### Recurrence relations and a prelude to spectral theory

This time I want to talk about one of the most basic and most important tools in the theory of orthogonal polynomials: the recurrence relations. This topic also demonstrates that the theory of orthogonal polynomials on the real line and on the unit circle differ significantly. Let’s start with OPRL! (That is, orthogonal polynomials on the real line.)

1. Recurrence relations for OPRL. Let $\mu$ be a Borel measure on the real line with our usual assumptions. (For these, check the first post.) As usual, $p_n(x,\mu) = p_n(x)$ denotes the $n$-th orthonormal polynomials. (But, as in other posts, I will often write orthogonal, when I talk about them.) We know that every polynomial $\Pi_{n+1}(x)$ of degree $n+1$ is the linear combination of the orthogonal polynomials, that is

$\Pi_{n+1}(x) = \sum_{k=0}^{n+1} \alpha_k p_k(x),$

where the coefficients $\alpha_k$ can be computed as $\alpha_k = \int \Pi_{n+1}(x) p_k(x) d\mu(x)$. (With Hilbert space notation, we can write $\alpha_k = \langle \Pi_{n+1}, p_k \rangle$. This will be useful later on.)

Substitute $x p_n(x)$ into the place of $\Pi_{n+1}(x)$! This way we have

$\alpha_k = \int x p_n(x) p_k(x) d\mu(x) = \int p_n(x) x p_k(x) d\mu(x) = 0$

for all $k \in \{ 0,1,\dots, n-2 \}$! That is,

$x p_n(x) = \alpha_{n+1} p_{n+1}(x) + \alpha_n p_n(x) + \alpha_{n-1} p_{n-1}(x)$

holds. Glancing at the coefficients and introducing the notation $a_n := \langle x p_n(x), p_{n-1}(x)\rangle, b_n := \langle x p_n(x), p_n(x) \rangle$, it is easy to see that

$x p_n(x) = a_{n+1} p_{n+1}(x) + b_n p_n(x) + a_n p_{n-1}(x).$

This relation is the so-called three-term recursion. If you think about it, this is possible because the relation $\langle x f(x), g(x) \rangle = \langle f(x), x g(x) \rangle$ holds in $L^2(\mu)$, since $\mu$ is supported on the real line. If we are working with measures supported on the unit circle, the above relation takes the form $\langle z f(z), g(z) \rangle = \langle f(z), \overline{z}g(z) \rangle$. This foreshadows that a recurrence relation is also possible for the orthogonal polynomials with respect to measures supported on the unit circle (OPUC from now on).

Taking a closer look at $a_n$ (and keeping in mind that $p_n(x) = \gamma_n x^n + \dots, \gamma_n > 0$), we have

$a_n = \int x p_n(x) p_{n-1}(x) d\mu(x)$

$= \int x p_{n-1}(x) p_n(x) d\mu(x)$

$= \int \Big( \frac{\gamma_{n-1}}{\gamma_n}p_n(x) + \Pi_{n-1}(x) \Big) p_n(x) d\mu(x),$

where $\Pi_{n-1}$ is some polynomial of degree $n-1$. Using the orthonormality of $p_n$, we obtain that $a_n = \frac{\gamma_{n-1}}{\gamma_n}$, which is also positive.

The three-term recurrence relation makes it possible to represent the “multiplication with $x$” operator in $L^2(\mu)$ with a tridiagonal matrix. To be more precise, if the polynomials are dense in $L^2(\mu)$ (true if $\mu$ has compact support) and we introduce the operator

$M_x: L^2(\mu) \to L^2(\mu), f(x) \to x f(x)$

(also a well-defined object if, for example, $\mu$ has compact support), then the “matrix” of $M_x$ in the orthonormal basis $\{ p_n \}_{n=0}^{\infty}$ can be written as the symmetric tridiagonal matrix

$J = \begin{pmatrix} b_0 & a_0 & 0 & 0 & \dots \\ a_0 & b_1 & a_1 & 0 & \dots \\ 0 & a_1 & b_2 & a_2 & \dots \\ 0 & 0 & a_2 & b_3 & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}.$

