Editing Fourier inversion theorem (section)

==Applications==

[[File:Commutative diagram illustrating problem solving via the Fourier transform.svg|thumb|400px|Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform.]]

In [[Fourier transform#Applications|applications of the Fourier transform]] the Fourier inversion theorem often plays a critical role. In many situations the basic strategy is to apply the Fourier transform, perform some operation or simplification, and then apply the inverse Fourier transform. 

More abstractly, the Fourier inversion theorem is a statement about the Fourier transform as an [[operator (mathematics)|operator]] (see [[Fourier transform#Fourier_transform_on_function_spaces|Fourier transform on function spaces]]). For example, the Fourier inversion theorem on <math>f \in L^2(\mathbb R^n)</math> shows that the Fourier transform is a unitary operator on <math>L^2(\mathbb R^n)</math>.