Carl de Boor
This chapter promotes, details and exploits the fact that (univariate) splines, i.e., smooth piecewise polynomial functions, are weighted sums of B-splines.
6.1 PIECEWISE POLYNOMIALS
A piecewise polynomial of order k with break sequence ξ (necessarily strictly increasing) is, by definition, any function f that, on each of the half-open intervals [ξj ξj+1), agrees with some polynomial of degree < k. The term ‘order’ used here is not standard but handy.
Note that this definition makes a piecewise polynomial function right-continuous, meaning that, for any x, f(x) = f(x+):= limh↓0 f(x + h). This choice is ...