Definition:Spline Function

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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