This week I gave an introductory talk at the Tenth Summer School on Potential Theory about potential theoretic methods in the study of orthogonal polynomials. This post is based upon my notes for that lecture. The aim of that talk was to prove a theorem about the weak-* convergence of normalized counting measures for zeros of orthogonal polynomials. This theorem was mentioned in an earlier post, therefore this can be thought of as an extension of those posts.

**1. Introduction.** Our goal is to show that potential theoretic methods can be used effectively for studying orthogonal polynomials. The following is based upon [VanAssche] Chapter 1.2.

Let be a finite Borel measure on the real line. Suppose that its support contains infinitely many points and

As usual, we are going to study the orthogonal polynomials with respect to . Let denote the -th *monic orthogonal polynomial* with respect to . It is easy to see that every polynomial can be written as a linear combination of (monic) orthogonal polynomials. They have a very important property: they are extremal in a sense, which is made precise in the following theorem.

**Theorem 1.1.** (-extremality of monic orthogonal polynomials) Let be a Borel measure with finite moments containing infinitely many points in its support. If denotes the -th monic orthogonal polynomial, then

**Proof.** Let be a monic polynomial of degree . It can be written as , where is some polynomial of degree . It follows that

and the minimum is clearly attained for . ♦

A lot is known about the zeros of the orthogonal polynomials (for example, see my first and second post), some of them are collected in the following theorem.

**Theorem 1.2. **Let be the -th monic orthogonal polynomial with respect to . Then the zeros of are real and simple, moreover they are contained in the convex hull of . ♦

According to these facts, we can assume without the loss of generality that the zeros of are denoted with , moreover

For these points we can define the normalized counting measure as

where is the Dirac probability measure concentrated at the point . Our aim is to prove the following theorem.

**Theorem 1.3.** (Erdős-Turán) Let be an absolutely continuous probability measure supported on with and let denote the monic orthogonal polynomials. If almost everywhere on , then

and

where is a measure on defined as

denotes the normalized counting measure of the zeros of , and ”” denotes weak-* convergence of measures. ♦

This theorem was proven first by Pál Erdős and Pál Turán. The proof I am going to show was given by Walter Van Assche, and it can be found in his book. In the next part we shall collect some important tools and then we prove Theorem 3.

**2.1. Potential, energy, capacity, equilbrium measure. **Let be a compact subset of the complex plane and let be a finite Borel measure supported on . The *potential* of is defined as

The potential is a lower semicontinuous function, i.e. .

The *energy of the measure* is defined as

With the notation (i.e. the Borel probability measures supported on ), the* energy of the set* is defined as

This quantity is also called *Robin’s constant*. The *logarithmic capacity* of is defined as

The capacity can be computed explicitly for some sets, for example, an interval on the real line has capacity . Sets with zero logarithmic capacity are called *polar sets*. A property is said to hold *quasi everywhere* for a set if it holds for all points in , where is a polar set. (It is important that *quasi everywhere* implies *almost everywhere* in the Lebesgue sense, but not the other way around.)

Given a weakly convergent sequence of measures, the following well-known theorem holds.

**Theorem 2.1. **(Principle of descent and lower envelope theorem) Let and be finite Borel measures on the complex plane with support lying in a compact set and assume that as , where ”” denotes weak-* convergence. Then

(This is the ”principle of descent” part.) Moreover, we have

(This is the ”lower envelope” part.) ♦

For the proof, see [Saff-Totik] Chapter I.6 Theorems 6.8 and 6.9. Another important and frequently used theorem is the principle of domination.

**Theorem 2.2.** (Principle of domination) Let and be two positive and finite Borel measures with compact support on with finite logarithmic energy and suppose that . If for some constant the inequality

holds -almost everywhere, then it holds for all . ♦

For the proof of the principle of domination, see [Saff-Totik] Chapter II, Theorem 3.2. If is not polar, then there is a unique measure denoted with such that

This measure is called the *equilibrium measure* of . The equilibrium measure is known explicitly only for a few sets, for example the interval , on which the equilibrium measure takes the form

which is the measure appearing in Theorem 2.4. According to the following well-known theorem, the potential of an equilibrium measure behaves nicely on the complex plane.

**Theorem 2.3. **(Frostman’s theorem) Let be a compact set. Then

(i) for all ,

(ii) quasi everywhere on . ♦

For the proof, see [Ransford] Theorem 3.3.4. These properties appearing in Frostman’s theorem characterize the equilibrium measure in a sense.

**Theorem 2.4. **(Characterization of the equilibrium measure) Let be a compact set and let be a probability measure of finite energy on . If quasi everywhere on then is the equilibrium measure . ♦

For the proof, see [Saff-Totik] Section I.3, Theorem 3.3.

**Exercise 2.5.** Show that Theorem 2.4.is false if we do not require that is supported on !

Combining Frostman’s theorem with the principle of descent we can establish the main lemma on which we will base the proof of Theorem 1.3.

