Fourier Transform

Definition

Let $f\in L^1({\mathbb{R}}^n)$. We define the fourier transform of $f$ to be $\hat{f}(\xi )={\int }_{{\mathbb{R}}^n}f(x)e^{-2\pi ix\cdot \xi }dx\forall \xi \in {\mathbb{R}}^n$ If $f\in L^1({\mathbb{R}}^n)$ and $\hat{f}\in L^1({\mathbb{R}}^n)$, then we have the following fourier inversion formula $f(x)={\int }_{{\mathbb{R}}^n}\hat{f}(\xi )e^{2\pi ix\cdot \xi }d\xi \forall x\in {\mathbb{R}}^n$

$e^{2\pi ix\cdot \xi }=\cos (2\pi ix\cdot \xi )+i\sin (2\pi ix\cdot \xi )$. If we fix $\xi$ and plot it with respect to $x_1$ and $x_2$, the graph resembles plane waves travelling in the $\xi$ direction with speed $|\xi |$. What the fourier inversion tells us is if we have a "nice'' function then we can represent it as a superposition of plane waves.

Note that