Matrices of this form are called Jacobi matrices. (Not to be confused with Jacobian matrices!) These objects connect the field of orthogonal polynomials with spectral theory of operators. We’ve just seen that if we have a measure on the real line with finite moments, then there is an associated Jacobi matrix via the three-term recursion for the orthogonal polynomials. The beautiful thing is, the opposite direction is also true. (As I read this sentence right before I publish the post makes me think “what is not beautiful about orthogonal polynomials?”.) That is (under suitable conditions of course), given a Jacobi matrix, we can construct a measure for which the orthogonal polynomials are given by the three-term recursion obtained from our known Jacobi matrix. It is a very important and interesting fact that this measure is not necessarily unique.  This correspondence between measures and Jacobi matrices is so important that I will give three different proofs for this in the following posts, but for now, I will continue with the recurrence relations for OPUC.

If you want more OPRL, check the book [Freud] or [Simon]. Although the latter book deals with OPUC, it has a short chapter on OPRL, which is a dense but thorough survey of results. The former book is very hard to find, but I recommend it if you can get your hands on it.

2. Recurrence relations for OPUC. Now let’s study measures on the unit circle! (In this section we closely follow the book [Simon].) Let $\nu$ be a measure supported on the unit circle $\partial \mathbb{D}$ (but not necessarily the whole unit circle), and suppose it has finite moments. We know that there exists a system of polynomials $\{ \varphi_n \}_{n=0}^{\infty}$ with $\varphi_n(z)= \kappa_n z^n + \dots, \kappa_n > 0$ such that these polynomials form an orthonormal system. There is an associated system of polynomials $\{ \Phi_n \}_{n=0}^{\infty}$, where $\Phi_n := \kappa_{n}^{-1} \varphi_n$. These are called the monic orthogonal polynomials, and we shall give the recurrence relation in their term.

On the unit circle, the operator “multiplication by $z$” has a very interesting property. Namely, if $f,g \in L^2(\nu)$, then $\langle z f(z), g(z) \rangle = \langle f(z), \overline{z} g(z) \rangle$ holds. Since we are working on the unit circle, we have

$\langle zf(z), zg(z) \rangle = \langle f(z), g(z) \rangle$,

which is quite special. (You should notice that it does not work on the real line.) Similarly to the OPRL case, this can be exploited to obtain a recurrence relation.

First we define the map $R_n: L^2(\nu) \to L^2(\nu)$ as

$(R_n f)(z) := z^n \overline{f(z)}$.

This may seem a bit random, but if we apply it for a polynomial $\Pi_n(z) = \sum_{k=0}^{n} a_k z^k$, we see that

$(R_n \Pi_n)(z) = z^n \overline{\sum_{k=0}^{n} a_k z^k} = z^n \sum_{k=0}^{n} \overline{a_k} z^{-k} = \sum_{k=0}^{n} \overline{a_{n-k}} z^k$.

In other words, $R_n$ takes the coefficients of $\Pi_n$, conjugates them and writes them down in reverse order. A standard notation in the literature for $R_n$ is $R_n \Pi_n = \Pi_n^*$, but this is confusing. If, for example, you have a polynomial of degree $n-1$, you no longer have reversed and conjugated coefficients. To avoid confusion, the star notation is used only when the degree of the polynomial is the same as the index of this mapping.

First thing we shall notice is that $R_n$ is an involution. (In other words, we have $R_n R_n f = f$.) More important is that it behaves nicely with respect to the inner product. That is, we have

$\langle R_n f, R_n g \rangle = \overline{\int z^n \overline{f(z)}} z^n \overline{g(z)} d\mu(z) = \int \overline{g(z)} f(z) d\mu(z) = \langle g, f \rangle$.

(Following physicist traditions, our inner product is conjugate linear in the first variable and linear in the second variable.) It follows immediately that $R_n$ preserves orthogonality.

