# C6.3 Approximation of Functions - Material for the year 2020-2021

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Some experience with (a) numerical analysis and (b) complex variables would be helpful, but neither is required.

16 lectures

### Assessment type:

- Written Examination

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 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.

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.

L. N. Trefethen, *Approximation Theory and Approximation Practice, Extended edition, SIAM 2019 (chapters 1-11, 13-19, 23-24)*

This course will be based on the textbook by Nick Trefethen, *Approximation Theory and Approximation Practice*, published by SIAM in 2019. The lectures and examination will be closely tied to the book, and the problems assigned will be taken from the book. Trefethen's text provides references to many other books and articles that can be read to expand understanding of the course material so a longer reading list is not included here.