Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

Fourier analysis is the study of how general functions can be decomposed into trigonometric or exponential functions with definite frequencies. There are two types of Fourier expansions:

  • Fourier series: If a (reasonably well-behaved) function is periodic, then it can be written as a discrete sum of trigonometric or exponential functions with specific frequencies.
  • Fourier transform: A general function that isn’t necessarily periodic (but that is still reasonably well-behaved) can be written as a continuous integral of trigonometric or exponential functions with a continuum of possible frequencies.

The reason why Fourier analysis is so important is that many (although certainly not all) of the differential equations that govern physical systems are linear, which implies that the sum of two solutions is again a solution. Therefore, since Fourier analysis tells us that any function can be written in terms of sinusoidal functions, we can limit our attention to these functions when solving the differential equations. And then we can build up any other function from these special ones. This is a very helpful strategy, because it is invariably easier to deal with sinusoidal functions than general ones.

Fourier series

Consider a function $f(x)$ that is periodic on the interval $0 ≤ x ≤ L$, then Fourier’s theorem states that $f(x)$ can be written as $$f(x)={a_0}+\sum_{n=1}^{\infty}\left[a_n \cos\left(\frac{2n\pi x}{L}\right)+b_n \sin \left(\frac{2n\pi x}{L}\right)\right]$$ where the constant coefficients $a_n$ and $b_n$ are called the Fourier coefficients of $f$ and is given by $$a_0=\frac{1}{L}\int_0^L f(x)\mathrm{d}x$$ $$a_n=\frac{2}{L}\int_0^L f(x)\cos\left(\frac{2\pi nx }{L}\right)\mathrm{d}x$$ $$b_n=\frac{2}{L}\int_0^L f(x)\sin\left(\frac{2\pi nx }{L}\right)\mathrm{d}x$$


Fourier Transform:

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