OPUC ON ONE FOOT

BARRY SIMON*

Abstract. We present an expository introduction to orthogonal polynomials on the unit circle.

1. Introduction

Orthogonal polynomials are the Rodney Dangerfield [108] of analysis. Because of the impact of Stieltjes’ great 1895 paper on F. Riesz, Nevanlinna, and Hilbert’s school, the moment problem and the closely related subject of orthogonal polynomials on the real line (OPRL) were central in the revolution in analysis from 1900–1920 and provided critical precursors to the Hahn-Banach theorem, the Riesz-Markov theorem, the spectral theorem, and the theory of selfadjoint extensions. But in recent years, too often the subject is dismissed as “classical” and not worthy of further study.

With developments in random matrix theory and combinatorics (e.g., [4, 5, 6, 7, 12, 43, 62, 79]), it is clear that orthogonal polynomials still have a lot to contribute. From one point of view, what makes them relevant is that they are the simplest of inverse spectral problems — indeed, Gel’fand-Levitan [26] explicitly note that their approach to inverse theory for Schrödinger operators is motivated by OPRL. Recently, OPUC ideas have provided a matrix realization of Lax pairs for the (defocusing) AKNS equation [64].

What is true for OPRL is even more true for orthogonal polynomials on the unit circle (OPUC). While the closely related area of positive harmonic functions on 𝔻 = {z ∈ ℂ | |z| < 1} drew the attention of Carathéodory, Fejér, Herglotz, F. Riesz, Schur, and Toeplitz in the 1910’s, the subject was only invented by Szegő in about 1920, especially in his deep 1920–1921 paper [95]. So OPUC never had its era of centrality but has had a steady but small following over the years. Traditionally, the book references for the subject were Szegő’s book [97], which has only one full and several partial chapters on OPUC, Geronimus’ book [33] and review [32], and a chapter in Freud [25], which are very dated. With a major development published only in 2003 (the CMV matrix of Section 5 below), it is hard not to be dated. Motivated by this dearth of review literature and by the opportunity to use Schrödinger operator techniques in a new setting, I published two volumes [88, 89] on the subject. Many friends asked if there wasn’t some way to learn about the subject in less than 1100 pages, and this expository note is the result.

Date: February 2, 2005.

2000 Mathematics Subject Classification. 42C05, 30E05, 42A70.

Key words and phrases. orthogonal polynomials, Verblunsky coefficients, Szegő’s theorem.

∗ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125. E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592.

Throughout, we use 𝔻 for the unit disk in ℂ, and ∂𝔻 for the unit circle. Our inner products, ⟨f, g⟩, are linear in g and antilinear in f. Significant missing material involve some explicit examples — these are discussed in Section 1.6 of [88]: my favorite are the Rogers-Szegő polynomials (Example 1.6.5). This article undergoes a kind of phase transition in the middle of Section 5 in that before there, most results have proofs or at least sketches given, and afterwards, there aren’t many proofs. This is because the earlier material is more central and also because the later proofs are lengthier.

To put OPUC in context, recall some basics of OPRL. Since the fascinating issues of indeterminate moment problems (see [1, 85]) are irrelevant to OPUC, we will assume all measures have compact support:

(1) If μ is a probability measure on ℂ (i.e., positive with μ(ℂ) = 1) with compact support and X_{n}(z) are the monic orthogonal polynomials (i.e., X_{n}(z) = z^{n}+ lower order, X_{n} ⊥ z^{ℓ}, ℓ = 0, …, n − 1),

zX_{n}(z) = X_{n+1}(z) + ∑_{j=0}^{n} a_{j}^{(n)}X_{j}(z)  (1.1)

a_{j}^{(n)} = \frac{⟨X_{j}, zX_{n}⟩}{∥X_{j}∥^{2}}  (1.2)

What makes OPRL special is that multiplication by x is selfadjoint, so if we use P_{n} in place of X_{n} for OPRL and ρ for μ,

⟨P_{j}, xP_{n}⟩ = ⟨xP_{j}, P_{n}⟩ = 0 j = 0, …, n − 2

and thus (1.2) becomes

xP_{n}(x) = P_{n+1}(x) + b_{n+1}P_{n}(x) + a_{n}^{2}P_{n−1}(x)  (1.3)

for Jacobi parameters, a_{n}, b_{n}; n = 1, 2, . . .. If p_{n} = P_{n}/∥P_{n}∥ are the orthonormal OPRL, the matrix elements of multiplication by x in p_{n} basis have the form:

J = (\begin{matrix}b_{1} & a_{1} & 0 & 0 & … \\ a_{1} & b_{2} & a_{2} & 0 & … \\ 0 & a_{2} & b_{3} & a_{3} & … \\ … & … & … & … & … \end{matrix})  (1.4)

(2) There is a one-one correspondence between bounded J’s (i.e., sup_{n}|a_{n}|+|b_{n}| < ∞) and ρ on ℝ with compact but infinite support. This is sometimes called Favard’s theorem.

