NONNEGATIVE SPLINES, IN PARTICULAR OF DEGREE-5

We investigate splines from a variational point of view, which have th e following properties: (a) they interpolate given data, (b) they stay nonnegative, when the data are positive, (c) for a given integer k > 0 they minimize the functional f(x) := integral(a)(b) x((k)) (t)(2)dt nonnegative, interpolating x is an element of W-2(k)[a, b]. We extend known results for k = 2 to larger k, in particular to k = 3 and we fin d general necessary conditions for solutions of this restricted minimi zation problem. These conditions imply that solutions are splines in a n augmented grid. In addition, we find that the solutions are in C2k-2 [a, b] and consist of piecewise polynomials in II2k-1 with respect to the augmented grid. We find that for general, odd k greater than or e qual to 3 there will be no boundary arcs which means (nontrivial) subi ntervals in which the spline is identically zero, We show also that th e occurrence of a boundary are in an interval between two neighboring knots prohibits the existence of any further knot in that interval. Fo r k = 3 we show that between given neighboring interpolation knots, th e augmented grid has at most two additional grid points. In the case o f two interpolation knots (the local problem) we develop polynomial eq uations for the additional grid points which can be used directly for numerical computation. For the general (global) problem we propose an algorithm which is based on a Newton iteration for the additional grid points and which uses the local spline data as an initial guess. Ther e are extensions to other types of constraints such as two-sided restr ictions, also ones which vary from interval to interval. As an illustr ation several numerical examples including graphs of splines manufactu red by MATLAB- and FORTRAN-programs are given.