Nonnegative splines, in particular of degree five (Q1128221)

The authors investigate spline functions of odd degree with fixed knots \(t_1,\dots, t_n\), which interpolate at the knots positive data values and minimize the functional \[ \int^b_a f''(t)^2dt \] for \(f\in W^2_2\). In a preceding paper, the cubic case was treated in some detail. Now, special emphasis lies on the quintic case, and also some results for the general case are contained. As a main result necessary conditions for a solution are given, which imply that the solutions are splines in an augmented grid. For splines with no inner knots (the so-called local problem) polynomial equations are developed which can be used for numerical computations. For the general problem a Newton type algorithm is proposed. The paper contains a large number of numerical examples, tables and figures. It is dedicated to Professor Jochen W. Schmidt, Dresden, on the occasion of his 65th birthday.