Discrete Fourier Transform Properties
Properties1
Let’s denote the Discrete Fourier Transform as $\mathscr{F}_{N}$ or $\hat{\mathbf{a}}$ given $\mathbf{a} \in \mathbb{C}^{N}$.
Convolution
$$ \mathscr{F}_{N}(\mathbf{a} \ast \mathbf{b}) = \hat{\mathbf{a}} \hat{\mathbf{b}} = (\hat{a}_{0}\hat{b}_{0}, \dots, \hat{a}_{N-1}\hat{b}_{N-1}) $$
In this case, $\ast$ is the discrete convolution.
Explanation
The Discrete Fourier Transform also satisfies the properties that the Fourier Transform does.
Gerald B. Folland, Fourier Analysis and Its Applications (1992), p251 ↩︎