What happens if we apply $R_n$ to the orthogonal polynomials? Since $\Phi_n$ is the unique polynomial of degree $n$ (up to multiplication by a nonzero constant) which is orthogonal to $1, z, \dots, z^{n-1}$, it follows that $\Phi_n^* = R_{n} \Phi_n$ is orthogonal to $R_{n}1 = z^{n}, R_{n}z = z^{n-1}, \dots, R_{n} z^{n-1} = z$, and moreover, it is the unique polynomial of degree $n$ which is orthogonal to $z, z^2, \dots, z^n$.

If you look at the three-term recurrence long enough, it can be interpreted as a result of trying to determine the coefficients of the polynomial

$xp_n(x) - \frac{\gamma_{n}}{\gamma_{n+1}}p_{n+1}(x)$,

which is of degree $n$. Let’s do the same thing! We shall look for the coefficients of

$z\Phi_n(z) - \Phi_{n+1}(z)$,

which is also of degree $n$. Using our special relations for the unit circle, we have

$\langle z \Phi_n(z), z^k - \Phi_{n+1}(z), z^k \rangle = \langle z \Phi_n(z), z^k \rangle - \langle \Phi_{n+1}(z), z^k \rangle$

$= \langle z \Phi_n(z), z^{k-1} \rangle - \langle \Phi_{n+1}(z), z^k \rangle$.

Since $\Phi_n$ and $\Phi_{n+1}$ is a monic orthogonal polynomial, this is zero if $k \in \{1, 2, \dots, n\}$. Therefore $z\Phi_n(z) - \Phi_{n+1}(z)$ is a constant multiple of $\Phi_n^*$! All and all, there is a constant $\alpha_n$ such that

$z \Phi_n(z) = \Phi_{n+1}(z) + \alpha_n \Phi_n^*(z)$.

Since $\Phi_{n+1}$ and $\Phi_n^*$ is orthogonal, we obtain that

$\| \Phi_n \|^2 = \| z \Phi_n \|^2 = \| \Phi_{n+1} \|^2 + |\alpha_n|^2 \| \Phi_n \|^2$,

which implies

$\| \Phi_{n+1} \|^2 = (1-|\alpha_n|^2) \| \Phi_n \|^2$.

Since the quantity on the left side is positive, it follows that

$|\alpha_n| < 1$.

The constants $\alpha_n$ are called the Verblunsky coefficients. As far as I know, they have been attributed to Verblunsky first by Barry Simon, who, according to him, “has the strongest personality, therefore the name stuck”. This was said at a lecture given by him at the Isaac Newton Institute for Mathematical Sciences at an instructional school dedicated to periodic, almost periodic and random operators. I recommend these lectures to watch, since they are wicked interesting and fun if you like the subject.

To summarize, if we are given a measure $\nu$ supported on the unit circle, we can obtain parameters $\alpha_n$ from the recurrence relation for orthogonal polynomials. Like in the OPRL case, it is an interesting question that is it true the other way? Given a sequence $\{ \alpha_n \}_{n=0}^{\infty}$ such that $|\alpha_n| < 1$, is there a measure with the same exact Verblunsky coefficients? The answer is affirmative, and contrary to the OPRL case, we always have uniqueness. This theorem is called Verblunsky’s theorem. [Simon] gives four proofs of this, although I will only give one in a later post. This is one of the many reasons for which I love the book [Simon]. There are several proofs for many theorems, for example, six proof can be found for the simple theorem stating that the zeros of $\Phi_n$ are contained in the open unit disk.

3. Conclusion. The purpose of this post was to set the stage for the spectral theory of orthogonal polynomials. In the next few posts, I will give three proof of the spectral theorem for OPRL (i. e. given a set of polynomials satisfying a three-term recursion, there is a measure for which the orthogonal polynomials coincide with our polynomials given by the recursion), and one for OPUC.

[Freud] Géza Freud, Orthogonal polynomials, Pergamon Press, 1971
[Simon] Barry Simon, Orthogonal polynomials on the unit circle, part I, AMS Colloquium Publications, 2004