Orthonormal Wave Packet Representation

The Fourier representation of a square integrable function $f$ ( $\in L^2(- \infty ,\infty )$ ) consists of the integral

$$f(t)=\int^\infty_{-\infty}\hat f(\omega )\frac{e^{i\omega t}}{\sqrt{2\pi}}d\omega\,,$$

where

$$\hat f(\omega )=\int^\infty_{-\infty} \frac{e^{-i\omega t}}{\sqrt{2\pi}} f(t)dt\,.$$

The virtue of this representation is that the basis functions

$$u_\omega (t)=\frac{e^{i\omega t}}{\sqrt{2\pi}}~~\qquad~~-\infty <\omega < \infty$$

are translation invariant, i.e.,

$$u_\omega (t+a)=e^{i\omega a}u_\omega (t)\,.$$

It sould be noted that in reality translation invariance is a limiting feature, one that manifests itself after one has taken the limit of some parametrized family of integrable functions, for example,

$$u_\omega (t)\equiv \lim_{T\to\infty}u_\omega (t,T)=\lim_{T\to\infty}e^{-t^2/T^2} \frac{e^{i\omega t}}{\sqrt{2\pi}}~.$$

Thus, although in the limit these basis functions are ``Dirac delta function'' orthonormalized,

$$\int^\infty_{-\infty} \overline{u}_{\omega '} (t)u_\omega (t)dt = \delta (\omega '-\omega )\,,$$

they are not square integrable ( $\not\in L^2(-\infty , \infty )$ ), i.e.,

$$\int^\infty_{-\infty} \vert u_\omega (t)\vert^2 dt = \infty\,.$$

This disadvantage can be overcome if one does not insist on the basis function being translation invariant, i.e. on going to the limit. The benefit accrued consists not only of the basis elements being square integrable, and hence orthonormal in the standard sense, but of the representation being in the form of an infinite series instead of an infinite integral. This means that the Hilbert space of square integrable functions is discrete-dimensional: any element is a linear combination of a countable number of basis elements. A Hilbert space which has a basis which is countable is said to be a separable Hilbert space (with the implication that there are Hilbert spaces which are nonseparable, i.e. do not have a basis which is countable). A separable Hilbert space has the property that any of its elements can be approximated with arbitrary accuracy by a partial Fourier-type sum.

However, we shall find that the largest benefit of a discrete basis representation consists of the fact that it allows one to view the behaviour of a given function, say $f(t)$ , and its Fourier transform $\hat f(\omega)$ from a single point of view: the basis elements reveal the structure of the given function simultaneously in the Fourier $(\omega )$ domain and in the time domain, or in the space domain, whichever the case may be. In practical terms this means that we shall resolve the given function $f(t)$ into a superposition of orthonormal wavepackets which are localized both in the frequency domain and in the time domain, i.e., they have a (mean) frequency and a (mean) location in time. Roughly speaking, each wave packet has the best of both arenas: one foot in the frequency domain and the other foot in the time domain.

By contrast, the Fourier integral representation consists of the given function being resolved into a superposition of infinite wave trains, each one having a definite frequency, but because of their infinite extent, having no definite location. This representation reveals the structure of the function in the Fourier domain, but not in the time domain.