What is the definition of the bandwidth of a signal? I was reading over the famous Nyquist–Shannon sampling theorem and I was trying to understand what the term bandwidth meant rigorously (exactly) with no ambiguity. In this site the define it as follow:


  2) Bandwidth is the range of frequencies -- the difference between the highest-frequency signal component and the lowest-frequency signal component -- an electronic signal uses on a given transmission medium. Like the frequency of a signal, bandwidth is measured in hertz (cycles per second). This is the original meaning of bandwidth, although it is now used primarily in discussions about cellular networks and the spectrum of frequencies that operators license from various governments for use in mobile services.


which to me it means that given some periodic function:

$$ f(x) = \frac{1}{2}a_0 + \sum^{\infty}_{n=1} a_n cos(nx) + \sum^{\infty}_{n=1} b_n sin(nx) $$

we just need to extract the frequencies that are the largest and subtract them. For the sake of an example consider:

$$ f(x) = cos(2 \pi x) + cos(4 \pi x ) + sin( 3 \pi x) + sin( 7 \pi x) $$

the frequency of a sin/cos is $f = \frac{n}{2 \pi}$, thus, components of $f$ with the largest and smallest $n$ give the frequencies we want. So the band width $B$ of $f$ is:

$$ B = f_{max} - f_{min} $$

$$ B = \frac{n_{max} }{2 \pi} - \frac{n_{min}}{2 \pi} = \frac{7 }{2 \pi} - \frac{2}{2 \pi} $$

is this algorithm correct? Do I have the correct understanding of bandwitdh in the context of signals and systems?

I think less like an engineer and more like a mathematician so it was important for me to know exactly what bandwidth means in the context of Fourier series since its a totally unambiguous example to me (since all period functions can be expressed like that).

Also, as I was about to tag the question the definition of band width according to this site came up as:


  the difference between the upper and lower frequencies in a contiguous set of frequencies.


this reminded me that a lot of the time I see signals represented in what this community calls "frequency domain", as sometimes a continuous line (i.e. $f(c) = lim_{x \rightarrow c}f(x)$). This seems to be really unintuitive to me because the Fourier series is a countable summation but a continuous frequency domain plot would imply its a uncountable summation of the periodic function. In the light of this, how is "contiguous set of frequencies" a good definition? Or what does "contiguous" mean? 

As you can see I just want to understand what bandwidth means in the end in a mathematical sense.
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.