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I Know that Fourier transform states that any non-periodic function could be described as summation of sines and cosines by saying that $$F(w)=\int_{-\infty }^{\infty }f(x)e^{^{-iwt}}dt$$

And this was derived from Fourier series by saying that any non-periodic function is a periodic one provided that the period goes to infinity, and by saying that you can say that any function could be described as a bunch of sines and cosines and that's why we transform our function to the frequency domain

But I did know recently that the Fourier transform is a projection of our function on the orthogonal basis which is $e^{^{-iwt}}$ and by saying that you mean that we show that the frequency inside our function by projecting it on another basis function

So by that we say that we can represent the frequency inside our function by projecting it or by saying that is a Fourier series which has an infinite period

My question is: which one of these ways is the correct one to think about Fourier transform?

Abdelrahman
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First of all, let us make a distinction between Fourier transform and Fourier series. The former, the Fourier transform, is what is written in the question: $$ F(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i \omega t} \mathrm{d} t $$ The anti-transform expresses the function $f(x)$ as an integral, roughly, summing sines and cosines with a continuous distribution of frequencies: $$ f(t) = N \int_{-\infty}^{\infty} F(\omega) e^{i \omega t} \mathrm{d} t $$ (here $N$ is a normalization factor). This $f(x)$ is not periodic at all.

The latter, the Fourier series, is the sum of sinusoidals and cosinusoidals with frequencies that are multiple of a given frequency. Something like this: $$ f(t) = \sum_n a_n e^{i \omega_0 n t} $$ Here the frequencies are multiples of $\omega_0$. The resulting $f(t)$ is periodic, with period $2\pi/\omega_0$.

[...] we can represent the frequency inside our function by projecting it or by saying that is a Fourier series which has an infinite period

For sure, the second is wrong. If the function is periodic, we can express it in Fourier series. If we are interested in a function on an interval, we can pretend that it is periodic, by repeating it. If the function is defined on the whole real axis, then we must use the Fourier transform. In both cases, we are projecting the function (our vector) on an orthonormal basis.

Very roughly, we can imagine that the sum in the Fourier series approaches an integral when the step between frequencies vanishes. This happens when $\omega_0$ tends to 0, which means that the period tends to infinite. In this limit, we can very roughly say that the Fourier series approaches an integral, so it becomes similar to a Fourier antitransform... But this is more science fiction than mathematics.

