September 26, 2011

Uncertainty principles for unitary group representations

Posted by Dirk under Math, Signal and image processing | Tags: , , |
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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.

 

September 20, 2011

The uncertainty principle for the one-dimensional wavelet transform

Posted by Dirk under Math, Signal and image processing | Tags: , , |
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1. The affine group behind the wavelet transform

Continuing my previous post on the uncertainty principle for the windowed Fourier transform, we now come to another integral transform: The wavelet transform.

In contrast to the windowed Fourier transform (which analyzes a function with respect to position and frequency) the wavelet transform analyzes a function with respect to position and scale. For a given analyzing function {\psi} and a signal {f}, the wavelet transform is (for {a\neq 0}, {b\in{\mathbb R}}):

\displaystyle W_\psi f(a,b) = \int_{\mathbb R} f(x) \tfrac{1}{\sqrt{|a|}}\psi(\tfrac{x-b}{a})dx.

In the same way, the windowed Fourier transform could be written as inner products of {f} with a translated and modulated window function, the wavelet transform can be written as inner products of {f} with translated and scaled functions {\psi}. And again, these modifications which happen to the analyzing function come from a group.

Definition 1 The affine group {G_{\text{aff}}} is the set {({\mathbb R}\setminus\{0\})\times {\mathbb R}} endowed with the operation

\displaystyle (a,b)(a',b') = (a\,a',a\,b' + b).

Indeed this is group (with identity {(1,0)} and inverse {(a,b)^{-1} = (a^{-1},a^{-1}b)}). The name affine group stems from the fact that the group operation behaves like the composition of one dimensional affine linear functions: For {f(x) = ax+b} and {g(x) = a'\,x+b'} we have {f(g(x)) = (a\,a')\, x + a\,b' + b}.

The affine group admits a representation on the space of unitary operators on {L^2({\mathbb R})}:

\displaystyle \Pi(a,b)f(x) = \tfrac{1}{\sqrt{|a|}}\psi(\tfrac{x-b}{a})

(note the normalizing factor {1/\sqrt{|a|}}).

2. The affine uncertainty principle

I am not sure who has to credited for the group theoretical background behind wavelets however, the two-part paper “Transforms associated to square integrable group representations” by Grossmann, Morlet and Paul has been influential (and can be found, e.g. in the compilation “Fundamental papers in wavelet theory” by Heil and Walnut.

As done by Stephan Dahlke and Peter Maass in “The Affine Uncertainty Principle in One and Two Dimensions” and can proceed in analogy to the windowed Fourier transform and the corresponding Weyl-Heisenberg group and compute the infinitesimal operators: Take the derivative of the representation with the respect to the group parameters and evaluate at the identity:

\displaystyle T_a f(x) := \frac{d}{da}[\Pi(a,b)f(x)|_{(a,b) = (1,0)} = -\tfrac{1}{2}f(x) - xf'(x)

and

\displaystyle T_b f(x) := \frac{d}{db}[\Pi(a,b)f(x)|_{(a,b) = (1,0)} = -f'(x).

Again, these operators are skew adjoint and hence, multiplying by {\mathrm{i}} gives self-adjoint operators.

These operators {\mathrm{i} T_b} and {\mathrm{i} T_a} do not commute and hence, applying Robertson’s uncertainty principle gives an inequality. The commutator of {\mathrm{i} T_a} and {\mathrm{i} T_b} is

\displaystyle [\mathrm{i} T_a,\mathrm{i} T_b] f(x) = f'(x).

Robertson’s uncertainty principle reads as

\displaystyle \tfrac{1}{2}|\langle f',f\rangle| \leq \|(\mathrm{i} T_a - \mu_1 I)f\|\,\| (\mathrm{i} T_b - \mu_2 I)f\| \ \ \ \ \ (1)

and with some manipulation this turn to (for {\|f\|=1})

\displaystyle \tfrac{1}{4}|\langle f',f\rangle|\leq (\|f'\|^2 - \mu_2^2)(\|xf'\|^2 -\tfrac{1}{4} - \mu_1^2). \ \ \ \ \ (2)

Again, one can derive the functions for which equality in attained and these are the functions of the form

\displaystyle f(x) = c(x-\mathrm{i} \lambda) ^{-1/2 + \mathrm{i}\lambda\mu_2+\mathrm{i}\mu_1}

for real {\lambda}. (By the way, these functions are indeed wavelets and sometimes called Cauchy-wavelets because of their analogy with the Cauchy kernel from complex analysis.)

