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The (modified) trigonometric functions $\{0, \cos(kx), \sin(kx)\}$ serves as a basis for periodic function. I have also seen (but not rigorously) that the Fourier transform can also be seen as an extension to the Fourier Series in the sense that we are expressing a signal $x(t)$ in terms of the uncountable basis $\{e^{i(kt)}\}_{k \in \mathbb{R}}$.

Question 2 is my actual question, but question 1 is relevant towards it I think.

Question 1: I have never worked with an uncountable basis before, is $\{e^{i(kt)}\}_{k \in \mathbb{R}}$ really a basis, say under some restrictions on $x(t)$?

Question 2: Suppose that it is a basis. I'm considering the case of a time-frequency decomposition of a signal. Can we still think of the time-varying-complex-exponentials as a basis? Here we think that the frequency spectrum changes with time, so maybe something like $$ x(t) = \sum_{k \in \mathbb{R}} c_k(t) \cdot e^{i(kt + \theta(t))} $$ I am a bit uncomfortable when thinking of the above as a basis because the decomposition of a signal as a sum of its basis may not be unique. For example, if $x(t) = e^{i(2t)}$ can also be represented as $x(t) = e^{i(3t - \theta(t))}$, where $\theta(t):= t$.

Question 3: If not a basis, what then? Is there a comfortable way to think of Fourier transforms for a mathematician?

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  • $\begingroup$ Also: Should this be asked in MathSE or here? $\endgroup$ Commented Nov 16, 2023 at 14:08
  • $\begingroup$ Most of us here are not mathematicians. Personally I have no clue what makes a Mathematician "comfortable" in this regard. For a simple engineer like me the topic is not much of a problem: If your frequency variable is continuous (I guess that's "uncountable") we use integrals. If frequency is discrete (e.g. Discrete FT) you use sums. Otherwise it's more or less the same. $\endgroup$
    – Hilmar
    Commented Nov 16, 2023 at 14:50
  • $\begingroup$ There are some clues here : sp4comm.org/webversion.html including 3.4.3 Infinite Sequences. Not sure it helps though. $\endgroup$
    – audionuma
    Commented Nov 16, 2023 at 14:58

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Question 1: Yes. In fact, the Fourier series becomes countable, although infinite, under restriction, say $[0,2\pi]$, with some exceptions. See this: Schauder basis: Relation to fourier series.

Question 2: One way to think of the fourier transform as bijective universal map $F:\mathbb R^n \to \mathbb C^n$ or even $F:\mathbb C^n \to \mathbb C^n$; everything that can be represented in time can be represented in the frequency. This begs the question, what would you use as a basis for all complex, or even real signals/function? Without restriction, it would have to be uncountable and infinite.

The uncomfortableness that you mention, I think, is universally felt by most people working in the fourier domain (or with complex numbers generally). The periodicity introduced by complex numbers makes things difficult in many ways (easier in others ways). However, one way to make things easier is by thinking about first in terms of its real and imagincary parts$\{1,\cos(kx),i\sin(kx)\}$ as per Euler's formula, or in terms of the frequency and phase $\{1,|e^{ixk}|,\arg(e^{ixk})\}$. Through the latter basis it can be seen that the non-uniqueness problem can be reduced to phase (modulo 2$\pi$) and frequency sign. This ambiguity can be dealt further with more advanced methods.

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  • $\begingroup$ Somehow I missed the fact that I meant $\{e^{i(kt)}\}$ as a basis for non-periodic function, that is what Im asking for actually, is the uncountable set of complex exponentials a basis for non-periodic functions? But I also forgot that you can just work with phase modulo $2\pi$ $\endgroup$ Commented Nov 16, 2023 at 20:22
  • $\begingroup$ Yes, and that's what I was trying to get across in paragraph 2. Those listed in paragraph three are alternative formulations, by Euler's formula and by changing to cartesian to spherical basis respectively. $\endgroup$
    – Davey
    Commented Nov 16, 2023 at 20:25

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