Convolution

1. of functions

Given a pair of functions \(f\) and \(g\) their convolution, written \(f \ast g\), is given by the integral
\[ (f \ast g)(x) = \int f(t)\,g(x-t)\,\mathrm{d}t \]
2. of sequences

Given a pair of sequences \(f\) and \(g\) their convolution, written \(f \ast g\), is given by the series
\[ (f \ast g)_j = \sum_i f_i\,g_{j-i} \]