Synopsis for C12.3a: Approximation of Functions

Level: M-Level. Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 2A89)

Recommended Prerequisites

None

Overview

How can a function f(x) be approximated over a prescribed domain by a simpler function like a polynomial or a rational function? Such questions were at the heart of analysis in the early 1900s and later grew into a mature subject of approximation theory. Recently they have been invigorated as problems of approximation have become central to computational algorithms for differential equations, linear algebra, optimization and other fields. This course, based on Trefethen's new text in which results are illustrated by Chebfun computations, will focus in a modern but still rigorous way on the fundamental results of interpolation and approximation and their algorithmic application.

Synopsis

Chebyshev interpolants, polynomials, and series. Barycentric interpolation formula. Weierstrass approximation theorem. Convergence rates of polynomial approximations. Hermite integral formula and Runge phenomenon. Lebesgue constants, polynomial rootfinding. Orthogonal polynomials. Clenshaw-Curtis and Gauss quadrature. Rational approximation.

1. L. N. Trefethen, Approximation Theory and Approximation Practice