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 $T$.
Knot
The points $T := \set {t_0, t_1, t_2, \ldots, t_{n - 1}, t_n}$ of $S$ are known as the knots.
Uniform Spline
$S$ is a uniform spline if and only if $T$ is a normal subdivision.
That is, if and only if the knots of $S$ are equally spaced.
Degree
The degree of $S$ is the maximum degree of the polynomials $P_k$ fitted between $t_k$ and $t_{k + 1}$.
Smoothness
Definition:Spline Function/Smoothness
Also known as
Some sources refer to this as a polynomial spline.
Others just call it a spline.
Examples
Example: Cubic Spline
- How to find the coefficients $a$, $b$, $c$, and $d$ in the cubic spline function $ax^3 + bx^2 + cx + d$
Also see
- Results about splines can be found here.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): spline-fitting
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): approximation theory
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): approximation theory