This last post on uncertainty principles will be probably the hardest one for me. As said in my first post, I supervised a Master’s thesis and posed the very vague question

“Why are the uncertainty principles for the windowed Fourier transform and the wavelet transform so different?”

I had different things in mind:

• The windowed Fourier transform can be generalized to arbitrary dimensions easily. Especially, the underlying Weyl-Heisenberg group can be generalized to arbitrary dimensions. Interestingly, the uncertainty principle carries over almost exactly: For the windowed Fourier transform in ${d}$ dimensions, the uncertainty principle reads as

$\displaystyle \text{Var}_g\cdot\text{Var}_{\hat g}\geq \tfrac{d^2}{4}$

and again, this inequality is sharp for the multivariate Gaussians. A generalization of the wavelet transform is by no means canonical. The sprit in one dimension was to use translation and scaling. However, in higher dimensions there are a lot more geometric transformations you can apply: rotations, anisotropic scalings and shearing. Here one has to identify a suitable group of actions and try to carry all things over. The most naive way, which uses isotropic scaling and rotation does lead to uncertainty relations but no function will make these inequalities sharp…

• The lower bound in the Heisenberg uncertainty principle is fixed (for normed ${g}$). However, the lower bound in the affine uncertainty (equation (1) in my previous post) is not fixed (for normed ${f}$). Indeed ${|\langle f',f\rangle|}$ can be arbitrarily small. Hence, a function which makes the inequality sharp may not lead to the minimum product of the corresponding operator variances. For other wavelet-like transformations (i.e. they include some kind of scaling) this is the same.
• The Heisenberg uncertainty principle has a clear and crisp interpretation involving the product of the variances for a function and its Fourier transform. There is no such thing available for the affine uncertainty principle. (In fact, this question was not addressed in the thesis but in the paper “Do uncertainty minimizers attain minimal uncertainty” and the Diploma thesis by Bastian Kanning).

The outcome was the (german) thesis Unschärferelationen für unitäre Gruppendarstellungen (Uncertainty relations for unitary group representations) by Melanie Rehders. As the question is so vague, there could not be one simple answer, but as a result of the thesis, one could say in a nutshell:

“The uncertainty principles are so different because the groups underlying wavelet-like transforms are semidirect products of a matrix group and ${\mathbb{R}^d}$ and hence, the identity can not be an infinitesimal generator and hence, not be a commutator”.

In this post I’ll face the challenge to give some meaning to this sentence.

1. The abstract structure behind

Let me introduce the players in a diagram which I redraw from the thesis:

As you see, we need several algebraic structures (as well as analytical ones).

2. From group representations to integral transforms

First, we need a locally compact group ${G}$, and naturally, this comes with a left invariant measure ${\mu}$, which is called Haar measure. With these tool we can intergrate complex valued functions defined of the group: ${\int_G f(x)d\mu}$ and we may also form the spaces ${L^p(G)}$.

Having the space ${L^2(G)}$, we can define a special representation of the group (remember that a group representation is a description of the group in terms of linear transformations of a vector space, in other words, a group homomorphism from ${G}$ to the space ${GL(V)}$ of linear mappings on some vector space ${V}$). The special representation we use the the so called left regular representation on the space of unitary operators on the space ${L^2(G)}$ (denoted by ${\mathcal{U}(L^2(G))}$. This representation is the mapping ${\pi:G\rightarrow \mathcal{U}(L^2(G))}$ defined by

$\displaystyle \pi(a) f(x) = f(a^{-1}x).$

One easily checks, that this is a homomorphism and the unitarity follows from the left invariance of the Haar measure. One could say, that the group ${G}$ acts on the functions ${f}$ in ${L^2(G)}$ in a unitary way. We now may define an integral transform as follows: For ${\psi\in L^2(G)}$ define

$\displaystyle V_\psi f(a) = \langle f,\pi(a)\psi\rangle_{L^2(G)}. \ \ \ \ \ (1)$

You may compare with the previous two posts, that this gives precisely the windowed fourier transform (for the Weyl-Heisenberg group) and the wavelet transform (for the affine group).

