Numerisk approximation

**Valid from:** Spring 2021**Decided by:** Professor Thomas Johansson**Date of establishment:** 2020-09-24

**Division:** Numerical Analysis**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course code:** NUMN19**Teaching language:** English

The overall goal of the course is to provide an introduction to classical results and numerical algorithms within approximation theory and prepare the participants for further studies in mathematics and computationally oriented subjects. The purpose is further to develop the participants' ability to solve problems, communicate mathematical reasoning, assess mathematical algorithms and translate them into effective code.

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be able to motivate and exemplify the need for approximations of functions, both from the theoretical and the computational point of view
- be able to describe how to find good approximations with respect to different norms, in particular the 1-, 2- and supremum-norms, and give an account of the difficulties in each of these cases,
- be able to give an account of the relation between the topology of the approximation space and the existence and uniqueness of best approximations,
- be able to formulate the main theorems of approximation theory, especially the characterisation theorems and the Weierstrass theorem, and outline their proofs.

*Competences and Skills*

For a passing grade the doctoral student must

- be able to identify the relevant approximation algorithm for a given situation, and write a computer program which implements it
- be able to present solutions and numerical results for problems such as the above ones in written and oral form,
- be able to, with adequate terminology and in a logical and well-structured manner, explain the design of the numerical methods and algorithms included in the course.

*Judgement and Approach*

For a passing grade the doctoral student must be able to argue for the importance of approximation theory as a tool in mathematics,computational technology and related subjects.

The course treats: • The approximation problem: Norms, approximation spaces, the Weierstrass theorem. • Theory of best approximation in Euclidean spaces: Existence, uniqueness, characterisation theorems, duals. • Construction of best approximations: Orthogonality, Chebyshev polynomials, Haar spaces, the exchange algorithm.

Iske, A.: Approximation Theory and Algorithms for Data Analysis. Springer, 2019. ISBN 9783030052270.

**Types of instruction:** Lectures, miscellaneous.
Theoretical and practical assignments.

**Examination formats:** Oral exam, written assignments**Grading scale:** Failed, pass**Examiner:**

**Course coordinators:** **Web page:** http://www.ctr.maths.lu.se/course/NewNumApprox/