SPLINE FUNCTIONS AND APPLICATIONS.

The document begins with a definition of a spline function, some general remarks about the nature and uses of such functions, an illustrative example, and a simple algebraic representation for any spline function. The important subclass of natural splines is defined in chapter 2 and the theorem of unique interpolation by natural splines and the well known minimal property of the smoothest interpolating natural spline is proven. Chapter 3 is concerned with the approximation of linear functionals. It includes Peanos theorem on remainders, A. Sards theory of best approximation of linear functionals, and I.J. Schoenbergs discovery that finding the best approximation to a given functional in Sards sense is equivalent to applying the functional to the corresponding natural spline interpolating function. Chapter 4 develops the properties of divided differences and of the B-splines, a special type of spline functions with limited support that constitute a basis for the class of spline functions and are important for computational purposes. Chapter 5 concerns a numerical algorithm for obtaining the natural spline interpolating function and also considers the smoothing natural splines resulting from Schoenbergs adaptation of E. T. Whittakers smoothing method. Author