> [!tldr] > The **continuous Fourier transform (CFT)** of a function $x(z)$ on $\mathbb{R}$ is another function $X$ (on $\mathbb{R}$) given by $X(f):= \int x(z)\exp(-2\pi izf) ~ dz.$It describes the original function in the **frequency domain**. ### Fourier Transform and Convolution > [!theorem|*] The Convolution Theorem > If $\mathcal{F}$ is the Fourier transform operator and $*$ indicates convolution, then $\mathcal{F}(u *v)=\mathcal{F}u \cdot \mathcal{F}v.$ ^64adae