What is the definition of the bandwidth of a signal?
It's hard to give a definition of bandwidth without mentioning Fourier transforms.

I hope the following definition makes the greatest sense for a mathematician. Consider the space of Fourier transform functions $X(\omega)$ whose domain of support is a compact interval in $\omega$ (a single piece or two symmetric pieces rather than being scattered). Then we shall consider two cases:

case one is for real valued functions $x(t)$ whose Fourier transforms $X(\omega)$ are double-sided and conjuage symmetric about $\omega=0$ For such a function $x(t)$ the bandwidth is the length of the domain of support that resides in the positive $\omega$.

case two is for complex valued (analytic) functions $x(t)$ whose Fouerier transforms are single-sided and has zero value for $\omega&lt;0$. For such a function $x(t)$ the bandwidth is the length of the domain of support that resides in the positive $\omega$.

Eventhough these two bandwidths are the same, the necessary Nyquist sampling rates are different; specifically it's twice larger for the real signals than the complex valued case.