The Moment Problem and Orthogonal Polynomials

Classical Analysis

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 2 group A. The teaching period is from November 15 to December 17, 2010 and from January 3 to January 28, 2011, in all 9 weeks.

Lectures by Christian Berg: Tuesday 10.15-12 (Aud 9), Thursday 13.15-15 (Aud 9) . First time: November 16.

Exercises by Leonel Robert: Thursday 15.15-17 (A109). First time: November 18.

The grading follows the 7-level scale and is based on 3 sets of problems which are given during the course and to be handed to Christian Berg on the following Thursdays before 17.30.

Problems for December 9: Problem set I (Available on November 26)

Problems for January 6: Problem set II (Available on December 16)

Problems for January 20: Problem set III (Available on January 11, 2011)

  • Teaching material: The course is based on lecture notes: "Moment problems and orthogonal polynomials". They are written in English. The notes consists of a preface (3 pages), which gives an introduction to the subject, and 5 chapters. The first one is preliminaries on measure theory and I will not lecture on that but use the results without proofs. I will concentrate on Chapter 2, which introduces the basic theory of orthogonal polynomials.

    The lectures will be in English unless all participants agree that they can be given in Danish.

    You can also read a small introductory paper to the subject (if you read Danish), which was published in NORMAT Volume 54, no 3 (2006), p. 116-133. You can download a pdf-version here: Introduction (Note: The page numbers you see in the file do not agree with the right pagenumbers given above)

    Plan for the course

    (modifications will be given as time goes by):

    Week 46:

  • Lectures: After a short introduction I will go to section 2.1, which will be the subject of the week.

    You should browse through Chapter 1 yourself: The main result is the Riesz representation theorem 1.2.2, which establishes a one-to-one correspondence between Radon measures and positive linear functionals. The notion of support of a Radon measure is also important (section 1.3) as well as section 1.4 on vague and weak convergence. Some of this will be taken up in the exercise class.

    On Tuesday I told a little about why we consider measures \mu on R with moments of any order: Because then we can use the Gram-Schmidt procedure on 1,x,x^2,... to make an orthonormal system (P_n), where P_n is a polynomial of degree n, in the Hilbert space L^2(R,\mu), and this sequence is the orthogonal polynomials associated with \mu. The classical orthogonal polynomials are Hermite, Laguerre and Legendre polynomials. They are associated with the normal law on R, the exponential law on [0,\infty[ and the uniform law on [0,1] respectively. These three measures are determinate. However, in the indeterminate case, another measure having the same moments as \mu will lead to the same orthonormal polynomials. This will be taken up later. On Thursday I did most of Section 2.1 with special focus on Theorem 2.1.4, Theorem 2.1.7 and the examples of Stieltjes (pages 39-41). You should be able to read the rest of section 2.1 yourself.

  • Exercise class: Leonel will remind you about Radon measures, the Riesz representation theorem (without proof), and the support of a Radon measure. He will discuss vague and weak convergence of Radon measures and explain Theorem 1.4.2 and Corollary 1.4.3. He will discuss in detail the results in Definition 1.4.7, Theorem 1.4.8, Corollary 1.4.9, Theorem 1.4.11.

    Week 47:

  • Lectures: 2.2. I lectured on most of the section, but did not go through the proof of the technical Lemma 2.2.11. You can skip that proof, but the result is used in theorem 2.2.13, which will be taken up by Leonel next week.

  • Exercise class: 1) Discuss the proof of Theorem 2.1.10. 2) Discuss that the measure on ]0,\infty[ with density t^m d_q(t) (page 41) is indeterminate. 3) Solve the exercises E 2.1.1-E 2.1.3. (In E 2.1.2 you shall be inspired by the proof of Theorem 2.1.7). 4) Show that (4n+3)!, (n!)^2, n!+1/(n+1) are moment sequences.

    week 48:

  • Lectures: 2.3 and 2.4.

  • Exercise class: 1) Discuss the proof of Theorem 2.2.13. 2) Solve Exercise E 2.2.2, E 2.3.1, E 2.3.2.

    week 49:

  • Lectures: 2.5.

  • Exercise class: Solve the exercises E 2.4.1, 2.4.2, 2.4.4, 2.4.6. To 2.4.2 there is the following hint: Prove first log a=\int_0^\infty (e^{-u}-e^{-au})/u du for a>0 by integrating the formula 1/a= \int_0^\infty e^{-ax}dx. Note that you are not allowed split the first integral in two.

    week 50:

  • Lectures: 2.6. I did not quite finish section 2.6. The last thing I proved was Corollary 2.6.14. The subsections 2.6.15-2.6.17 will be left for the exercise class on January 6 and I will lecture on 2.6.18 on January 4.

  • Exercise class: Solve E 2.4.3. (Only first part. You are not yet able to find s\in P which is determinate, but such that E^2s is indeterminate)

    E 2.4.8. (It should not be necessary to look at the reference. Write the formula for the n'th Fibonacci number given in E 2.3.2 as F_n=(1/\sqrt{5})\phi^n(1-q^n) with q=(\hat\phi)/\phi=(1-\sqrt{5})/(1+\sqrt{5}). Then you shall write 1/F_{n+2} as sum of an infinite series using the geometric series 1/(1-r)=1+r+r^2+... and find a measure with masses at a sequence of points having the moments 1/F_{n+2}.

    E 2.5.1, E 2.5.2.

    week 1:

  • Lectures: I will start with section 2.6.18 about Chebyshev polynomials. I will hand out some pages about the classical orthogonal polynomials. It is good to know that all the orthogonal polynomials can be found on the website orthogonal polynomials This website contains many systems we will not discuss. The classical ones are also available at Maple. You can download the worksheet ortopol.mw by saving it as worksheet and take it into maple. ortopol.mw I lectured about the following systems: Chebyshev of first and second kind, Hermite and Laguerre. I told what the measure is for Jacobi, but just showed the formulas available for Jacobi polynomials by using the above internetlink. I also mentioned the several important special cases of Jacobi: If the parameters \a=alpha and \b=beta are equal we get the Gegenbauer polynomials. If this common value is -1/2, 0, 1/2 we get the polynomials T_n, Legendre and U_n (up to a normalizing constant). The normalization of the classical orthogonal polynomials varies a little from book to book. There are always orthonormal and monic versions, but sometimes also other principles have been guiding when a normalization is chosen.

  • Exercise class: Discuss Lemma 2.6.15 and Theorem 2.6.16. Solve then E 2.6.1, E 2.6.5, E 2.6.7.

    week 2:

  • Lectures: 2.7. I covered up to Theorem 2.7.10.

  • Exercise class: Prepare the following Extra exercises

    week 3:

  • Lectures: Tuesday: I was only able to lecture up to Theorem 2.7.12, where I will start on Thursday. I finished section 2.7 and started section 2.8.

  • Exercise class: Give the necessary details to finish the proof of Theorem 2.7.7 concerning the convergence of \sum |Q_n(\lambda)|^2, by using the hint to apply E 2.6.5 stated on page 101 line 9 from the bottom.

    Do E 2.7.3 and the following Extra exercises

    week 4:

  • Lectures: No lecture on Tuesday because of exams in other courses. On Thursday I will prove Theorem 2.8.2 and 2.8.5. Then I will give some survey about results one can approach on the basis of the notes. In this part I will not give any proofs.

  • Exercise class: No exercise class.
    berg@math.ku.dk/ November 10, 2010.