Definition:Convolution of Real Sequences

Definition 1
Let $\sequence f : \N \cup \{0\} \to \R$ and $\sequence g : \N \cup \{0\} \to \R$ be real sequences.

The convolution of $f$ and $g$ is defined as:
 * $\displaystyle \sequence {f_i} * \sequence {g_i} := \sum_{j \mathop = 0 }^i f_j g_{i - j}$

Definition 2
Let $\sequence f : \Z \to \R$ and $\sequence g : \Z \to \R$ be functions similar to sequences.

The convolution of $f$ and $g$ is defined as:
 * $\displaystyle \sequence {f_i} * \sequence {g_i} := \sum_{j \mathop \in \Z } f_j g_{i - j}$