NMAK13016U The Moment Problem and Orthogonal Polynomials

Volume 2013/2014
Education
MSc Programme in Mathematics
Content
1.Short survey of Radon measures
2. Moments
3. Extensions of linear functionals
4. Positive definite sequences
5. The moment problems of Stieltjes and Hausdorff
6. A Hilbert space approach to the one-dimensional moment problem
7. Orthogonal polynomials
8. Möbius transformations and Hellinger's circles
9. Denseness of polynomials
10. The theorems of Carleman
Learning Outcome
Knowledge:

After the course the students should know weak and vague convergence of measures on Euclidean spaces and the relation to convergence of the corresponding moments. They should know how different moment sequences can be characterized in terms of positive definiteness. They should know the basics about orthogonal polynomials such as the three term recurrence relation and interlacing of zeros. They should learn that a moment problem can be determinate or indeterminate and how to distinguish the two cases.

Skills:

The students should be able to calculate moments of concrete measures,
master the formulas defining some of the classical systems of orthogonal polynomials, be able to decide vague or weak convergence of concrete sequences of measures, apply the given criteria to decide the nature of the moment problem. 

Competences:

The students should get insight in a classical subject, which has been important in the development of mathematics of the 20'th century. The students should get a deeper understanding of linear algebra, measure theory and real and complex analysis, which are the three subjects which are vital in the course.
KomAn, An2, MI (measure and integration theory)
4 hours of lectures and 2 hours of exercises for 9 weeks
  • Category
  • Hours
  • Exam
  • 30
  • Lectures
  • 36
  • Preparation
  • 122
  • Theory exercises
  • 18
  • Total
  • 206
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Written examination, 24 hours
Two sets of problems shall be solved during the course, counting for 50% of the grading. A 24-hours take home exam shall count for the remaining 50% of the grading.
Aid
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner.
Re-exam
30 minutes oral examination with several internal examiners.
Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.