Synopsis

The course provides an introduction to the theory of approximation of real functions of one variable. Mostly, we will look at approximation of complicated functions by simple functions such a polynomials, but we will also look a little at approximation using rational polynomials and of periodic functions. Given a function and a set of possible approximating functions we can ask: how do we construct a good approximation, what do we mean by a good approximation, is there an approximation which is better than any other, how do we estimate how well we are approximating the orginal function?

Syllabus

Introduction to approximation. Construction of interpolating polynomials and their properites: Lagrange interpolatioon, divided differences, Newton interpolation, hermite interpolation, quadrature using interpolating polynomials. Polynomial approximation in the L_2 norm, orthogonal polynomials, their properties and characterisation, Chebyshev polynomials, Gauss Quadrature. Polynomial approximation in the L_infinity norm, uniform approximation, de la Vallee Pousin and Oscillation theorems, exchange algorthm and its convergence. Approximation of periodic functions, order of convergence of approximations. B splines, approximation with splines and their convergence properties, Peano Kernel Theorem. Approximation using rational polynomials. Texts: The following books are both still in print or avaialble in the library:

Reading List

• M J D Powell, 1981, Approximation theory and methods (CUP, reprinted 1988)

• P J Davis, 1963, Interpolation and approximation (Dover reprint, 1975)