18. Smoothing spline (p. 21)

This chapter promotes, details and exploits the fact that (univariate) splines, i.e., smooth piecewise polynomial functions, are weighted sums of B-splines. 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 arbitrary, but has become standard. Keep in mind that, at its break ξj, the piecewise polynomial function f has, in effect, two values, namely its limit from the left, f(ξj−), and its limit from the right, f(ξj+) = f(ξj). The set of all piecewise polynomial functions of order k with break sequence ξ is denoted here Π<k,ξ. 1 2. B-splines defined B-splines are defined in terms of a knot sequence t: = (tj), meaning that · · · ≤ tj ≤ tj+1 ≤ · · ·. The jth B-spline of order 1 for the knot sequence t is the characteristic function of the half-open interval [tj.. tj+1), i.e., the function given by the rule 1, if tj ≤ x < tj+1; Bj1(x): = Bj,1,t(x):= 0, otherwise. Note that each of these functions is piecewise constant, and that the resulting sequence (Bj1) is a partition of unity, i.e., In particular