generated by the mother wavelet 1/2πi1/x+i. The Poisson integral is generated by the mother wavelet 1/π1/x2+1	The averaging operator f′(b)=1/2a∫b−ab+a f(t) dt is defined by the mother wavelet χ[−1,1](t), to average the modulus of f(t) we use all elements of the unit ball in L∞[−1,1].
The covariant transform maps vectors to functions on G or, in the induced case, to functions on the homogeneous space G/H.	Functions are mapped from the real line to the upper half-plane parametrised by either the ax+b-group or the homogeneous space SL2(ℝ)/K.	Functions are mapped from the real line to the upper half-plane parametrised by either the ax+b-group or the homogeneous space SL2(ℝ)/A.
Annihilating action on the mother wavelet produces functional relation on the image of the covariant transform	The operator −dA −i dN=I+(x+i)d/dx annihilates the mother wavelet 1/2πi1/x+i, thus the image of wavelet transform is in the kernel of the Cauchy–Riemann operator −LA+iLN=i a(∂b+i∂a). Similarly, for the Laplace operator.	The mother wavelet v0=χ[−1,1] satisfies the equality χ[−1,1]=χ[−1,0]+χ[0,1], where both terms are again scaled and shifted v0. The image of the wavelet transform is suitable for the stopping time argument and the dyadic squares technique.
An invariant pairing ⟨ ·,· ⟩ generates the contravariant transform [Mw0 f] ⟨ f(g),(g)w0 ⟩ for	The contravariant transform with the invariant Hardy pairing on the ax+b group produces boundary values of functions on the real line.	The covariant transform with the invariant sup pairing produces the vertical and non-tangential maximal functions.
The composition Mv ∘ WF of the covariant and contravariant transforms is a multiple of the identity on irreducible components.	SIO is a composition of the Cauchy integral and its boundary value.	The Hardy–Littlewood maximal function is the composition of the averaging operator and the contravariant transform from the invariant sup pairing.
The Hardy space is an invariant subspace of the group representation.	The Hardy space consists of the limiting values of the Cauchy integral. SIO is bounded on this space.	The Hardy–Littlewood maximal operator is bounded on the Hardy space Hp .

Figure 6.1: The correspondence between different elements of harmonic analysis.

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