# Fourier Analysis/The Fourier transform

Let ${\displaystyle u\in L^{1}(\mathbb {R} ^{n})}$. Then the Fourier transform of ${\displaystyle u}$ is defined to be the function
${\displaystyle {\hat {u}}:\mathbb {R} ^{n}\to \mathbb {R} ,{\hat {u}}(\xi ):=\int _{\mathbb {R} ^{n}}e^{-2\pi ix\cdot \xi }u(x)dx}$.