Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series. It showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat propagation. The subject of Fourier analysis encompasses a vast spectrum of mathematics.

The decomposition process itself is called a Fourier transform. The transform often gives a more specific name which depends upon the domain and other properties of the function being transformed. The original concept of Fourier analysis has been extended over time to apply to more abstract and general solutions and the general field is often known as harmonic analysis.

Fourier analysis has many scientific applications including areas of physics, partial differential equations, number theory, combinatorics, signal processing, imaging, probability theory and many more. This wide applicability stems from many useful properties of the transforms linear operators, invertibility, exponential functions, convolution theorem and the discrete version. It was also useful as a compact representation of a signal.

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