(3) If A is a bounded selfadjoint operator on a separable Hilbert space, H, and φ is a cyclic unit vector (i.e., {A^{n}φ}_{n=0}^{∞} span H), one can use the spectral theorem to find a measure dρ on [−∥A∥, ∥A∥] with ∫ x^{n} dρ = ⟨φ, A^{n}φ⟩ and then the OPRL for this measure to find a semi-infinite Jacobi matrix unitarily equivalent to A with φ mapped to (1 0 0 …)^{t}. This realization is unique, that is, the a_{n}’s and b_{n}’s are intrinsic to the pair (A, φ). It was Stone who emphasized this point of view that the study of Jacobi matrices was the same as the study of selfadjoint operators with a distinguished cyclic vector.

(4) A key role is played by the Stieltjes transform of ρ, that is, the function, m, on ℂ\supp(dρ) given by

m(z) = ∫ \frac{dρ(x)}{x − z}  (1.5)

(5) The Jacobi parameters can also be captured from m(z) via a continued fraction expansion (of Stieltjes) at ∞:

m(z) = \frac{1}{−z + b_{1} − \frac{a_{1}^{2}}{−z + b_{2} − a_{2}^{2} …}}  (1.6)

We will not discuss applications of OPUC in detail but note its important applications to linear prediction and filtering theory. The basics are due to Wiener [107], Kolmogorov [51], Krein [52, 53], and Levinson [57]. The ideas have been especially developed by Kailath [45, 46, 47].

The title of this article is based on an incident reported in the Talmud [99] that someone asked the famous first-century rabbi Hillel to describe Judaism to him while he stood on one foot. Hillel’s answer was: “Do not do unto others that which is hateful to you. The rest is commentary. Go forth and study.” This article is OPUC on one foot. [88, 89] are commentary.

It is a pleasure to thank M. Aizenman for pushing me to write such an article. I’d like to thank S. Denisov, F. Gesztesy, L. Golinskii, D. Lubinsky, F. Marcellán, P. Nevai, and G. Stolz for useful input. This paper was started while I was a visitor at the Courant Institute and completed during my stay as a Lady Davis Visiting Professor at Hebrew University, Jerusalem. I’d like to thank P. Deift and C. Newman for the hospitality of Courant and H. Farkas and Y. Last for the hospitality of the Mathematics Institute at Hebrew University.

2. The Szegő Recursion

OPUC is the study of probability measures on ∂𝔻, that is, positive measures, μ, with

μ(∂𝔻) = 1  (2.1)

The Carathéodory function (after [15]) of μ is defined on 𝔻 by

F (z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ)  (2.2)

This analog of (1.5) is an analytic function on 𝔻 which obeys

F (0) = 1  z ∈ 𝔻 ⇒ Re F (z) > 0  (2.3)

The Schur function (after [84]) is then defined by

F (z) = \frac{1 + zf(z)}{1 − zf(z)}  (2.4)

and is an analytic function mapping 𝔻 to \overline{𝔻}, that is,

sup_{z∈𝔻} |f(z)| ≤ 1  (2.5)

(f(z) ≡ e^{iθ_{0}} is included and produced by μ, a point mass at e^{iθ_{0}}).

(2.2) sets up a one-one correspondence between probability measures μ and analytic functions obeying (2.3) — this is essentially a form of the Herglotz representation (see [83, pp. 247]) and can be realized via

dμ = w-lim_{r↑1} Re F (re^{iθ}) \frac{dθ}{2π}  (2.6)

or by

F (z) = 1 + 2 ∑_{n=1}^{∞} c_{n}z^{n}  (2.7)

where c_{n} are the moments of μ given by

c_{n} = ∫ e^{−inθ} dμ(θ)  (2.8)

(2.4) sets up a bijection between f’s obeying (2.5) and F ’s obeying (2.3).

