Fourier Analysis/The Fourier transform

Definition (Fourier transform):

Let $u\in L^{1}(\mathbb {R} ^{n})$. Then the Fourier transform of $u$ is defined to be the function

${\hat {u}}:\mathbb {R} ^{n}\to \mathbb {R} ,{\hat {u}}(\xi ):=\int _{\mathbb {R} ^{n}}e^{-2\pi ix\cdot \xi }u(x)dx$.

Proposition (Fourier transform is defined on L^1):