Definition:Convolution of Real Sequences

Definition 1
Let $\sequence f$ and $\sequence g$ be real sequences.

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

Definition 2
Let $f: \Z \to \R$ and $g: \Z \to \R$ be mappings from the integers to the real numbers.

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