We call a measure trivial if it is supported on a finite set and nontrivial otherwise. We will mainly be interested in nontrivial measures. μ is trivial if and only if its Schur function is a finite Blaschke product

f(z) = e^{iθ_{0}} ∏_{j=1}^{n−1} \frac{z − z_{j}}{1 − z̄_{j}z}  (2.9)

with z_{1}, …, z_{n−1} ∈ 𝔻. Here n is the number of points in the support of dμ. Later (see the remark after Theorem 7.1) we will interpret (2.9) in terms of OPUC.

If μ is a nontrivial probability measure on ∂𝔻, we define the monic orthogonal polynomials Φ_{n}(z; dμ) (or Φ_{n}(z) if dμ is understood) by:

Φ_{n}(z) = z^{n}+ lower order ∫ e^{−ijθ}Φ_{n}(e^{iθ}) dμ(θ) = 0 j = 0, 1, 2, …, n−1 (2.10)

so in L^{2}(∂𝔻, dμ), ⟨Φ_{n}, Φ_{m}⟩ = 0 if n ≠ m. The orthonormal polynomials φ_{n} are defined by

φ_{n}(z) = \frac{Φ_{n}(z)}{∥Φ_{n}∥}  (2.11)

where ∥ · ∥ is the L^{2}-norm. {φ_{n}}_{n=0}^{∞} is an orthonormal set in L^{2}. It may not be a basis (e.g., dμ(θ) = dθ/2π where φ_{n}(z) = z^{n} and \overline{z^{j}}, j = 1, …, are orthogonal to all φ_{n}). We will discuss this further below (see Theorem 2.2).

If dμ is trivial, say supp(dμ) = {z_{j}}_{j=1}^{k}, we can still define Φ_{n}, φ_{n} for n = 0, 1, …, k −1. We can even define Φ_{k} (but not φ_{k}) as the unique monic polynomial of degree k with ∥Φ_{k}∥ = 0, that is,

Φ_{k}(z) = ∏_{j=1}^{k}(z − z_{j})  (μ trivial)  (2.12)

Clearly, (2.10) and the fact that the polynomials of degree at most n have dimension n + 1 implies

deg(P) ≤ n, P ⊥ z^{j}, j = 0, …, n − 1 ⇒ P = cΦ_{n}  (2.13)

On L^{2}(∂𝔻, dμ), define the anti-unitary map, ^{∗,n}, by

f^{∗,n}(e^{iθ}) = e^{inθ} \overline{f(e^{iθ})}  (2.14)

One mainly considers^{∗,n} on the set of polynomials of degree n which is left invariant:

P (z) = ∑_{j=0}^{n} c_{j}z^{j} ⇒ P^{∗,n}(z) = ∑_{j=0}^{n} c̄_{j}z^{n−j} = z^{n} \overline{P (1/z̄)}  (2.15)

Henceforth, following a standard, but unfortunate, convention, we drop the “ , n” and just use P^{∗}, hoping the n is implicit. Note that 1^{∗} = z^{n}, depending on n!

Since ^{∗} is anti-unitary, (2.13) implies

deg(P) ≤ n, P ⊥ z^{j}, j = 1, …, n ⇒ P = cΦ_{n}^{∗}  (2.16)

Since ⟨f, zg⟩ = ⟨z^{−1}f, g⟩, it is easy to see that Φ_{n+1} − zΦ_{n} ⊥ z^{j} for j = 1, 2, …, n. Since Φ is monic, this difference is of degree n, so (2.16) implies

Φ_{n+1}(z) = zΦ_{n}(z) − ᾱ_{n}Φ_{n}^{∗}(z)  (2.17)

for complex numbers α_{n}, called the Verblunsky coefficients (in the older literature, also called reflection, Schur, Szegő, or Geronimus coefficients). (2.17) is called Szegő recursion after its first occurrence in Szegő’s book [97]. In the engineering literature, it is called the Levinson algorithm after its rediscovery in linear prediction theory [57]. The choice of minus and ᾱ_{n} rather than α_{n} will be made clear by Geronimus’ theorem (see Theorem 3.1). Since Φ_{n} is monic, (2.15) implies Φ_{n}^{∗}(0) = 1, so (2.17) at z = 0 implies

α_{n} = −\overline{Φ_{n+1}(0)}  (2.18)

Theorem 2.1. We have

∥Φ_{n+1}∥^{2} = (1 − |α_{n}|^{2})∥Φ_{n}∥^{2}  (2.19)
∥Φ_{n}∥ = ∏_{j=0}^{n−1}(1 − |α_{j}|^{2})^{1/2}  (2.20)

For any nontrivial μ, we have α_{j}(dμ) ∈ 𝔻 for all j. If μ is trivial with n points in its support, then α_{0}(dμ), …, α_{n−2}(dμ) ∈ 𝔻 and α_{n−1}(dμ) ∈ ∂𝔻.

