Lecture 5 Analytic Functions

We saw in the first section that an inspiring geometry of cycles can be recovered from the properties of SL₂(ℝ). In this section we consider a realisation of the function theory within Erlangen approach [170, 172, 175, 176]. The covariant transform will be our principal tool in this construction.

5.1 Induced Covariant Transform

The choice of a mother wavelet or fiducial operator F from Section 4.1 can significantly influence the behaviour of the covariant transform. Let G be a group and H be its closed subgroup with the corresponding homogeneous space X=G/H. Let be a representation of G by operators on a space V, we denote by H the restriction of to the subgroup H.

The following is the main motivating example.

The image of induced covariant transform have the similar property:

Thus it is enough to know the value of the covariant transform only at a single element in every coset G/H in order to reconstruct it for the entire group G by the representation χ. Since coherent states (wavelets) are now parametrised by points homogeneous space G/H they are referred sometimes as coherent states which are not connected to a group [219], however this is true only in a very narrow sense as explained above.

Functions on the group G, which have the property v(gh)=v(g)χ(h) (2), provide a space for the representation of G induced by the representation χ of the subgroup H. This explains the choice of the name for induced covariant transform.

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.

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

There is a statement which extends Corollary 6 from differential operators to integro-differential ones. We will formulate it for the wavelets setting.

Clearly, the Corollary 6 is a particular case of the Corollary 11 with a distribution a, which is a combination of derivatives of Dirac’s delta functions. The last Corollary will be illustrated at the end of Section 1.2.

5.2 Induced Wavelet Transform and Cauchy Integral

We again use the general scheme from Subsection 3.2. The ax+b group is isomorphic to a subgroups of SL₂(ℝ) consisting of the lower-triangular matrices:

The corresponding homogeneous space X=SL₂(ℝ)/F is one-dimensional and can be parametrised by a real number. The natural projection p:SL₂(ℝ)→ ℝ and its left inverse s: ℝ→ SL₂(ℝ) can be defined as follows:

Thus we calculate the corresponding map r: SL₂(ℝ)→ F, see Subsection ??:

Therefore the action of SL₂(ℝ) on the real line is exactly the Möbius map (1):

We also calculate that

To build an induced representation we need a character of the affine group. A generic character of F is a power of its diagonal element:

Thus the corresponding realisation of induced representation (5) is:

The only freedom remaining by the scheme is in a choice of a value of number κ and the corresponding functional space where our representation acts. At this point we have a wider choice of κ than it is usually assumed: it can belong to different hypercomplex systems.

One of the important properties which would be nice to have is the unitarity of the representation (6) with respect to the standard inner product:

A change of variables x=au+b/cu+d in the integral suggests the following property is necessary and sufficient for that:

A mother wavelet for an induced wavelet transform shall be an eigenvector for the action of a subgroup H′ of SL₂(ℝ), see (1). Let us consider the most common case of H′=K and take the infinitesimal condition with the derived representation: dZnw₀ =λ w₀, since Z (13) is the generator of the subgroup K. In other word the restriction of w₀ to a K-orbit should be given by e^{λ t} in the exponential coordinate t along the K-orbit. However we usually need its expression in other “more natural” coordinates. For example [195], an eigenvector of the derived representation of dZn should satisfy the differential equation in the ordinary parameter x∈ℝ:

The equation does not have singular points, the general solution is globally defined (up to a constant factor) by:

To avoid multivalent functions we need 2π-periodicity along the exponential coordinate on K. This implies that the parameter m=−iλ is an integer. Therefore the solution becomes:

The corresponding wavelets resemble the Cauchy kernel normalised to the invariant metric in the Lobachevsky half-plane:

Therefore the wavelet transform (3) from function on the real line to functions on the upper half-plane is:

Introduction of a complex variable z=u+i v allows to write it as:

According to the general theory this wavelet transform intertwines representations Fκ (6) on the real line (induced by the character a^κ of the subgroup F) and Km (8) on the upper half-plane (induced by the character e^{i m t} of the subgroup K).

