The Moment Problem and Orthogonal Polynomials

The Moment Problem and Orthogonal Polynomials

The most important mathematician in connection with the moment problem is:

Thomas Jan Stieltjes, (1856-1894)

If you want to know something about Stieltjes or other mathematicians, then look at MacTutor History of Mathematics Archive

The course lies in Blok 2A. The teaching period is from November 14 to January 27, in all 9 weeks.

There will be lectures Monday 15-17 (Aud 10), Thursday 9-11, 13-14 (Aud 10). In the week January 2-January 7 there will be no lectures. The week is set aside for individual work with a project to be defined later. There will be assigned written problems, which shall be handed in at the following dates: November 24, December 8 and January 26. The grading (pass/nonpass or following the 13-scale depending on the wish of the students) will be based on the project (50%) and the best 2 of the 3 written problems, each of which will count 25%.

Problems for November 24: E 1.1.1, E 1.1.3, E 1.2.2, E 1.4.1

Problems for December 8: E 2.1.2, E 2.2.2, E 2.3.1, E 2.4.2.

Hint for 2.1.2: Seek inspiration in the proof of Theorem 2.1.7. Hint for 2.4.2: Start by proving the formula
$\log a=\int_0^\infty \frac{e^{-u}-e^{-ua}}{u}du$ for $a>0$

Problems for January 26: E 2.6.2, E 2.6.4, E 2.7.1

  • Teaching material: The course is based on lecture notes with the same title as the course. They are written in English. They were handed out at the beginning of the course on November 14. If you are interested in a copy then please contact me. The lecture notes are in English and the lectures will be in English unless all participants agree that they can be given in Danish.
  • Course description: To get an idea about the content of the course see Introduction
  • The study-board has sent the following message concerning evaluation of the course: Studienævnet for Matematiske Fag ønsker at gennemføre en evaluering af de studerendes holdninger i blokkens 6. uge, nærmere bestemt 2-9/1 2006. Der bruges et elektronisk evalueringsskema, som bliver tilgængeligt for de studerende i ISIS. De studerende får adgang ved at logge ind på http://isis.ku.dk/evaluering/respondent.
  • Plan for the course:

    Week 46: Introduction and Chapter 1. Focus is on the main results 1.1.10 and the Riesz representation theorem 1.2.2 without going into details with the proofs. The notion of support of a Radon measure is important (section 1.3) and section 1.4 on vague and weak convergence is important.

    Week 47: 2.1 and 2.2.

    week 48: 2.3, 2.4, 2.5

    On November 28 I essentially finished 2.3 and 2.4, but I still have to prove Lemma 2.4.7. You have to study this yourself. I finished 2.5.

    week 49: 2.6

    week 50: 2.7

    week 1: Project on different classes of orthogonal polynomials-there will be no lectures that week. Information about the project can be obtained by contacting me.

    week 2: 2.8

    week 3: 2.9, 3.1

    week 4: 3.2, 3.3.

    week 5: The course is over

    berg@math.ku.dk/ December 12, 2005.