Spline Toolbox

B-form

The B-form has become the standard way to represent a spline during its construction, since the B-form makes it easy to build in smoothness requirements across breaks and leads to banded linear systems. The B-form describes a spline as a weighted sum

of B-splines of the required order , with their number, , at least as big as plus the number of polynomial pieces that make up the spline. Here,

is the th B-spline of order for the knot sequence . In particular, is piecewise-polynomial of degree < , with breaks , is nonnegative, is zero outside the interval , and is so normalized that  

ppform

Knot Multiplicity