In functional analysis a convolution is a mathematical operation on two functions \(f\) and \(g\) that produces a third function denoted \(f*g\) which is defined as the integral of the product of the two functions after one is reflected about the \(y\)-axis and shifted.
Definition: For functions \(f\) and \(g\), the convolution \(f*g\) is defined as the integral transform \[ (f*g)(t):=\int _{-\infty}^\infty f(s)g(t-s) \, ds=\int _{-\infty}^\infty f(s-t)g(s) \, ds. \] Since sequences can just be thought of as discrete functions from \(\mathbb{N}\) there is an equivalent discrete definition of the convolution of two sequences.
Definition: For sequences \(x:=\{ x_{n} \}_{n=0}^\infty\) and \(y:=\{ y_{n} \}_{n=0}^\infty\) the convolution \(z:=x*y\) is defined by \[ z_{n}:=\sum_{i=0}^n x_{i}y_{n-i}=\sum_{i=0}^nx_{n-i}y_{i}. \]
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