### 40.1 Overview

B-splines are commonly used as basis functions to fit smoothing curves to large data sets. To do this, the abscissa axis is broken up into some number of intervals, where the endpoints of each interval are called breakpoints. These breakpoints are then converted to knots by imposing various continuity and smoothness conditions at each interface. Given a nondecreasing knot vector t = {t_0, t_1, …, t_{n+k-1}}, the n basis splines of order k are defined by

B_(i,1)(x) = (1, t_i <= x < t_(i+1)
(0, else
B_(i,k)(x) = [(x - t_i)/(t_(i+k-1) - t_i)] B_(i,k-1)(x)
+ [(t_(i+k) - x)/(t_(i+k) - t_(i+1))] B_(i+1,k-1)(x)


for i = 0, …, n-1. The common case of cubic B-splines is given by k = 4. The above recurrence relation can be evaluated in a numerically stable way by the de Boor algorithm.

If we define appropriate knots on an interval [a,b] then the B-spline basis functions form a complete set on that interval. Therefore we can expand a smoothing function as

f(x) = \sum_i c_i B_(i,k)(x)


given enough (x_j, f(x_j)) data pairs. The coefficients c_i can be readily obtained from a least-squares fit.