Approximations and expansions Splines Theorem

Curry–Schoenberg theorem

Statement

Let $k \ge 1$ be an integer, the order, so that polynomial pieces have degree at most $k - 1$ . Let $\xi_1 < \xi_2 < \cdots < \xi_{l+1}$ be breakpoints, and for $i = 2, \dots, l$ let $\nu_i$ be an integer with $0 \le \nu_i \le k$ . Let $S$ be the space of functions $s$ on $[\xi_1, \xi_{l+1})$ such that:

on each interval $[\xi_i, \xi_{i+1})$ , $s$ agrees with a polynomial of degree at most $k - 1$ ;
at each interior breakpoint $\xi_i$ , the two adjacent polynomial pieces have equal derivatives of orders $0, 1, \dots, \nu_i - 1$ at $\xi_i$ .

Let $n = k + \sum_{i=2}^{l} (k - \nu_i)$ . Let $t_1 \le t_2 \le \cdots \le t_{n+k}$ be a knot sequence with $t_k = \xi_1$ , $t_{n+1} = \xi_{l+1}$ , $t_1 \le \dots \le t_k$ , $t_{n+1} \le \dots \le t_{n+k}$ , and in which each interior breakpoint $\xi_i$ occurs exactly $k - \nu_i$ times among $t_{k+1}, \dots, t_n$ . Then the B-splines $B_{1,k}, \dots, B_{n,k}$ of order $k$ for the knots $t$ , restricted to $[\xi_1, \xi_{l+1})$ , form a basis of $S$ . In particular, $\dim S = n$ .

Remarks

Each $B_{j,k}$ is a piecewise polynomial of degree at most $k - 1$ , and a knot of multiplicity $m$ costs it exactly $m$ orders of smoothness, so each $B_{j,k}$ lies in $S$ . The dimension of $S$ is $n$ by counting: $l$ pieces give $kl$ coefficients, and the conditions at $\xi_i$ remove $\nu_i$ of them, which are independent. Linear independence of the B-splines can be proved with the dual functionals of de Boor and Fix, and $n$ independent elements of an $n$ -dimensional space form a basis. Example: for $k = 4$ and $\nu_i = 3$ at every interior breakpoint, $S$ is the space of cubic splines of class $C^2$ , and $n = l + 3$ . The end knots $t_1, \dots, t_k$ and $t_{n+1}, \dots, t_{n+k}$ can be chosen freely in the stated ranges; a common choice is to make them all equal to the endpoints. The intervals are half-open because B-splines are right-continuous. With the common choice of end knots, all of them vanish at $\xi_{l+1}$ , and one extends a spline there by its left limit.

Going deeper

For a student

The theorem turns the abstract space of piecewise polynomials into a concrete space with a well-conditioned basis of small support. It is why spline software stores a spline as a knot vector and a list of B-spline coefficients. Interpolation and least squares fitting with splines become banded linear systems in this basis. Raising the multiplicity of a knot by one lowers the smoothness there by one, and multiplicity $k$ allows a jump.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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B-spline concept
Basis concept
Spline concept
Curry–Schoenberg theorem statement

Tag	`DHC96EYCN6`
URI	`m2://DHC96EYCN6`
Kind	statement, theorem
Writing	W0 stub
MSC	41A15, 65D07
Status	This copy of the site has no build data. Run `m2 status DHC96EYCN6`.
About	B-spline, Spline
Uses	B-spline, Spline, Basis