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Fourier analysis began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions. The attempt to understand functions (or other objects) by breaking them into basic pieces that are easier to understand is one of the central themes in Fourier analysis. Fourier analysis is named after Joseph Fourier who showed that representing a function by a trigonometric series greatly simplified the study of heat propagation. If you want to find out more, look up fourier synthesis and the fourier transform.
It is to convert a function into a sum of sine (or cosine) functions so as to simplify its analysis.
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Fourier analysis shows that the saw wave is constructed through manipulation of a sine wave, I can't remember the maths behind it but it's definitely a sine wave.
Physics would be one of a few examples of fourier transform. One would also use it when they are using engineering so, yeah that is basically it as far as the fourier transform is concerned.