# John-von-Neumann-Gastprofessur: Prof. Nick Trefethen ^{}, Oxford University ^{}

## Von-Neumann Lecture Course: Approximation Theory and Approximation Practice

A course consisting of eight 90-minute weekly lectures**beginning Monday 10 May 2010; 14:15–15:45, room MI 00.07.011.**This is a mathematics course for students and researchers interested in numerical computation. Familiarity with Matlab is essential. Some familiarity with approximation theory is desirable but not essential. The course is built on an unusual book being completed by the lecturer with the title "Approximation Theory and Approximation Practice: A 21st-Century Treatment in the Form of 32 Executable Chebfun M-Files". It aims to teach both old and new ideas of univariate approximation of functions in a fresh and computational way, illustrating everything through the system (http://www.maths.ox.ac.uk/chebfun/). Both theorems and algorithms will be emphasized: for the former, always with reference to their originator whether in 1912 or 2004; for the latter, always in a hands-on and exploratory fashion.

- Chebyshev points and interpolants
- Chebyshev polynomials and series
- Barycentric interpolation formula
- Weierstrass Approximation Theorem
- Analyticity and convergence rates
- The Gibbs phenomenon
- The Runge phenomenon
- Best approximation and the Remez algorithm
- Lebesgue constants
- Clenshaw-Curtis and Gauss quadrature
- Polynomial roots and colleague matrices
- Approximations based on a conformal map
- Rational functions

**For Bachelor and Master students**: The lecture has 3 ECTS. Successful attendance depends on the written solutions for the exercises given as homework assignements each week.

## Solutions to Assignments

solns1.m | solns1.pdf |

solns2.m | solns2.pdf |

solns3.m | solns3.pdf |

solns4.m | solns4.pdf |

solns5.m | solns5.pdf |

solns6.m | solns6.pdf |

solns7.m | solns7.pdf |

## Additional Reference

C. Runge, Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten^{}