Proof. (2.17), unitarity of multiplication by z, and Φ_{n}^{∗} ⊥ Φ_{n+1} imply

∥Φ_{n}∥^{2} = ∥zΦ_{n}∥^{2} = ∥Φ_{n+1} + ᾱ_{n}Φ_{n}^{∗}∥^{2} = ∥Φ_{n+1}∥^{2} + |α_{n}|^{2}∥Φ_{n}∥^{2}

which implies (2.19). Induction and Φ_{0} = 1 implies (2.20). By (2.19), |α_{j}| < 1 in the nontrivial case and for j = 0, …, n−2 in the trivial case. Since ∥Φ_{n}∥ = 0 ≠ ∥Φ_{n−1}∥ in the trivial case, (2.19) implies |α_{n−1}| = 1. □

Since it arises often, we define

ρ_{j} = (1 − |α_{j}|^{2})^{1/2}  |α_{j}|^{2} + ρ_{j}^{2} = 1  (2.21)

One can use (2.20) to relate completeness of {φ_{n}}_{n=0}^{∞} to the Verblunsky coefficients:

Theorem 2.2. For any nontrivial measure, the following are equivalent:

(a) lim_{n→∞} ∥Φ_{n}∥ = 0

(b) ∑_{j=0}^{∞}|α_{j}|^{2} = ∞

(c) {φ_{n}}_{n=0}^{∞} are a basis for L^{2}(∂𝔻, dμ)

Remark. We will see later that there is an additional equivalence via Szegő’s theorem (see (8.9)). The equivalence of a Szegő condition to completeness is due to Kolmogorov [51] and Krein [52, 53].

Sketch. By (2.20), (a) ⇔ (b). If

P_{[k,ℓ]} = projection in L^{2}(∂𝔻, dμ) onto span{z^{m}}_{m=k}^{ℓ}  (2.22)

we have that

∥Φ_{n}∥ = ∥(1 − P_{[0,n−1]})z^{n}∥  (2.23)

= ∥(1 − P_{[1,n]})1∥  (2.24)
= ∥(1 − P_{[0,n−1]})z^{−1}∥  (2.25)

where (2.23) follows from the definition of Φ_{n}, (2.24) by applying ^{∗,n} to z^{n} and P_{[0,n−1]}, and (2.25) by using the fact that multiplication by z^{−1} is unitary. It follows that

∥(1 − P_{[0,∞)})z^{−1}∥ = lim_{n→∞} ∥Φ_{n}∥  (2.26)

so (a) ⇔ z^{−1} ∈ span{φ_{n}}_{n=0}^{∞}. If z^{−1} ∉ span{φ_{n}}_{n=0}^{∞}, clearly they are not complete. If z^{−1} ∈ span{φ_{j}}_{j=0}^{∞}, an argument (see the proof of Theorem 1.5.7 in [88]) taking powers of z^{−1} shows z^{−ℓ} ∈ span{φ_{n}}_{n=0}^{∞} for all ℓ, so {φ_{n}}_{n=0}^{∞} are complete. □

Let 𝔻^{∞,c} denote the set of complex sequences {α_{j}}_{j=0}^{N} where either N = ∞ and |α_{j}| < 1 for all j, or else N < ∞ and α_{0}, …, α_{N−1} ∈ 𝔻 while α_{N} ∈ ∂𝔻. In the topology of componentwise convergence, 𝔻^{∞,c} is compact (and is a compactification of 𝔻^{∞}). The map, S, from μ ↦ {α_{j}(dμ)}_{j=0}^{N} is a well-defined map from M_{+,1}(∂𝔻), the probability measures on ∂𝔻, to 𝔻^{∞,c}. By (2.17), the α’s determine the Φ_{n}’s. Since ∫ Φ_{n}(z) dμ = δ_{n0}, the Φ_{n}’s determine the moments inductively, and so dμ, since {z^{ℓ}}_{ℓ=−∞}^{∞} span a dense set of C(∂𝔻). Thus S is one-one. Moreover,

Theorem 2.3 (Verblunsky’s Theorem [102]). S is onto.

