Welcome to Week 2! In the first week we got to know the general approximation problem in normed vector spaces. This week, we will specifically look at the problem of approximating continuous functions by polynomials with respect to the ∞ norm and the 2 norm. We will encounter the Chebyshev equioscillation theorem and show that L approximation is unique. With a view towards (weighted) L2 approximation, we will discuss linear independence of polynomials.

Intended Learning Outcomes

At the end of the week you should be able to:

  • derive the characterisation and uniqueness of best L polynomial approximation;
  • explain the role of Chebyshev polynomials with respect to best approximation;
  • explain linear independence of polynomials and L2 best approximation.

Tasks and Materials

  • The first problem sheet is available and will be discussed this week, please have a look at it before class!
  • The material of this week is covered in Chapters 8.3 and 9 in the book by Süli and Mayers, where more details can be found.
  • Problem sheet 2 has been made available (with solutions to follow later) but will be discussed in Week 3