A strictly time-limited signal $x(t)$, meaning that $x(t)$ is exactly $0$ outside an interval of duration $T$, must have a Fourier transform $X(f)$ whose support is $(-\infty, \infty)$. Similarly, if the Fourier transform $Y(f)$ of a real-valued signal $y(t)$ is strictly band-limited, meaning that $Y(f)$ is exactly $0$ outside a frequency band $[-W,W]$, then $y(t)$ must have support $(-\infty, \infty)$. More succinctly, strictly time-limited signals cannot be strictly band-limited, and strictly band-limited signals cannot be strictly time-limited. So, if strictness is demanded, the notion of a time-bandwidth product is meaningless; it is always infinite.
On the other hand, suppose that we are a little more lax in what we mean by time-limited and band-limited. Then it is possible to make some progress. One result of this type is what is called the Landau-Pollak theorem.
The late U.S. Supreme Court Justice Potter Stewart once rendered
a decision in which he memorably said
``I may not know how to define it legally, but I know it when I see it.''
He was, of course, speaking of pornography, but the notion of
bandwidth of a signal is very similar. Every engineer understands
the notion of bandwidth, perhaps only at an intuitive level or maybe with more erudition and detail for a specific purpose, but a definition of bandwidth
that is satisfactory for all purposes remains as elusive as ever.
A signal $s(t)$ is said to be strictly time-limited if $s(t)$ is 0
outside a time interval of finite length, e.g.
$s(t) = 0$ if $t < 0$
or $t > T$. It is said to be
strictly band-limited if its Fourier transform $S(f)$
is 0 outside a frequency interval of finite length, e.g.
$S(f) = 0$
if $\vert f\vert > W$, or $S(f) = 0$ if $\vert f\vert > f_c+W/2$,
or $\vert f \vert < f_c - W/2$
corresponding respectively to lowpass or bandpass signals of bandwidth $W$.
Unfortunately, a signal
cannot be both strictly time-limited and strictly band-limited.
Now, let $s(t)$ be a unit-energy strictly time-limited function.
In particular, suppose that $s(t) = 0$ if $t < 0$ or if $t > T$.
Let $\eta$ denote a very small positive number, that is,
$0 < \eta \ll 1$. Let $W$ be a number such that
$$
\int_{-W}^W \vert S(f)\vert^2 \, df > 1 - \eta,
$$
that is, almost all of the energy in $s(t)$ is in the
frequency band $[-W, W]$. We say that $s(t)$ is
essentially band-limited to $W$ Hz or an essentially
low-pass signal of bandwidth $W$ Hz. Now, suppose that
$\{s(t)\}$ denotes a collection of unit-energy signals that are
all strictly time-limited to $[0, T]$ and all essentially band-limited
to $[-W, W]$. How many mutually orthogonal signals does the set
$\{s(t)\}$ contain? The answer is given by a result called
the Landau-Pollak Theorem. The number of orthogonal
signals is less than $2WT/(1 - \eta)$. Since $\eta$ is very
close to 0, the denominator is just slightly less than 1, and so we
conclude that the number of orthogonal signals strictly
time-limited to $[0, T]$ and essentially band-limited to
$W$ Hz is just a tad larger than $M = 2WT$.
Duality says that if the signal is strictly band-limited to
$W$ Hz, then it is essentially time-limited to a duration of
$T$ seconds (that is, ($1-\eta$) of the energy is in that duration)
and there are just a few more than $2WT$ (real-valued) orthogonal signals with this property.
As one might expect, it is possible to ease the "strictly" restriction in one domain to essentially restricted and arrive at the result that
If $2WT$ is defined as the time-bandwidth product of the set of signals that are essentially time-limited to $T$ seconds of duration and also essentially band-limited to $[-W,W]$ Hz of bandwidth, then the set contains just a smidgen more than $2WT$ orthogonal signals.
Now, all the signals in the space of essentially time-limited and essentially band-limited signals can be expressed as a linear combination of the $2WT$ orthogonal complex-valued signals in the space, and the coefficients of the linear combination are the coordinates or representation of a given signal in the space with respect to the basis consisting of these $2WT$ orthogonal signals. So, we can represent (and reproduce if need be from this representation) any such signal from its coordinates with respect to the basis, that is, from $2WT$ complex-valued numbers.
A different notion of time duration and bandwidth of an arbitrary unit-energy signal $x(t)$ comes from noting that $|x(t)|^2$ is a pdf (note that $|x(t)|^2 \geq 0$ for all $t$, and $\int |x(t)|^2 dt = 1$ which are the two defining properties of pdfs). Similarly, $|X(f)|^2$ is also a pdf since $\int |X(f)|^2 df$ is the signal energy which has value $1$. Some folks, especially physicists, like to use the standard deviations of these pdfs as the (root-mean-square) measures $\Delta t$ and $\Delta f$ of the time duration and the bandwidth respectively. I don't like $\Delta t$ and $\Delta f$ because (for example) if these pdfs are Gaussian, then only $68\%$ of the signal energy is contained in the frequency band $[-\Delta f, \Delta f]$ which is a rather poor approximation compared to the (say) $99\%$ essential bandwidth described earlier (when $\eta = 0.01$) that us poor engineers are more concerned with, but ymmv. Anyway, to cut a long story short, the product $\Delta t\cdot\Delta f$ is bounded below )cf. this answer by Matt L. This is a minor variant of the Heisenberg Uncertainty Principle beloved of physicists and OverLordGoldDragons, but I say Meh! Essential notions are more useful for engineers than RMS notions.