By the way: These functions are necessarily complex valued. If one restricts oneself to real valued functions there is a simpler inequality, which one may call “real valued affine uncertainty”. First, observe that {\langle f',f\rangle = 0} for real valued {f}, and hence, the left hand side in (1) is zero (which make the inequality a bit pointless). Using that for real valued {f} we have {\langle T_a f,f\rangle = 0}, and that {\|T_a f\|^2 = \|xf'\| - \tfrac{1}{4}\|f\|^2} together with {\|(\mathrm{i} T_b -\mu_2)f\|^2\neq 0} for {f\neq 0} we obtain (with {\mu_1=0}) from (1)

\displaystyle \tfrac{1}{2} \|f\| \leq \|xf'\|.

Since we know that equality is only attained for the Cauchy wavelets (which are not real valued we can state:

Corollary 2 (Real valued affine uncertainty) For any real valued function which is in the domain of {[T_a,T_b]} it holds that

\displaystyle \tfrac{1}{2} \|f\| < \|xf'\|.

As some strange curiosity, one can derive this “real valued affine uncertainty principle” by formal integration by parts and Cauchy-Schwarz inequality totally similar to the Heisenberg uncertainty principle (as I’ve done in my previous post):

\displaystyle \begin{array}{rcl} \|g\|_2^2 & =& \int_{\mathbb R} 1\cdot|g(x)|^2dx\\ & = &-\int_{\mathbb R} x\tfrac{d}{dx}|g(x)|^2dx\\ & = &-2\int_{\mathbb R} xg(x)g'(x)dx\\ & \leq &2\int_{\mathbb R} |xg'(x)|\,|g(x)|dx\\ & \leq &2\Big(\int_{\mathbb R} |xg'(x)|^2dx\Big)^{1/2}\Big(\int_{\mathbb R} |g(x)|^2dx\Big)^{1/2}. \end{array}

Dividing by {2\|g\|} gives the “real valued affine uncertainty” (but only in the non-strict way).

 

September 16, 2011

The uncertainty principle for the windowed Fourier transform

Posted by Dirk under Math, Signal and image processing | Tags: , , |
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Some years ago I became fascinated by uncertainty principles. I got to know them via signal processing and not via physics, although, from a mathematical point of view they are the same.

I recently supervised a Master’s thesis on this topic and the results clarified a few things for me which I used to find obscure and I’d like to illustrate this here on my blog. However, it takes some space to introduce notation and to explain what it’s all about and hence, I decided to write a short series of posts, I try to explain, what new insights I got from the thesis. Here comes the first post:

1. The Fourier transform and the windowed Fourier transform

Let’s start with an important tool from signal processing you all know: The Fourier transform. For {f:{\mathbb R}\rightarrow{\mathbb C}} the Fourier transform is

\displaystyle \hat f(\omega) = (2\pi)^{-1/2}\int_{\mathbb R} f(x) \mathrm{e}^{-\mathrm{i} x\omega} dx.

(I was tempted to say “whenever the integral is defined”. However, the details here are a little bit more involved, but I will not go into detail here; {hat f} is defined for {L^2}-functions, for {L^1} functions and even for tempered distributions…) Roughly speaking, the Fourier transform decomposes a signal into its frequency components, which can be seen from the Fourier inversion formula:

\displaystyle f(x) = (2\pi)^{-1/2}\int_{\mathbb R} \hat f(\omega) \mathrm{e}^{\mathrm{i} x\omega} d\omega,

i.e. the (complex) number {f(\omega)} says “how much the frequency {\omega} (i.e. the function {x\mapsto \mathrm{e}^{\mathrm{i} x\omega}}) contributes to {f}”. In the context of signal processing one often speaks of the “time representation” {f} and the “frequency representation” {\hat f}.

One drawback of the Fourier transform, when used to analyze signals, is its “global” nature in that the value {\hat f(\omega)} depends on every value of {f}, i.e. a change of {f} in a small interval results a change of all of {\hat f}. A natural idea (which is usually attributed to Gabor) is, to introduce a window function {g} which is supposed to be a bump function, centered at zero, then translate this function and “localize” {f} by multiplying it with {g(\cdot-t)}. The resulting transform

\displaystyle G_gf(\omega,t) = (2\pi)^{-1/2}\int_{\mathbb R} f(x)g(x-t)\mathrm{e}^{-ix\omega}dx

is called windowed Fourier transform, short-time Fourier transform or (in the case of {g(x) = \mathrm{e}^{-x^2/2}}) Gabor transform.

