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.