The Fourier transform (Jean-Baptiste Joseph Fourier, 1768 – 1830)

If you have any periodic signal, i.e. a function of time or space (e.g. light, sound, information, etc.) the Fourier transform provides you with the frequency spectrum of that signal.

For any integrable function \(f(x,y)\) with real numbers \(x\) and \(y\) its Fourier transform can be defined by \[ \mathcal{F}(u,v)= \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f(x,y)\, \mathrm{e}^{-2\pi{\rm i}(ux+vy)}\, \mathrm{d}x\mathrm{d}y. \] To get back the function \(f\) from \(\mathcal{F}\), the so-called Fourier synthesis, you can calculate \[ f(x,y) = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \mathcal{F}(u,v)\, \mathrm{e}^{+2\pi{\rm i}(ux+vy)}\, \mathrm{d}x\mathrm{d}y. \] The definition in two or three variables is commonly used for optics and Quantum Mechanics, where it helps to switch between the wave function of position to the wave function of momentum. Besides, this two dimensional definition is particularly suitable for the discussion of moiré patterns, since there you examine mainly two-dimensional images. There exist of course definitions over \(\mathbb{R}^n\) or even over some (compact) manifolds \(\mathcal{M}\) for Lebesgue-measurable functions. In many applications a descrete version is useful, the so called DFT.

The Fourier transfomation appears again and again in a striking manner. For example, if you send X-rays through a crystal and then take a picture on a photoplate, you will get points that exactly match the Fourier transform. In a way, nature performs a Fourier transformation. That's really cool, isn't it?

Further reading and links:
[1]  Lehar, Steven: An Intuitive Explanation of Fourier Theory.
[2]  Sanderson, Grant: But what is the Fourier Transform? A visual introduction. Video on
[3]  Wikipedia, the free encyclopedia: Fourier transform.
[4]  Encyclopedia of Mathematics: Fourier transform.
[5]  Mathematik-Online-Kurs: Fourier-Analysis. Uni Stuttgart.
[6]  Werner, Dirk: Funktionalanalysis. Springer, Berlin 1995.

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