5.3 The Cauchy-Riemann (Dirac) and Laplace Operators

Ladder operators L^±=±i A +B act by raising/lowering indexes of the K-eigenfunctions w_m,κ (9), see Subsection 3.3. More explicitly [195]:

There are two possibilities here: m±κ is zero for some m or not. In the first case the chain (12) of eigenfunction w_m,κ terminates on one side under the transitive action (16) of the ladder operators; otherwise the chain is infinite in both directions. That is, the values m=∓κ and only those correspond to the maximal (minimal) weight function w_{∓κ ,κ} (x)=1/(x±i)^κ ∈ L₂(ℝ), which are annihilated by L^±:

By the Cor. 6 for the mother wavelets w_{∓κ ,κ} , which are annihilated by (13), the images of the respective wavelet transforms are null solutions to the left-invariant differential operator D_±=L^L±:

This is a conformal version of the Cauchy–Riemann equation. The second order conformal Laplace-type operators Δ₊=L^L−L^L+ and Δ₋=L^L+L^L− are:

For the mother wavelets w_m,κ in (13) such that m=∓κ the unitarity condition κ+κ=2, see (7), together with m∈ℤ implies κ=∓ m=1. In such a case the wavelet transforms (11) are:

for w_−1,1 and w_1,1 respectively. The first one is the Cauchy integral formula up to the factor 2πi √ℑ z. Clearly, one integral is the complex conjugation of another. Moreover, the minimal/maximal weight cases can be intertwined by the following automorphism of the Lie algebra sl₂:

As explained before f^±(w) are null solutions to the operators D_± (14) and Δ_± (15). These transformations intertwine unitary equivalent representations on the real line and on the upper half-plane, thus they can be made unitary for proper spaces. This is the source of two faces of the Hardy spaces: they can be defined either as square-integrable on the real line with an analytic extension to the half-plane, or analytic on the half-plane with square-integrability on an infinitesimal displacement of the real line.

For the third possibility, m±κ≠ 0, there is no an operator spanned by the derived representation of the Lie algebra sl₂ which kills the mother wavelet w_m,κ. However the remarkable Casimir operator C=Z²−2(L⁻L⁺+L⁺L⁻), which spans the centre of the universal enveloping algebra of sl₂ [321]*§ 8.1 [240]*§ X.1, produces a second order operator which does the job. Indeed from the identities (12) we get:

Thus we get dCκ w_m,κ =0 for κ=2 or 0. The mother wavelet w_0,2 turns to be the Poisson kernel [114]*Ex. 1.2.17. The associated wavelet transform

consists of null solutions of the left-invariant second-order Laplacian, image of the Casimir operator, cf. (15):

Another integral formula producing solutions to this equation delivered by the mother wavelet w_m,0 with the value κ=0 in (17):

Furthermore, we can introduce higher order differential operators. The functions w_{∓ 2m+1,1} are annihilated by n-th power of operator dL^±κ with 1≤ m≤ n. By the Cor. 6 the the image of wavelet transform (11) from a mother wavelet ∑₁ⁿ a_m w_{∓ 2m,1} will consist of null-solutions of the n-th power D_±ⁿ of the conformal Cauchy–Riemann operator (14). They are a conformal flavour of polyanalytic functions [16].

We can similarly look for mother wavelets which are eigenvectors for other types of one dimensional subgroups. Our consideration of subgroup K is simplified by several facts:

• The parameter κ takes only complex values.
• The derived representation does not have singular points on the real line.

For both subgroups and N′ this will not be true. The further consideration will be given in [195].

5.4 The Taylor Expansion

Consider an induced wavelet transform generated by a Lie group G, its representation and a mother wavelet w which is an eigenvector of a one-dimensional subgroup H′⊂ G. Then by Prop. 5 the wavelet transform intertwines with a representation H′ induced by a character of H′.

