Definition:Spline Function

Definition
Let $\closedint a b$ be a closed real interval.

Let $T : = \set {a = t_0, t_1, t_2, \ldots, t_{n - 1}, t_n = b}$ form a subdivision of $\closedint a b$.

Let $S: \closedint a b \to \R$ be a continuous function on $\closedint a b$ whose values on $t_0, t_1, \ldots, t_n$ are known.

On each of the intervals $\closedint {t_k} {t_{k + 1} }$, let $P_k: \closedint {t_k} {t_{k + 1} }: \R$ be a polynomial function such that:


 * for $t$ on each of $t_k < t < t_{k + 1}$: $\map S t = \map {P_k} t$

The function $S: \closedint a b \to \R$ is known as a spline function on $T$.

Also known as
Some sources refer to this as a polynomial spline.

Others just call it a spline.