To have convenient properties for the integral transform one need some more conditions

1. Irreducibility, i.e. that the only subspaces of ${L^2(G)}$ which are invariant under every ${\pi(a)}$ are ${\{0\}}$ and ${L^2(G)}$.
2. Square integrability, i.e. that there exists a non-zero function ${\psi\in L^2(G)}$ such that

$\displaystyle \int_G |\langle \psi,\pi(a)\psi\rangle|^2 d\mu < \infty;$

these functions ${\psi}$ are called admissible.

We have the following theorem: \href{Grossmann, Morlet, Paul}} Let ${\pi}$ be a unitary, irreducible, and square integrable representation of a locally compact group ${G}$ on ${L^2(G)}$ and let ${\psi}$ be admissible. Then it holds that the mapping ${V_\psi}$ defined in (1) is a multiple of an isometry. Especially, ${V_\psi}$ has a left-inverse which is (up to a constant) given by its adjoint.

This somehow clarifies the arrow from “group representation” to “integral transform”.

3. From group representations to Lie algebra representations

For a closed linear group ${G}$, i.e. a closed subgroup of ${GL(d,\mathbb{R})}$, one has the associated Lie-Algebra ${\mathfrak{g}}$ defined with the help of the matrix exponential by

$\displaystyle \mathfrak{g} = \{ X\ |\ \text{for all}\ t:\ \exp(tX)\in G\}.$

The corresponding Lie-bracket is the commutator:

$\displaystyle [X,Y] = XY-YX.$

If we now have a representation of our group ${G}$ on some Hilbert space ${H}$ (you may think of ${H = L^2(G)}$ but here we may have any Hilbert space), we may ask if there is an associated representation of the Lie-Algebra ${\mathfrak{g}}$. Indeed there is one which is called the derived representation. To formulate this representation we need the following subspace of ${H}$:

$\displaystyle H_\pi^\infty = \{f\in H\ |\ a\mapsto \pi(a)f\ \text{is a}\ C^\infty\ \text{mapping}\}.$

Theorem 1 Let ${\pi}$ be a representation of a closed linear group ${G}$ in a Hilbert space ${H}$. The mapping ${d\pi}$ defined by

$\displaystyle d\pi(X)f = \lim_{t\rightarrow 0}\frac{\pi(\exp(tX))f - f}{t}$

is a representation of the Lie-Algebra ${\mathfrak{g}}$ on the space ${H_\pi^\infty}$.

This clarifies the arrow from “group representations” to Lie algebra representations.

4. Lie-algebra representations and uncertainty relations

We are now ready to illustrate the abstract path from Lie-algebra representations to uncertainty relations. This path uses the so called infinitesimal generators:

Definition 2 Let ${G}$ be a closed linear group with Lie algebra ${\mathfrak{g}}$ and let ${{X_1,\dots,X_m}}$ be a basis of ${\mathfrak{g}}$. Let ${\pi}$ be a representation of ${G}$ on a complex Hilbert space ${H}$ and let the derived representation ${d\pi}$ be injective. Then, the operators ${T_j= \mathrm{i} d\pi(X_j)}$ are called the infinitesimal generators of ${G}$ with respect to the representation ${\pi}$.

These infinitesimal generators are always self-adjoint. Hence, we may apply Robertson’s uncertainty principle for every two infinitesimal generators for which the commutator does not vanish.

The abstract way, described in the Sections 2, 3 and 4 is precisely how we have derived the Heisenberg uncertainty principle and the affine uncertainty principle in the two previous posts. But now the question remains: Why are they so different?

