Approximation of functions and data

Abstract

This chapter introduces a number of methods for obtaining spline approximations to given functions, or more precisely, to data obtained by sampling a function. In Section 5.1, we focus on local methods where the approximation at a point x only depends on data values near x. Connecting neighbouring data points with straight lines is one such method where the value of the approximation at a point only depends on the two nearest data points. In order to get smoother approximations, we must use splines of higher degree. With cubic polynomials we can prescribe, or interpolate, position and first derivatives at two points. Therefore, given a set of points with associated function values and first derivatives, we can determine a sequence of cubic polynomials that interpolate the data, joined together with continuous first derivatives. This is the cubic Hermite interpolant of Section 5.1.2. In Section 5.2 we study global cubic approximation methods where we have to solve a system of equations involving all the data points in order to obtain the approximation. Like the local methods in Section 5.1, these methods interpolate the data, which now only are positions. The gain in turning to global methods is that the approximation may have more continuous derivatives and still be as accurate as the local methods. The cubic spline interpolant with so called natural end conditions solves an interesting extremal problem. Among all functions with a continuous second derivative that interpolate a set of data, the natural cubic spline interpolant is the one whose integral of the square of the second derivative is the smallest. This is the foundation for various interpretations of splines, and is all discussed in Section 5.2. Two approximation methods for splines of arbitrary degree are described in Section 5.3. The first method is spline interpolation with B-splines defined on some rather arbitrary knot vector. The disadvantage of using interpolation methods is that the approximations have a tendency to oscillate. If we reduce the dimension of the approximating spline space, and instead minimize the error at the data points this problem can be greatly reduced. Such least squares methods are studied in Section 5.3.2. We end the chapter by a discussing a very simple approximation method, the Variation Diminishing Spline Approximation. This approximation scheme has the desirable ability to transfer the sign of some of the derivatives of a function to the approximation. This is 99