Definition:Extension of Sequence

Definition

As a sequence is, by definition, also a mapping, the definition of an extension of a sequence is the same as that for an extension of a mapping:

Let:

$\left \langle {a_k} \right \rangle_{k \mathop \in A}$ be a sequence on $A$, where $A \subseteq \N$.
$\left \langle {b_k} \right \rangle_{k \mathop \in B}$ be a sequence on $B$, where $B \subseteq \N$.
$A \subseteq B$
$\forall k \in A: b_k = a_k$.

Then $\left \langle {b_k} \right \rangle_{k \mathop \in B}$ extends or is an extension of $\left \langle {a_k} \right \rangle_{k \mathop \in A}$.

Negative Integers

A sequence on $\N$ can be extended to the negative integers.

Let $\left \langle {a_k} \right \rangle_{k \mathop \in \N}$ and $\left \langle {b_k} \right \rangle_{k \mathop \in \N}$ be sequences on $\N$.

Let $a_0 = b_0$.

Let $c_k$ be defined as:

$\forall k \in \Z: c_k = \begin{cases} a_k & : k \ge 0 \\ b_{-k} : & k \le 0 \end{cases}$

Then $\left \langle {c_k} \right \rangle_{k \mathop \in \Z}$ extends (or is an extension of) $\left \langle {a_k} \right \rangle_{k \mathop \in \N}$ to the negative integers.