[88] has four proofs of this theorem (Theorems 1.7.11, 3.1.3, 4.1.5, and 4.2.8); see Section 3 below. Given that S is a bijection, it is easy to see that it is a homeomorphism if M_{+,1}(∂𝔻) is given the vague (i.e., C(∂𝔻)-weak ^{∗}) topology.

Applying ^{∗} (actually, ^{∗,n+1}) to (2.17) yields

Φ_{n+1}^{∗}(z) = Φ_{n}^{∗}(z) − α_{n}zΦ_{n}(z)  (2.27)

Using (2.19) and (2.11), we get the recursion relations for φ_{n} written in matrix form

\binom{φ_{n+1}(z)}{φ_{n+1}^{∗}(z)} = A(z, α_{n})\binom{φ_{n}(z)}{φ_{n}^{∗}(z)}  (2.28)

where

A(z, α) = ρ^{−1} (\begin{matrix}z & −ᾱ \\ −zα & 1 \end{matrix})  (2.29)

Notice that det A = z, so by inverting A, we get inverse recursion relations. We note the one for Φ_{n−1}:

Φ_{n−1}(z) = \frac{ρ_{n−1}^{−2}[Φ_{n} + ᾱ_{n−1}Φ_{n}^{∗}]}{z}  (2.30)

Note that, by (2.18), […] vanishes at zero, so the right side of (2.30) is a polynomial of degree n − 1. This implies:

Theorem 2.4 (Geronimus [31]). Let μ, ν be two probability measures on ∂𝔻 so that for some N_{0}, Φ_{N_{0}}(z; dμ) = Φ_{N_{0}}(z; dν). Then Φ_{j}(z; dμ) = Φ_{j}(z; dν) for j = 0, 1, …, N_{0} −1, α_{j}(dμ) = α_{j}(dν) for j = 0, 1, …, N_{0} −1, and φ_{j}(z; dμ) = φ_{j}(z; dν) for j = 0, 1, …, N_{0}.

Remark. As noted in a footnote in Geronimus [31] and rediscovered by Wendroff [105], the result for OPRL requires equality for P_{N_{0}} and P_{N_{0}−1} and, in particular, it often happens that P_{N_{0}}(x, dγ) = P_{N_{0}}(x, dρ), but no other P_{j}’s are equal.

Proof. By (2.18), Φ_{N_{0}} at 0 determines α_{N_{0}−1}, and so ρ_{N_{0}−1}, and thus Φ_{N_{0}−1} by (2.30). By induction, all α_{j}, j ≤ N_{0} − 1, and Φ_{j}, j ≤ N_{0}, are equal and so, by (2.20), ∥Φ_{j}∥, and so φ_{j}. □

As a final aspect of Szegő recursion, we turn to the Christoffel-Darboux formula (proven by Szegő [97] for OPUC; Christoffel [16] and Darboux [18] had a similar formula for OPRL), which is an analog of an iterated Wronskian formula for ODE’s. With A given by (2.29), one finds, by matrix multiplication, that

A(ζ, α_{n})^{∗} (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix}) A(z, α_{n}) = (\begin{matrix}−zζ̄ & 0 \\ 0 & 1 \end{matrix})  (2.31)

so

\overline{φ_{n+1}^{∗}(ζ)} φ_{n+1}^{∗}(z) − \overline{φ_{n+1}(ζ)} φ_{n+1}(z)
= \overline{φ_{n}^{∗}(ζ)} φ_{n}^{∗}(z) − zζ̄ \overline{φ_{n}(ζ)} φ_{n}(z)
= (1 − zζ̄) \overline{φ_{n}(ζ)} φ_{n}(z) + [ \overline{φ_{n}^{∗}(ζ)} φ_{n}^{∗}(z) − \overline{φ_{n}^{∗}(ζ)} φ_{n}(z)]

which, iterated to n = 0 (where […] = 0), yields

Theorem 2.5 (Szegő [97]; CD Formula for OPUC).

