Uniqueness of Smoothest Interpolants

(1) inf{‖f ‖p : f ∈ C [0, 1], f(ti) = ei, i = 1, . . . , m}, where ‖ · ‖p is the usual L [0, 1] norm, 1 ≤ p ≤ ∞, and to characterize f for which the above infimum is attained. If m ≤ n, then this problem has a simple solution. There are algebraic polynomials of degree at most n−1 satisfying the interpolation data, and any of these polynomials obviously solve our problem. If m > n then, in general, the above infimum is not attained within C[0, 1]. We are in the wrong space. For 1 < p ≤ ∞ we should be in W (n) p [0, 1], the standard Sobolev space of real-valued functions on [0, 1] with n − 1 absolutely continuous derivatives and nth derivative existing a.e. as a function in L[0, 1]. It is here that our solution is to be found. For p = 1 the situation is slightly more complicated, but known (see e.g., Fisher, Jerome [8], de Boor [5]). This extremal problem has played an important role in the development of spline theory. In the case p = 2 the unique solution of this extremal problem is a natural spline of degree 2n − 1 (with simple knots t1, . . . , tm). This oftquoted result, see de Boor [3], Schoenberg [18], was fundamental in the history of spline theory. For p = ∞ a solution (not necessarily unique) is a perfect spline of degree n with at most m − n − 1 knots, see Karlin [9], de Boor [4]. For p = 1 there is a solution which is a spline of degree n − 1 with at most m − n knots. In general, for 1 < p < ∞, this problem has a unique solution. It is attained by the unique function f satisfying f(ti) = ei, i = 1, . . . , m, whose nth derivative has the form