The so-called commutator tables of the Lie-algebras shed some light on this:

Example 1 (The Heisenberg algebra) The associated Lie algebra to the Weyl-Heisenberg group is the real vector space ${{\mathbb R}^2 \times \mathrm{i} {\mathbb R}}$ with the Lie bracket

$\displaystyle [(\omega,b,i\phi),(\omega',b',\mathrm{i}\phi')] = (0,0,\mathrm{i}(\omega b' - b'\omega)).$

A basis of this Lie algebra is ${(1,0,0)}$, ${(0,1,0)}$, ${(0,0,\mathrm{i})}$ and the three commutators are

$\displaystyle [(1,0,0),(0,1,0)] = (0,0,2\mathrm{i})$

$\displaystyle [(1,0,0),(0,0,\mathrm{i})]=(0,0,0)$

$\displaystyle [(0,1,0),(0,0,\mathrm{i})] = (0,0,0).$

Two facts are important: There is an element which commutes with every other element. In other words: The center of the algebra is one-dimensional and spanned by one of the basis elements. If we remember the three infinitesimal generators ${\mathrm{i} T_\omega}$, ${\mathrm{i} T_b}$ and ${\mathrm{i} T_\tau}$ for the windowed Fourier transform, we observe that they obey the same commutator relations (which is not a surprise…).

Example 2 (The “affine Lie algebra”) The Lie algebra of the affine group ${({\mathbb R}\setminus\{0\})\times {\mathbb R}}$ (with composition ${(a,b)(a',b') = (aa',ab'+b)}$) is ${{\mathbb R}\times {\mathbb R}}$ with Lie bracket

$\displaystyle [(x,y),(x',y')] = (0,xy'-x'y).$

A basis of the Lie algebra is ${(1,0)}$, ${(0,1)}$ and the commutator is

$\displaystyle [(1,0),(0,1)] = (0,1).$

Here, there is no element which commutes with everything, i.e. the center of the Lie algebra is trivial. Of course, the commutator relation resembles the one for the infinitesimal generators ${\mathrm{i} T_a}$ and ${\mathrm{i} T_b}$ for the wavelet transform.

5. Higher dimensional wavelets

Wavelets in higher dimensions are a bit tricky. If one thinks of groups acting on ${{\mathbb R}^d}$ which consist of translation and some thing as dilation one observes that one basically deals with semidirect products of a subgroup ${D}$ of ${GL(d,{\mathbb R})}$ and ${{\mathbb R}^d}$: For ${A\in D}$ and ${b\in{\mathbb R}^d}$ one may transform a function ${f:{\mathbb R}^d\rightarrow{\mathbb C}}$ as

$\displaystyle \pi(A,b)f(x) = |\det(A)|^{-1/2}f(A^{-1}x-b). \ \ \ \ \ (2)$

Indeed this the so called quasiregular representation of the semidirect product of ${D}$ and ${{\mathbb R}^d}$. Two important examples of 2-dimensional wavelet-like transformations are:

Example 3 The “standard” 2-dimensional wavelet transform. One takes the group

$\displaystyle D = \left\{ \begin{bmatrix} a_1 & -a_2\\ a_2 & a_1 \end{bmatrix}\ :\ a_1,a_2\neq 0 \right\}$

which is a combination of rotation and isotropic scaling. Another parametrization is:

$\displaystyle \left\{ a \begin{bmatrix} \cos(\phi) &-\sin(\phi)\\ \sin(\phi) & \cos(\phi) \end{bmatrix}\ :\ a> 0,\ \phi\in [0,2\pi{[}\right\}$

where ${a}$ is the scaling factor and ${\phi}$ is the rotation angle.

Example 4 The shearlet transform bases of the group

$\displaystyle D = \left\{ \begin{bmatrix} a & \sqrt{a}\,s\\ 0 & \sqrt{a} \end{bmatrix}\ : s\in {\mathbb R},\ a>0 \right\}$

which consists of anisotropic scaling by ${a}$ and “shear” by ${s}$.

Doing some more algebra, one observes that the center of the associated Lie algebra of the semidirect product of the form (2) is always trivial and hence, the identity never appears as a commutator. This neat observation shows, that no wavlet-like transformation which bases on a group structure can ever have any uncertainty relation which behaves like

$\displaystyle c\|f\|\leq \text{some product of variances of operators}$

as in the Heiseberg case.

Although this may not be a groundbreaking discovery, this observation and the whole underlying algebra somehow cleared my view on this issue.