I was studying about fourier series, fourier transforms and the sampling theorem recently, when the following question came to my mind.
Suppose I have a function $f : \mathbb R \longrightarrow \mathbb R$, then to communicate this function to another person completely, I would have to communicate its value at all real numbers, so I have to communicate $|\mathbb R|$ numbers (here I am considering a general function, not, for example polynomials which are specified uniquely by their coefficients)
However, suppose that the function is periodic, with period $p$, then I can communicate the values of the function at all points in $[0,p)$, which is still $|\mathbb R|$ numbers.
Alternatively, I can take its fourier series and communicate only its coefficients, and the entire function is uniquely specified, thus having to communicate only $|\mathbb N|$ numbers.
Thus the entire information about $|\mathbb R|$ numbers is captured in $|\mathbb N|$ numbers, which does not seem very intuitive.
One thing that I thought afterwards was that all functions which I have seen being represented as fourier series are nice and mostly continuous functions, which can be uniquely specified by their values on any dense subset of $\mathbb R$, for example the rationals, which have the same cardinality as the naturals, and hence all the data in a continuous function can be represented in only countable number of values. However, I am not very sure about how "ill behaved" a function has to be before its fourier series representation stops existing.
The same question holds for the sampling theorem, that the information about the frequency spectrum of the signal (even though band limited) is captured in countably many numbers (the sampled function)
Is my argument correct? Is there a better way to visualise how all the data contained in uncountably many numbers be captured in countably many numbers?
The reason for this situation that you view as an anomaly is that two functions that are equal almost everywhere w.r.t Lebesgue measure have the same Fourier series or Fourier transform. Each equivalence class of functions that are equal a.e. Lebesgue has size $2^{2^{\aleph_0}}$. So you can have $2^{\aleph_0}$ equivalence classes, with each equivalence class having size $2^{2^{\aleph_0}}$, and still end up with a total set of functions of size $2^{2^{\aleph_0}}$.