(1 − zζ̄) ∑_{j=0}^{n} \overline{φ_{n}(ζ)} φ_{n}(z) = \overline{φ_{n+1}^{∗}(ζ)} φ_{n+1}^{∗}(z) − \overline{φ_{n+1}(ζ)} φ_{n+1}(z)  (2.32)

If z = ζ and lie in 𝔻, we have various positivity facts that imply (the first since φ_{0}(z) = 1):

|φ_{n}^{∗}(z_{0})| ≥ (1 − |z_{0}|^{2})^{1/2}  for z_{0} ∈ 𝔻  (2.33)
lim |φ_{n+1}^{∗}(z_{0})| = ∞ ⇔ ∑_{j=0}^{∞}|φ_{j}(z_{0})|^{2} = ∞  for z_{0} ∈ 𝔻  (2.34)

3. Verblunsky’s and Geronimus’ Theorems

In this section, we will prove Verblunsky’s theorem (Theorem 2.3) and also a celebrated theorem of Geronimus. Our approach follows Section 3.1 of [88] which claims a new proof of Geronimus’ theorem assuming Verblunsky’s theorem. But in preparing this article, we realized the argument can be slightly modified to also prove Verblunsky’s theorem.

To state Geronimus’ theorem, we need to describe the Schur algorithm [84]. Given a Schur function, f, define

γ_{0}(f) = f(0)  f(z) = \frac{γ_{0} + zf_{1}(z)}{1 + γ̄_{0}zf_{1}(z)}  (3.1)

If γ_{0} ∈ ∂𝔻 (i.e., f(z) ≡ γ_{0}), we do not define f_{1}. Otherwise, f_{1} defined by (3.1) is also a Schur function since w → (γ_{0} + w)/(1 + γ̄_{0}w) is a biholomorphic bijection of 𝔻 to 𝔻 if |γ_{0}| < 1, and g a Schur function with g(0) = 0 implies g(z)/z is a Schur function (the Schwarz lemma).

(3.1) is called the Schur algorithm. It can be iterated, that is, we define γ_{n}(f), the Schur parameters, and f_{n+1}, the Schur iterates, inductively by

γ_{n}(f) = f_{n}(0)  f_{n}(z) = \frac{γ_{n} + zf_{n+1}(z)}{1 + γ̄_{n}zf_{n+1}(z)}  (3.2)

If, for some n, f_{n}(z) = e^{iθ_{0}}, we set γ_{n} = e^{iθ_{0}} and stop. In this way, we map any Schur function, f, to a sequence in 𝔻^{∞,c}. We can now state Geronimus’ theorem:

Theorem 3.1 (Geronimus’ Theorem). Let μ be a probability measure on ∂𝔻, f its Schur function, and γ_{n}(dμ) ≡ γ_{n}(f) the Schur parameters of f. Then

γ_{n}(dμ) = α_{n}(dμ)  (3.3)

This gives a continued fraction expansion of F whose coefficients are α_{n}, and so is an analog of (1.6). This formula explains why we took a minus and conjugate in (2.17). The procedure of dropping a Verblunsky coefficient from the start can be understood by using the recursion relations and the relation of F to the OPUC (see Theorem 4.4 below). This approach to proving Theorem 3.1, due to Peherstorfer [69], is discussed in Section 3.3 of [88].

(3.1)/(3.2) can be rewritten and then iterated following Schur [84]:

f(z) = γ_{0} + (1 − γ̄_{0}f)zf_{1}  (3.4)
= γ_{0} + ∑_{j=1}^{n−1}[ ∏_{k=0}^{j−1}(1 − γ̄_{k}f_{k})]z^{j}γ_{j} + ∏_{k=0}^{n−1}(1 − γ̄_{k}f_{k})z^{n}f_{n}  (3.5)

which implies that if f(z) = ∑_{n=0}^{∞} a_{n}(f)z^{n}, then

a_{n}(f) = γ_{n} ∏_{j=0}^{n−1}(1 − |γ_{j}|^{2}) + polynomial in (γ_{0}, γ̄_{0}, …, γ_{n−1}, γ̄_{n−1})  (3.6)

Plugging this into (2.4) and using (2.7) implies

c_{n}(dμ) = γ_{n−1} ∏_{j=0}^{n−2}(1 − |γ_{j}|^{2}) + polynomial in (γ_{0}, γ̄_{0}, …, γ_{n−2}, γ̄_{n−2}) (3.7)

(the polynomials are different but the leading terms are the same up to a shift of index).