Of course we can write the windowed Fourier transform in term of the usual Fourier transform as

\displaystyle G_gf(\omega,t) = \widehat{(f\,g(\cdot-t))}(\omega). \ \ \ \ \ (1)

In other words: The localization in time is precisely determined by the “locality” of {g}, that is, how well {g} is concentrated around zero. The better {g} is concentrated around {0}, the more “localized around {t}” is the information of {f}, the windowed Fourier transform {G_gf(\omega,t)} uses.

For the localization in frequency one obtains (by Plancherel’s formula and integral substitution) that

\displaystyle G_gf(\omega,t) = \mathrm{e}^{-\mathrm{i} x\omega}\widehat{(\hat f\,\hat g(\cdot-\omega)}(-x).

In other words: The localization in frequency is precisely determined by the “locality” of {\hat g}, that is, how well {\hat g} is concentrated around zero. The better {\hat g} is concentrated around {0}, the more “localized around {\omega}” is the information of {\hat f}, the windowed Fourier transform {G_gf(\omega,t)} uses.

Hence, it seems clear that a function {g} is well suited as a window function, if it both well localized in time and frequency. If one measures the localization of a function around zero by its variance

\displaystyle \text{Var}_g = \int_{\mathbb R} x^2|g(x)|^2 dx ,

then there is the fundamental lower bound on the product of the variance of a function and the variance of its Fourier transform, know under the name “Heisenberg uncertainty principle” (or, as I learned from Wikipedia, “Gabor limit”): For {\|g\|_{L^2}=1} it holds that

\displaystyle \text{Var}_g\cdot\text{Var}_{\hat g}\geq \tfrac14.

Proof: A simple (not totally rigorous) proof goes like this: We use partial integration, the Cauchy-Schwarz inequality and the Plancherel formula:

\displaystyle \begin{array}{rcl} \|g\|_2^2 & =& \int_{\mathbb R} 1\cdot|g(x)|^2dx\\ & = &-\int_{\mathbb R} x\tfrac{d}{dx}|g(x)|^2dx\\ & = &-2\int_{\mathbb R} xg(x)g'(x)dx\\ & \leq &2\int_{\mathbb R} |xg(x)|\,|g'(x)|dx\\ & \leq &2\Big(\int_{\mathbb R} |xg(x)|^2dx\Big)^{1/2}\Big(\int_{\mathbb R} |g'(x)|^2dx\Big)^{1/2}\\ & = &2\Big(\int_{\mathbb R} |xg(x)|^2dx\Big)^{1/2}\Big(\int_{\mathbb R} |\omega \hat g(\omega)|^2d\omega\Big)^{1/2}\\ & = &2(\text{Var}_g)^{1/2}(\text{Var}_{\hat g})^{1/2}. \end{array}

\Box

Moreover, the inequality is sharp for the functions {g(x) = C\mathrm{e}^{-\lambda x^2}} for {\lambda>0}. In this sense, these Gaussians are best suited for the windowed Fourier transform.

While this presentation was geared towards usability, there is a quite different approach to uncertainty principles related to integral transforms which uses the underlying group structure.

2. The group behind the windowed Fourier transform

The windowed Fourier transform (1) can also be seen as taking inner products of {f} with the family of functions {g(\cdot-t)\mathrm{e}^{-\mathrm{i} x\omega}}. This family is obtained from the single function {g} by letting the so-called Weyl-Heisenberg group act on it:

Definition 1 The Weyl-Heisenberg group {G_{WH}} is the set {{\mathbb R}\times{\mathbb R}\times S^1} endowed with the operation

\displaystyle (\omega,b,\tau)(\omega',b'\tau') = (\omega+\omega',b+b',\tau\tau'\mathrm{e}^{\mathrm{i} (\omega b'-\omega' b)/2)}).

The Weyl-Heisenberg group admits a representation of the space of unitary operators on {L^2({\mathbb R})}, that is a map {\Pi:G_{WH}\rightarrow U(L^2({\mathbb R}))}

\displaystyle \Pi(\omega,b,\tau)f(x) = \tau\mathrm{e}^{-\mathrm{i} \omega b/2}\mathrm{e}^{\mathrm{i}\omega x}f(x-b).

It indeed the operators {\Pi(\omega,b,\tau)} are unitary and it holds that

\displaystyle \Pi(\omega,b,\tau)\Pi(\omega',b',\tau')f(x) = \Pi((\omega,b,\tau)(\omega',b',\tau'))f(x).

Moreover, the mapping {(\omega,b,\tau)\mapsto \Pi(\omega,b,\tau)f} is continuous for all {f}, a property of the representation which is called strong continuity.

