What does 'essentially supported' mean in uncertainty principle?

Question

The uncertainty princple roughly says, in a heuristic way, that if a function $f$ is supported on a rectangle $T$, then its Fourier transform $\hat{f}$ is 'essentially supported' on the dual rectangle $T^*$. So, what does that mean? How to express it rigorously?

Answer 1

Not sure exactly what that's supposed to mean -- e.g., consider $y(t)=A\sin(2\pi\,\nu_0\,t), -\infty\leq t\leq\infty$ for a pure sine-wave tone (of amplitude $A$ and frequency $\nu_0$) as a function of time $t$. In the conjugate frequency-amplitude space that waveform's represented as the $\delta$-function $y=A\,\delta(\nu-\nu_0)$.

So you'd be saying that if the "rectangle" is the entire $-\infty\leq t\leq\infty$ axis, then the "dual rectangle" is a $\delta$ function? But if the waveform had two frequencies, $\nu_0$ and $\nu_1$, you'd now have two $\delta$ functions but the same $-\infty\leq t\leq\infty$. So that doesn't seem entirely sensible.

Conversely, suppose you had a finite square-wave sound $y(t)=A,\ -t_0\leq t\leq t_0$ (and $y(t)=0$ for all other $t$). Then that requires all frequencies $-\infty\leq\nu\leq\infty$ of various and sundry amplitudes, regardless of the width $2t_0$ of the square-wave, just to cancel out all sound $y(t)=0,\ t<-t_0,\ t>t_0$. So describing that kind of situation as a "rectangle" and "dual rectangle" doesn't seem sensible, either.

Where'd you get that "rectangle" and "dual rectangle" from? Maybe the source formulates that idea a little more precisely?