Nonlinear Sets of Spline Functions

Abstract

The use of spline functions dates back at least to the beginning of this century. Piecewise linear functions had been used already in connection with Peano’s existence proof for solution to the initial value problem of ordinary differential equations, although these functions were not called splines. Splines were first identified since Schoenberg’s fundamental paper was published in 1946. After this moment the theory of polynomial spline functions developed rapidly and was rich in contents. Numerous generalisations of spline have been introduced, but the families of functions used in these extensions have always been linear with respect to the free parameters. Spline functions have wide range of applications to numerical approximations. But to certain functions, for example, to the function with singular points, the interpolation by polynomial spline is not always efficient and reasonable. For the approximation of functions with singularities it would much better to use a nonlinear family of splines. Since 1973 many authors have investigated such nonlinear sets of splines functions, especially rational splines functions and obtained a lot of results. One might expect that the construction of such nonlinear splines and to apply these results to practice, will had to much difficulties, since we will have to solve some more complicated nonlinear simultaneous equations in the computing process. It will be shown that it is not true, and that the numerical methods with nonlinear classes of splines proceed almost along the same lines as do applications of classical linear splines, if appropriate nonlinear classes are selected.