In this light, the windowed Fourier transform can be written as

\displaystyle G_g f(\omega,t) = (2\pi)^{-1/2} \langle f,\Pi(\omega,b,\mathrm{e}^{\mathrm{i}\omega b/2})g\rangle.

Now there is a motivation for the uncertainty principle as follows: Associated to the Weyl-Heisenberg group there is the Weyl-Heisenberg algebra, a basis of which is given by the so called infinitesimal generators. These are, roughly speaking, the derivatives of the representation with respect to the group parameters, evaluated at the identity. In the Weyl-Heisenberg case:

\displaystyle T_\omega f(x) := \frac{d}{d\omega}[\Pi(\omega,b,\tau)f(x)|_{(\omega,b,\tau) = (0,0,1)} = \mathrm{i} xf(x)

and

\displaystyle T_b f(x) := \frac{d}{db}[\Pi(\omega,b,\tau)f(x)|_{(\omega,b,\tau) = (0,0,1)} = -f'(x)

and

\displaystyle T_\tau f(x) := \frac{d}{d\tau}[\Pi(\omega,b,\tau)f(x)|_{(\omega,b,\tau) = (0,0,1)} = \mathrm{i} f(x).

(In the last case, my notation was not too good: Note that {\tau = \mathrm{e}^{\mathrm{i} t}} with {t\in[0,2\pi[} and the derivative has to be taken with respect to {t}.)

All these operators are skew adjoint on {L^2({\mathbb R})} and hence the operators

\displaystyle \mathrm{i} T_\omega f(x) = -xf(x),\quad \mathrm{i} T_b f(x) = -\mathrm{i} f'(x),\quad \mathrm{i} T_\tau f(x) = -f(x)

are self adjoint.

For any two (possibly unbounded) operators on a Hilbert space there is a kind of abstract uncertainty principle (apparently sometimes known as Robertson’s uncertainty principle. It uses the commutator {[A,B] = AB-BA} of two operators:

Theorem 2 For any two self adjoint operators {A} and {B} on a Hilbert space it holds that for any {f} in the domain of definition of {[A,B]} and any real numbers {\mu_1} and {\mu_2} it holds that

\displaystyle \tfrac12|\langle [A,B] f,f\rangle|\leq \|(A-\mu_1 I)f\|\,\|(B-\mu_2)f\|.

Proof: The proof simply consists of noting that

\displaystyle \begin{array}{rcl} |\langle [A,B] f,f\rangle| &=& |\langle B f,Af\rangle -\langle Af,BS f\rangle|\\ & =& |\langle (B-\mu_2 I) f,(A-\mu_1 I)f\rangle -\langle (A-\mu_1 I)f,(B-\mu_2 I)S f\rangle|\\ & =& |2\text{Im} \langle (B-\mu_2 I) f,(A-\mu_1 I)f\rangle|\\ & \leq &2|\langle (B-\mu_2 I) f,(A-\mu_1 I)f\rangle|. \end{array}

Now use Cauchy-Schwarz to obtain the result. \Box

Looking closer at the inequalities in the proof, one infers in which cases Robertson’s uncertainty principle is sharp: Precisely if there is a real {\lambda} such that

\displaystyle (A-\mu_1 I)f = -\mathrm{i}\lambda (B-\mu_2 I)f. \ \ \ \ \ (2)

Now the three self-adjoint operators {\mathrm{i} T_\omega}, {\mathrm{i} T_b} and {\mathrm{i} T_\tau} have three commutators but since {\mathrm{i} T_\tau} is a multiple of the identity and hence commutes with the others. I.e. there is only one commutator relation:

\displaystyle [\mathrm{i} T_b,\mathrm{i} T_\omega] f = \mathrm{i} f.

Hence, using {A=\mathrm{i} T_b}, {B = \mathrm{i} T_\omega} and {\mu_1=\mu_2=0} in Robertson’s uncertainty principle gives (in sloppy notation)

\displaystyle \tfrac12 \|f\|^2\leq \|f'\|_2\|xf\|_2

which is exactly the Heisenberg uncertainty principle.

Moreover, by (2), equality happens if {f} fulfills the differential equation

\displaystyle -\mathrm{i} f'(x) = \mathrm{i}\lambda x f(x)

the solution of which are exactly the functions {f(x) = C\mathrm{e}^{-\lambda x^2/2}}.

Since the Heisenberg uncertainty principle is such an impressing thing with a broad range of implications (a colleague said that its interpretation, consequences and motivation somehow form a kind of “holy grail” in some communities), one may try to find other cool uncertainty principles be generalizing the approach of the previous sections to other transformations.

In the next post I am going to write about other group related integral transforms and its “uncertainty principles”.