On splines and their minimum properties

0. Introduction. It is the purpose of this note to show that the several minimum properties of odd degree polynomial spline functions [4, 18] all derive from the fact that spline functions are representers of appropriate bounded linear functionals in an appropriate Hilbert space. (These results were first announced in Notices, Amer. Math. Soc., 11 (1964) 681.) In particular, spline interpolation is a process of best approximation, i.e., of orthogonal projection, in this Hilbert space. This observation leads to a generalization of the notion of spline function. The fact that such generalized spline functions retain all the minimum properties of the polynomial splines, follows from familiar facts about orthogonal projections in Hilbert space. 1. Polynomial splines and their minimum properties. A polynomial spline function, s(x), of degree m ≥ 0, having the n ≥ 1 joints x1 < x2 < · · · < xn, is by definition a real valued function of class C(−∞,∞), which reduces to a polynomial of degree at most m in each of the n+1 intervals (−∞, x1), (x1, x2), . . ., (xn,+∞). The most general such function is given by