Doriano Brogioli
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    But the derivation I did know about Fourier transform is just by deriving it from the Fourier series by taking the period to tend to infinity – Abdelrahman Jan 03 '24 at 13:02
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    Well, maybe my comment on "science fiction" is a bit too much, but it is true that there are lots of subtleties more than bringing the period to infinite. For example: what is the space of functions that you can describe? For example, not $x^2$. So, I agree that you can intuitively go from the series to the integral by increasing the period, but I think that this fact does not help us too much. – Doriano Brogioli Jan 03 '24 at 13:08
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    So you say the correct way of looking at Fourier transform is just by projecting our main function on the basis $e^{-iwt}$ and by doing that we see the frequency inside our original function? And this apply to any transform like Laplace transform because there is no Laplace series to derive from it Laplace transform – Abdelrahman Jan 03 '24 at 13:35
  • In the Laplace transform, there is no orthonormal basis .. – LL 3.14 Jan 03 '24 at 13:48
  • @LL3.14, I really have a confusion, I know that Fourier transform represent our function by just a bunch of sines and I Know how he did that by considering transform generalization of the Fourier series, but I didn't know how does Laplace represent a function as exponential and sinusoids, so I searched and found something called projection of a function of a basis and I think about like a projection of a vector, so I rethought about Fourier transform and said it just a projection and so like Laplace transform, but this doesn't seem right, so can you provide me a derivation for Laplace transfor – Abdelrahman Jan 03 '24 at 14:22
  • The Laplace transform is somehow similar to the Fourier transform. It also represents the function as a sum (or integral) of some basis. However, in the case of Laplace transform, the basis is not orthogonal. – Doriano Brogioli Jan 03 '24 at 14:23
  • @DorianoBrogioli, so the main idea of Laplace and Fourier transform is just projection of function on some basis , not that Fourier transform came from Fourier series – Abdelrahman Jan 03 '24 at 14:26
  • For sure, the Fourier anti-transform, the Laplace anti-transform and the Fourier series represent some functions (not all the functions) as sums (or integrals) of some basis. The Fourier has the advantage, over the Laplace transform, that the basis is orthonormal, so the transform and the anti-transform are well defined and very similar. – Doriano Brogioli Jan 03 '24 at 14:28
  • Now, the question can be: "what is the relation between Fourier series and anti-transform?" Then, the answer is that, intuitively, and roughly, you reach the Fourier anti-transform as the limit of the Fourier series. But I feel that the two questions are separate and different. – Doriano Brogioli Jan 03 '24 at 14:29
  • @DorianoBrogioli, so all the transform (Laplace or Fourier) are just representing function by some basis, can you provide me a book that talks in detail about this subject and help me to understand how this transformation works – Abdelrahman Jan 03 '24 at 14:31
  • @DorianoBrogioli,so the main idea of Fourier transform is representing function by some basis, looking at Fourier transform as limit of Fourier series is just another way of looking at it, but the main way of thinking about Fourier transform is representing function in some basis by projecting it, is that correct? – Abdelrahman Jan 03 '24 at 14:34
  • Yes, this is what I think. Of course, it is more an opinion than a rigorous concept! Typically, the topic is explained in books on analysis, in particular where the Hilbert spaces are discussed. However, there are specific books like this: "Fourier and Laplace Transforms", Beerends, Morsche, van den Berg, van de Vrie. – Doriano Brogioli Jan 03 '24 at 14:37
  • It was nice to discuss with you. I hope that my answer was useful. – Doriano Brogioli Jan 03 '24 at 15:30
  • @DorianoBrogioli, I've seen many books and watched lots of videos about the derivation of Fourier transform and it's all about making Fourier transform as the limit of Fourier series, and I Know that the Fourier series is written as the summation of infinite basis,but what does it mean that Fourier transform is integral of the basis ? , isn't the basis must be n-dimensional ? – Abdelrahman Jan 05 '24 at 00:50
  • Obtaining the Fourier transform from the Fourier series is only an analogy, a qualitative idea. If you look at the rigorous theorems about Fourier transform, you see that they never rely on the analogy with Fourier series. The reason is exactly what you mention: in Fourier series, you have a discrete sum, while in Fourier transform you have an integral. The exp(i w t) terms are called the basis. There is not much more than this. There is no need to "derive" the Fourier transform: rather, it is useful to prove its properties. – Doriano Brogioli Jan 06 '24 at 15:16
  • @Mans With respect to the last paragraph I believe my MSO question on the Fourier inversion theorem illustrates the converse of "this is more science fiction than mathematics". But just to be clear, the Fourier series is not equivalent to the Fourier transform, rather the Fourier series coefficients are derived from the Fourier transform. In the case of a normal Fourier series, the Fourier series coefficients are derived from a truncated Fourier transform (see this answer I recently posted on Math StackExchange). – Steven Clark Feb 13 '24 at 20:47
  • @StevenClark, I've never said that the two are equivalent, I'm just saying that in any book I've read considering the Fourier transform, they derived the Fourier transform the same way, they derived the Fourier transform from Fourier series just by taking that the limit as the period goes to infinity, I don't know why you see this as a science fiction, in all the books I've read the derivation is done the same way – Abdelrahman Feb 28 '24 at 14:56
  • @Mans My point was the Fourier series for $f(x)$ as the period goes to infinity is a representation of the function $f(x)$, not the Fourier transform of $f(x)$. This was in response to the last paragraph in the answer above which seemed to claiming the Fourier series for $f(x)$ as the period goes to infinity "becomes similar to a Fourier transform". – Steven Clark Feb 28 '24 at 15:37
  • @StevenClark, Oh I've misunderstood you, you're correct in what you are saying, my attempt is to say that Fourier transform correspond to the $$a_{n}$$ in Fourier series $$ f(t) = \sum_n a_n e^{i \omega_0 n t} $$ ,, which describes the frequency content of the original function $f(t)$ – Abdelrahman Feb 28 '24 at 16:01
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These explanations are not necessarily meant to be rigorous, but may help intuition.

This said, the Fourier transform is indeed a projection onto an orthogonal basis (given that $\displaystyle\int_{-\infty}^\infty e^{i\omega t}e^{i\sigma t}dt=\delta(\omega-\sigma)$), while the idea of an infinite period involves a delicate conversion from discrete to continuous spectrum.

  • Why then all the derivations are done by representing Fourier transform as an series such that it's period goes to infinity? – Abdelrahman Mar 01 '24 at 03:49
  • @Mans: "but may help intuition". –  Mar 01 '24 at 08:23
  • You say that Fourier transform is projection on orthogonal basis, I don't know where Does that came from, the thing I Know (from the derivation) that fourier transformation came from Fourier series – Abdelrahman Mar 01 '24 at 20:36