The Fourier transform is a powerful tool that allows one to express very general types of functions on a symmetric domain (e.g. a locally compact abelian group) in terms of special functions, such as characters. Sometimes, when studying an expression involving some function , it is often useful to expand into Fourier coefficients (e.g. as Fourier series if the domain is the unit circle , or as Fourier integrals if the domain is a Euclidean space ).
Generally speaking, Fourier decomposition is advantageous under one of the following conditions:
Fourier phases already appear in the expression. If the expression already contains terms such as , then it is natural to try to manipulate it using, say, the Fourier inversion formula.
The expression one is studying would be easier to manipulate if was a character. For instance, suppose one was trying to understand an expectation , where were two random variables. For arbitrary , there is not much one can obviously do, but if were an exponential function such as , then it looks like there is a way to separate and from each other. So Fourier analysis could be a useful tool here.
The expression one is studying enjoys some sort of translation invariance. For instance, expressions involving convolution, constant-coefficient differential operators, addition of random variables, addition of sets in a group, or finding patterns such as arithmetic progressions, would fall into this category. Fourier analysis can be interpreted as the representation theory of the translation action, so it is reasonable to suspect that it will be a useful tool in translation-invariant settings.
The function has a particularly simple, or well understood Fourier transform. For instance, if lies in a Sobolev space , then its Fourier transform lies in a weighted space, which is a more elementary object than , so it can be profitable to work in the Fourier domain as much as possible when dealing with problems involving Sobolev spaces.
One expects the "worst case" to be when behaves like a character. In such cases, taking a Fourier-analytic perspective is likely to isolate the dominant features of , while making it easier to control all the other features.
(Counting additive quadruples in a set)
(Bilinear Hilbert transform)
(Kato local smoothing estimate for the Schrodinger equation)
(Central limit theorem)
(Product estimates for - the point is that it converts an oscillatory problem into a non-oscillatory one)
Not every translation-invariant expression is simplified by the Fourier transform. There are some expressions of "quadratic" expressions and higher which seem to require some sort of "higher-order" Fourier analysis (e.g. quadratic Fourier analysis) to control properly. The theory of higher-order Fourier analysis is not nearly as well developed as linear Fourier analysis, but is the subject of ongoing research.