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Why are Fourier series important? Are there any real life applications of Fourier series?

Fourier series is a way of representing a periodic function as a (possibly infinite) sum of sine and cosine functions. It is analogous to a Taylor series, which represents functions as possibly infinite sums of monomial terms.
A sawtooth wave represented by a successively larger sum of trigonometric terms.
For functions that are not periodic, the Fourier series is replaced by the Fourier transform. For functions of two variables that are periodic in both variables, the trigonometric basis in the Fourier series is replaced by the spherical harmonics. The Fourier series, as well as its generalizations, are essential throughout the physical sciences since the trigonometric functions are eigenfunctions of the Laplacian, which appears in many physical equations.
Real-life applications:
  1. Signal Processing. It may be the best application of Fourier analysis.
  2. Approximation Theory. We use Fourier series to write a function as a trigonometric polynomial.
  3. Control Theory. The Fourier series of functions in the differential equation often gives some prediction about the behavior of the solution of a differential equation. They are useful to find out the dynamics of the solution.
  4. Partial Differential equation. We use it to solve higher order partial differential equations by the method of separation of variables.

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