**Lemma 2.6. **(Brolin) Let be a compact subset of the complex plane, assume that , or in other words, is not polar. Suppose that the equilibrium measure of is supported on , i.e. . Let be a sequence of probability measures on and suppose that they converge weakly to a probability measure . If

holds, then the weak limit of is the equilibrium measure .

**Proof.** According to Theorem 2.4, it is enough to prove that quasi everywhere on , and our goal is to show this. The property

implies that (since is a probability measure)

Using Fatou’s lemma, Frostman’s theorem (i.e. on ) and Fubini’s theorem, we have

which gives

Since and is a probability measure, it follows immediately that

The lower envelope theorem (see Theorem 2.1) gives quasi everywhere, therefore

Since is supported on , for every fixed there is a arbitrary close to such that . Using that is lower semicontinuous (i.e. ), we have

On the other hand, the principle of domination (see Theorem 2.2) implies that for alllatex $ z \in K $,

and, according to Frostman’s theorem, quasi everywhere on , therefore

These imply that , and this is what we wanted to show. ♦

**2.2. Chebyshev polynomials. **Let be a compact subset of the complex plane containing infinitely many points and define the quantity

It is known that there exists a unique polynomial $Vyacheslav T_n(x,K) = T_n(x) $ of degree $Vyacheslav n $ such that $Vyacheslav \| T_n \|_K = m_n(K) $. (This property is the $Vyacheslav L^\infty $ analogue of Theorem 2.1. $Vyacheslav T_n $ is called the $Vyacheslav n $-th Chebyshev polynomial. It is also known, and shall prove useful for us, that

The Chebyshev polynomials on the interval $ [-1,1] $ are very well known and have been intensively studied. They can be written in the form

which is indeed a monic polynomial. Incidentally, they are also the orthogonal on for the measure , which is the equilibrium measure for .

**3. The proof of the main theorem. **Before we start with the proof of Theorem 2.4, we need one more lemma.

**Lemma 3.1. **(Ullman) Let be a positive measure on and let be a sequence of nonnegative -measurable functions. Suppose that

Then there exists a subsequence such that

C

**Proof.** Let be a positive sequence of real numbers such that and

Define

It is immediate that , and therefore there exists a subsequence such that $ \lim_{k \to \infty} g_{n_k}(x) = 0 $ -almost everywhere. (Recall that implies that every subsequence of has a subsequence which converges to -almost everywhere.) But this means that for large , we have

which implies

and this is what we needed to prove. ♦

**Proof of Theorem 1.4.** First we are going to show that . The -extremal property of (see Theorem 1.1.) implies that

where is the Chebyshev polynomial of . The convergence of the norm of the Chebyshev polynomials gives that

Now assume that

This means that there is a subsequence such that

Using Lemma 3.1 with we obtain a further subsequence such that

which, since , implies that Lebesgue almost everywhere. Since the equilibrium measure is mutually absolutely continuous with the Lebesgue measure, we have

Now, it is easy to see that the potential of the normalized counting measure defined as

(where are the zeros of the monic orthogonal polynomial ) is

Let be a subsequence of such that weakly converges to some limit . (Such a subsequence exists, since Helly’s selection theorem can be applied.) We get that

-almost everywhere.The lower envelope theorem (see Theorem 2.2) gives

which, according to Lemma 2.6, implies that is the equilibrium measure . But this is a contradiction, since then quasi everywhere! Therefore

which implies

Now we prove that . Let be an arbitrary subsequence of . Since is a probability measure supported on a compact set, we can apply Helly’s selection theorem, which gives a subsequence of such that for some . Applying the same reasoning as before, we obtain that

and

whose, with the application of Lemma 2.6 imply that is the equilibrium measure. Therefore holds. ♦

**4. Generalizations of Theorem 1.4. **The main theorem works not only for measures supported on , but for measures supported on compact subsets of the real line. To be more precise, the following theorem holds.

**Theorem 4.1. **Let be a Borel probability measure on the real line supported on a compact set , which has positive capacity. Let denote the -th monic orthogonal polynomial with respect to . Suppose that , where is absolutely continuous and is singular with respect to the equilibrium measure . If $ d\mu_1(x) = w(x) d\nu_K(x) $ and -almost everywhere, then

and

where is the normalized counting measure of the zeros of . ♦

It can be shown easily that if we do not require that is supported on the real line, then the statement of Theorem 4.1 regarding the weak-* convergence of the normalized counting measures is false.

**References.**

[Ransford] Thomas Ransford, Potential Theory in the Complex Plane, Cambridge University Press, 1995

[Saff-Totik] Edward B. Saff and Vilmos Totik, Logarithmic Potentials with External Fields, Springer-Verlag, 1997

[Stahl-Totik] Herbert Stahl and Vilmos Totik, General Orthogonal Polynomials, Cambridge University Press, 1992

[VanAssche] Walter Van Assche, Asymptotics for Orthogonal Polynomials, Lecture Notes in Mathematics, Springer-Verlag, 1987