Convolution

• (discrete) The convolution of two functions $f,g:\mathbb{Z}\to \mathbb{C}$ is a function $f\ast g:\mathbb{Z}\to \mathbb{C}$ defined as $(f\ast g)(n)=\sum _{m\in \mathbb{Z}}f(m)g(n-m).$
• (continuous) The convolution of two functions $f,g:\mathbb{R}\to \mathbb{C}$ is a function $f\ast g:\mathbb{R}\to \mathbb{C}$ defined as $(f\ast g)(x)={\int }_{-\infty }^{\infty }f(t)g(x-t)dt.$