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I'd like to know if there are any applications of Fourier analysis / Fourier series expansion in General relativity ? I mean how Fourier transform has applications in Quantum mechanics.

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In dealing with gravitational waves, yes. A quick Google search came up with lots of hits, e.g.

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Fourier Analysis has many applications in Nonlinear PDEs, for example, Nonlinear Schrödinger Equation, a very often used method is Hardy-Littlewood decomposition, to get the well-posedness(existence, uniqueness and some kind of dependence on the initial data) of the solution. A good book is "Nonlinear Dispersive Equation" by Terence Tao.

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