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I am currently reading Digital Image Processing by Gonzalez and Woods and in the the chapter of Filtering in Frequency Domain there is a huge mention about Fourier Transform.

The author writes,

" Fourier's Contribution in this field states that any periodic function can be expressed as the sum of sines and /or cosines of different frequencies each multiplied by a different coefficient (which we now call Fourier Series)"

Can anyone explain me on a basic mathematical level on how this modelling of every periodic function just with sines and/or cosines is possible?

Turing101
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    Your title asks a history question and your final paragraph asks us to give you a mathematical intro. Which is it you're actually after? – Arthur Dec 15 '19 at 08:56
  • Concerning the historical part, it would be more appropriate to post that question at our History of Science and Mathematics site, but actually that question has already been asked. – José Carlos Santos Dec 15 '19 at 09:07
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    Here's one viewpoint: The fourier basis functions are the eigenfunctions of the derivative operator $\frac{d}{dx}$ (assuming the derivative operator is defined on an appropriate space of functions). But the integration by parts formula tells us that the adjoint of $\frac{d}{dx}$ is $-\frac{d}{dx}$ (in a setting where the boundary terms vanish). So the operator $\frac{d}{dx}$ is skew-symmetric, hence it is normal. By the spectral theorem from linear algebra, we would expect (or at least hope) that the eigenfunctions of $\frac{d}{dx}$ form an orthonormal basis. – littleO Dec 15 '19 at 09:08
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    If you're after a quick introduction to what on earth a Fourier transform is, here is a nice youtube video by 3blue1brown. – Arthur Dec 15 '19 at 09:11

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