Sampling Theorem and Signals Explained to a Mathematician Let $f:\left(-\frac T2,\frac T2\right)\to\mathbb{R}$ for some $T&gt;0$. The Fourier coefficients of $f$ are $$\left\{\begin{matrix}a_0&amp;=&amp;\displaystyle\frac 1T\int_{-T/2}^{T/2}g(t)\;dt\\a_k&amp;=&amp;\displaystyle\frac 2T\int_{-T/2}^{T/2}g(t)\cos\left(k\frac{2\pi}Tt\right)\;dt\\b_k&amp;=&amp;\displaystyle\frac 2T\int_{-T/2}^{T/2}g(t)\sin\left(k\frac{2\pi}Tt\right)\;dt\end{matrix}\right.\;\;\;\;\;(k\in\mathbb{N})$$
and the Fourier polynom of degree $n\in\mathbb{N}$ itself is $$\mathcal{F}_n[f](x):=a_0+\sum_{k=1}^n\left(a_k\cos\left(k\frac{2\pi}Tx\right)+b_k\sin\left(k\frac{2\pi}Tx\right)\right)\tag{1}$$ Now, for any $g\in L^2$, where $$L^2:=\left\{g:(-\pi,\pi)\to\mathbb{C}:g\text{ is continuous almost everywhere}\right\},$$ the complex Fourier coefficients $c_k$ of $g$ satisfy Parseval's identity, i.e. $$\sum_{k\in\mathbb{Z}}\left|c_k\right|^2=\frac{1}{2\pi}\left\|g\right\|_{L^2}^2\tag{2}$$

Now, we can interpret $f$ as a $T$-periodic signal that we want to transmit over a channel. I'm asking myself the following questions:


What is the signal bandwidth?$^1$
What is the frequency spectrum?
What is the channel bandwidth?
What does the sampling theorem state?
I've read that the square of the $k$-th coefficient, i.e. $a_k^2+b_k^2$, is proportional to the "energy contained in this harmonic". I think this is somehow related to $(2)$. But I have no idea what is meant by energy.


I've read all the definitions, but (as I'm a computer science and mathematics student with almost no relation to physics) I wasn't able to answer the questions above by myself.

I know that $(4.)$ has something to do with the fact, that we need to use some kind of discrete Fourier transformation, i.e. we consider an equidistant grid $$x_j=-\pi+j\frac \pi N\;\;\;\text{for }j=0,\ldots,2N$$ and (maybe) use the composite trapezoidal rule to approximate $$c_k=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)e^{-ikx}\;dx\approx\frac{1}{2N}\sum_{j=0}^{2N-1}f\left(x_j\right)\exp\left(-ik\frac{2\pi}Tx_j\right)$$ So, I think the sampling theorem targets the question how huge do I need to choose $N$ in order to reconstruct (what exactly does that mean? If the Fourier polynom has infinitely many summands we cannot reconstruct the original function) the signal and at which size of $N$ it would be pointless to choose an even bigger $N$.



$^1\;\;$My lecture notes distinguish whether or not $\displaystyle\mathcal{F}[f]:=\lim_{n\to\infty}\mathcal{F}_n[f]$ is actually finite (i.e. $\exists k_0\in\mathbb{N}:\forall k\ge k_0:a_k=b_k=0$). They state, that "the signal bandwith is the difference between the lowest and the highest frequency"?  What does that mean? In the first place, I wasn't even sure if signal bandwidth is a property of $f$ or $\mathcal{F}[f]$. Considering $f$, I would not understand why $f$ should have multiple frequencies and how I would calculate them. So, I think we need to consider $\mathcal{F}[f]$. More precisely, I think we need to look at the "frequencies" of the sine and cosine functions in the $k$-th summand in $(1)$. Obviously, these frequencies are both equal to $$\mathcal{f}_k:=\frac kT$$ and independent of $f$. But, I still don't understand how I can determine the "highest" frequency.
"Highest frequency" is shorthand for the highest k/T for which a and b sub k are non-zero.

Considering f, one reason it might consist of the sum of multiple sinusoidal components is that this is a solution to many ordinary linear low-order differential equations.