If the mother wavelet is itself in the domain of the induced wavelet transform then the chain (16) of H′-eigenvectors w_m will be mapped to the similar chain of their images ŵ_m. The corresponding derived induced representation dH′ produces ladder operators with the transitive action of the ladder operators on the chain of ŵ_m. Then the vector space of “formal power series”:

is a module for the Lie algebra of the group G.

Coming back to the case of the group G=SL₂(ℝ) and subgroup H′=K. Images ŵ_m,1 of the eigenfunctions (10) under the Cauchy integral transform (16) are:

They are eigenfunctions of the derived representation on the upper half-plane and the action of ladder operators is given by the same expressions (12). In particular, the sl₂-module generated by ŵ_1,1 will be one-sided since this vector is annihilated by the lowering operator. Since the Cauchy integral produces an unitary intertwining operator between two representations we get the following variant of Taylor series:

For two other types of subgroups, representations and mother wavelets this scheme shall be suitably adapted and detailed study will be presented elsewhere [195].

5.5 Wavelet Transform in the Unit Disk and Other Domains

We can similarly construct an analytic function theories in unit disks, including parabolic and hyperbolic ones [191]. This can be done simply by an application of the Cayley transform to the function theories in the upper half-plane. Alternatively we can apply the full procedure for properly chosen groups and subgroups. We will briefly outline such a possibility here, see also [170].

Elements of SL₂(ℝ) can be also represented by 2× 2-matrices with complex entries such that, cf. Example 24:

This realisations of SL₂(ℝ) (or rather SU(2,ℂ)) is more suitable for function theory in the unit disk. It is obtained from the form, which we used before for the upper half-plane, by means of the Cayley transform [191]*§ 8.1.

We may identify the unit disk ⅅ with the homogeneous space SL₂(ℝ)/T for the unit circle T through the important decomposition SL₂(ℝ)∼ ⅅ×T with K=T—the compact subgroup of SL₂(ℝ):

where

Each element g∈SL₂(ℝ) acts by the linear-fractional transformation (the Möbius map) on ⅅ and T H₂(T) as follows:

In the decomposition (21) the first matrix on the right hand side acts by transformation (22) as an orthogonal rotation of T or ⅅ; and the second one—by transitive family of maps of the unit disk onto itself.

The representation induced by a complex-valued character χ_k(z)=z^−k of T according to the Section 3.2 is:

The representation 1 is unitary on square-integrable functions and irreducible on the Hardy space on the unit circle.

We choose [173, 175] K-invariant function v₀(z)≡ 1 to be a vacuum vector. Thus the associated coherent states

are completely determined by the point on the unit disk u=βα⁻¹. The family of coherent states considered as a function of both u and z is obviously the Cauchy kernel [170]. The wavelet transform [170, 173] W:L₂(T)→ H₂(ⅅ): f(z)↦ Wf(g)=⟨ f,v_g ⟩ is the Cauchy integral:

This approach can be extended to arbitrary connected simply-connected domain. Indeed, it is known that Möbius maps is the whole group of biholomorphic automorphisms of the unit disk or upper half-plane. Thus we can state the following corollary from the Riemann mapping theorem:

If a domain is non-simply connected, then the group of its biholomorphic mapping can be trivial [257, 27]. However we may look for a rich group acting on function spaces rather than on geometric sets. Let a connected non-simply connected domain D be bounded by a finite collection of non-intersecting contours Γ_i, i=1,…,n. For each Γ_i consider the isomorphic image G_i of the SL₂(ℝ) group which is defined by the Corollary 13. Then define the group G=G₁× G₂× … × G_n and its action on L₂(∂ D)= L₂(Γ₁)⊕ L₂(Γ₂)⊕ … ⊕ L₂(Γ_n) through the Moebius action of G_i on L₂(Γ_i).

Thus all classical objects of complex analysis (the Cauchy-Riemann equation, the Taylor series, the Bergman space, etc.) for a rather generic domain D can be also obtained from suitable representations similarly to the case of the upper half-plane [170, 175].