Applications Fourier inversion theorem

some problems, such differential equations, become easier solve when fourier transform applied. in case solution original problem recovered using inverse fourier transform.

in applications of fourier transform fourier inversion theorem plays critical role. in many situations basic strategy apply fourier transform, perform operation or simplification, , apply inverse fourier transform.

more abstractly, fourier inversion theorem statement fourier transform operator (see fourier transform on function spaces). example, fourier inversion theorem on



f
∈

l

2


(


r


n


)


{\displaystyle f\in l^{2}(\mathbb {r} ^{n})}

shows fourier transform unitary operator on




l

2


(


r


n


)


{\displaystyle l^{2}(\mathbb {r} ^{n})}

.