Motivated by the study of heat diffusion, Fourier asserted that an arbitrary function in could be represented by a trigonometric series

where

A good part of mathematical analysis developed since then was devoted to the attempt to make Fourier's statement precise. Despite the delicate problems of convergence, Fourier series are a powerful and widely used tool in mathematics, engineering, physics, and other areas. The existence of the Fast Fourier Transform has extended this use enormously in the past thirty years. Fourier expansions are not only useful to study single functions or function spaces, they can also be applied to study operators between function spaces. It is a well-known fact that the trigonometric basis diagonalizes translation invariant operators on the interval , identified with the torus.

However the Fourier system is not adapted to represent local information in time of a function or an operator, since the representation functions themselves are not at all localized in time, we have for all and . A local perturbation of may result in a perturbation of all expansion coefficients . Roughly speaking the same remarks apply to the Fourier transform. The Fourier transform is an ideal tool to study stationary signals and processes (where the properties are statistically invariant over time). However many physical processes and signals are nonstationary, they evolve with time, such as speech or music.

Let us take for instance a short segment of Mozart's Magic Flute (say thirty seconds and the corresponding number of samples, as they are stored on a CD). If we represent this piece of music as a function of time, we may be able to perceive the transition from one note to the next, but we get little insight about which notes are in play. On the other hand the Fourier representation may give us a clear indication about the prevailing notes in terms of the corresponding frequencies, but information about the moment of emission and duration of the notes is masked in the phases. Although both representations are mathematically correct, but one does not have to be a member of the Vienna Philharmonic Orchestra to find neither of them very satisfying. According to our hearing sensations we would intuitively prefer a representation which is local both in time and frequency, like music notation, which tells the musician which note to play at a given moment. Additionally such a local time-frequency representation should be discrete, so that it is better adapted to applications.

Dennis Gabor had similar considerations in mind, when he
introduced in 1946 in his ``Theory of Communication'' a method to represent
a one-dimensional signal in two dimensions, with
time and frequency as coordinates [Gab46].
Gabor's research in communication theory was driven by the question
how to represent locally as good as possible by a finite number of data
the information of a signal which is given a priori through
uncountably many function values . He was
strongly influenced by developments in quantum mechanics, in particular
by Heisenberg's *uncertainty principle* and by
the fundamental results of Nyquist and Hartley.
on the limits for the transmission of information over a channel.

Gabor proposed to expand a function into a series of elementary functions, which are constructed from a single building block by translation and modulation (i.e. translation in the frequency domain). More precisely he suggested to represent by the series

where the elementary functions are given by

for a fixed function and

We could also say that the in (0.2)
are obtained by shifting along a *lattice*
in the *time-frequency plane*.
If and its Fourier transform are essentially localized at
the origin, then is essentially localized at in the
time-frequency plane. Hence each such
elementary function essentially occupies a certain area (``logon'')
in the time-frequency plane. Each of the
expansion coefficients , associated to a certain area of the
time-frequency plane via , represents one *quantum of information*.
For properly chosen shift parameters the cover the
time-frequency plane, as demonstrated in Figure 2.

Gabor proposed to use the Gauss function and its translations and
modulations with shift parameters as elementary signals,
since they ``assure the best utilization of the information
area in the sense that they possess the smallest product of effective
duration by effective width'' [Gab46]. Recall that the
*uncertainty principle inequality* states that for all functions
and for all points
in the
time-frequency plane

where equality is achieved only by functions of the form

i.e., by modulated and translated Gaussians. The Fourier transform of the Gauss function is of the same analytic form, its sharpness is reciprocal.

It is obvious that time series and Fourier series are limiting cases of Gabor's series expansion. The first one may be obtained by letting in (0.3), in which case the approximate the delta distribution , in the second case, the become ordinary sine and cosine waves for .

The idea to represent a function in terms of the time-frequency
shifts of a single atom did not only originate in communication
theory but somewhat 15 years earlier also in quantum mechanics. In an
attempt to expand general
functions (quantum mechanical states) with respect to states with minimal
uncertainty, John von Neumann [vN55] introduced
in 1932 a set of *coherent states* on a lattice with lattice
constants in the *phase space* with position
and momentum as coordinates (here is the
*Planck constant*).
These states, associated with the *Weyl-Heisenberg group*
are in principle the same used by Gabor.
Therefore the system
is also called *Weyl-Heisenberg system*,
and the time-frequency lattice with
lattice constants is also referred
to as *von Neumann lattice*.
We recommend the book of Klauder and Skagerstam for an excellent
review on coherent states [KS85].

Only two years after Gabor's paper, Shannon published ``A
Mathematical Theory of Communication'' [Sha48].
It should be emphasized that the temporal coincidence
is not the only connection between Gabor theory and Shannon's
principles of information theory.
Both, Shannon and Gabor, tried to ``cover'' the time-frequency plane
with a set of functions, transmission signals for digital communication
in Shannon's case and building blocks for natural signals
in Gabor's case. While Gabor explicitly suggested
the Gaussian function and Weyl-Heisenberg structure, Shannon only
emphasized the relevance of orthonormal bases without explicitly suggesting
a signal set design. Yet, the determination of a *critical density*
(referred to as degrees of freedom per time and bandwidth in Shannon's
work) was one of the key mathematical prerequisites for Shannon's famous
Capacity Theorem. In summary, both Gabor
and Shannon worked about the same time on communication engineering
problems related to Heisenberg uncertainty
and phase space density, where at that time only
very few mathematicians, most prominently
von Neumann, had touched upon their basics.
Note, however, that Shannon's work certainly had a greater
impact on the engineering community than the work of Gabor.

Two questions arise immediately with an expansion of the form (0.6):

- Can any be written as superposition of such ?
- How can the expansion coefficients in (0.6) be computed?

While Gabor was awarded the Nobel Prize in Physics in 1971
for the conception of holography, his paper on ``Theory of Communication''
went almost unnoticed until the early 80's, when the work of Bastiaans
and Janssen refreshed the interest of
mathematicians and engineers in Gabor analysis.
The connection to wavelet theory^{1} and the increasing interest of
scientists in signal analysis and frame theory was then
very much influenced by the work
of I. Daubechies.
But before we proceed to the 80's let us go back to the 30's and 40's and
follow the development of Gabor theory from the signal analysis point of view.