John,
You can find some more info, in my 2012 FQXI essay, about the nature of Fourier superpositions.
The most important point is this; as it is usually measured, a Gaussian function has the minimum time-bandwidth product. Consider two scenarios, both involving multiplying noisy sinusoids, with a Gaussian "window" function:
Case 1) A single noisy sinusoid, is multiplied by the Gaussian window
Case 2) A sum of several, noisy sinusoids, all too close together in frequency, to be "resolved", once they are multiplied by the Gaussian window.
The Shannon Capacity Theorem states that there is a only a small number of bits of information that can be recovered, from any measurements of such signals. If that number is sufficiently small (low signal to noise ratio), then it will be impossible to distinguish the two cases, impossible to deduce, from any measurements, how many sinusoids were summed together, to create either signal.
In numerical analysis, describing such a signal via Fourier Transforms, is referred to as a non-parametric model; there is no fixed number of parameters, used to describe the signal. However, if an observer knows, a priori, that a certain parametric model, such as a single sinusoid, is in fact valid, then it may be possible to "decide" the values of those parameters (frequency, amplitude, phase), even though it cannot be deduced , from observations, that the single sinusoid is, in fact, the correct model.
Almost all modern communications signaling exploits such parametric models, precisely to enable them to decide upon the information content of messages, that would otherwise be undecidable. Thus, in many operational systems, frequency can be estimated to several orders of magnitude better accuracy, than the Fourier Uncertainty Principle specifies; because the uncertainty principle is not really about the accuracy of measurements, but rather, the inability to deduce the number of signal components (sinusoids) that ought to be measured in the first place.
Rob McEachern