(3.6) also shows that if γ_{j}(f) = γ_{j}(g) for j = 0, …, n−1, then the Schur function \frac{1}{2}(f − g) = O(z^{n}) so, by the Schwarz lemma,

γ_{j}(f) = γ_{j}(g), j = 0, …, n − 1 ⇒ |f(z) − g(z)| ≤ 2|z|^{n}  (3.8)

Lemma 3.2. The map from Schur functions to 𝔻^{∞,c} is one-one and onto.

Proof. (3.8) shows that if γ_{j}(f) = γ_{j}(g) for all j, then f = g on 𝔻. Given a sequence in 𝔻^{∞}, define the Schur approximates, f^{[n]}, by setting f_{n+1}^{[n]} to 0 in (3.2) and using {γ_{j}}_{j=0}^{n} to define f_{n}^{[n]}, f_{n−1}^{[n]}, …, f^{[n]}. By construction,

γ_{j}(f^{[n]}) = \begin{cases}γ_{j} & j ≤ n \\ 0 & j > n \end{cases}  (3.9)

Since γ_{j}(f^{[n]}) = γ_{j}(f^{[m]}) for j ≤ min(n, m), we have, by (3.8), that f^{[n]} converge uniformly on compacts and the limit clearly has the prescribed set of γ’s. Given a sequence in 𝔻^{∞,c}\𝔻^{∞}, suppose γ_{n+1} = e^{iθ_{0}} ∈ ∂𝔻, set f_{n+1} ≡ e^{iθ_{0}} and use (3.2) to define f with the prescribed γ’s. □

Proof of Theorems 2.3 and 3.1. (z^{n} − Φ_{n}) ⊥ Φ_{n} ⇒ ∥Φ_{n}∥^{2} = ⟨z^{n}, Φ_{n}⟩, so applying ∗,n,

⟨Φ_{n}^{∗}, 1⟩ = ∥Φ_{n}∥^{2} = ∏_{j=0}^{n−1}(1 − |α_{j}|^{2})  (3.10)

Taking the inner product of (2.17) with the function, 1, and using ⟨Φ_{n+1}, 1⟩ = 0, we see

⟨zΦ_{n}, 1⟩ = α_{n} ∏_{j=0}^{n−1}(1 − |α_{j}|^{2})  (3.11)

By (2.17) and induction, the coefficients of Φ_{j} are polynomials α_{0}, ᾱ_{0}, …, α_{j−1}, ᾱ_{j−1} and so, by induction, the moments c_{j+1} are polynomials in the same α’s. Then (3.11) becomes (a formula of Verblunsky)

c_{n+1} = α_{n} ∏_{j=0}^{n−1}(1 − |α_{j}|^{2}) + polynomial in (α_{0}, ᾱ_{0}, …, α_{n−1}, ᾱ_{n−1})  (3.12)

We will now prove Theorem 3.1 by induction and then Theorem 2.3 follows from Lemma 3.2. For n = 0, we have, by (3.12) and (3.7), that

c_{1} = α_{0} = γ_{0}  (3.13)

Suppose we know α_{j} = γ_{j} for j = 0, 1, …, n − 1. We fix those n values in 𝔻 and ask what values of c_{n+1} can occur. By (3.7), it is a solid disk in ℂ of radius ∏_{j=0}^{n−1}(1 − |α_{j}|^{2}) since γ_{n} can run through \overline{𝔻}. The center of the disk is some fixed point (given fixed {γ_{j}}_{j=0}^{n−1}).

By (3.12), it is also a subset of the disk of radius ∏_{j=0}^{n−1}(1 − |α_{j}|^{2}) with possibly another center. But since the sets are the same, the centers must be the same, and all α_{j} must occur. Once we know the centers and radii are the same, the equality of the formulae for c_{n+1} implies α_{n} = γ_{n}. □

4. Zeros, the Bernstein-Szegő Approximation, and Boundary Conditions

Our first goal in this section is to prove that the zeros of OPUC lie in 𝔻. There are six proofs of this in [88]. We pick the one that is shortest, using the same argument that led to (2.19).

Theorem 4.1. Φ_{n} has all its zeros in 𝔻 and Φ_{n}^{∗} has all its zeros in ℂ\\overline{𝔻}.

Proof. (Landau [56]) Let Φ_{n}(z_{0}) = 0 and define P (z) = Φ_{n}(z)/(z − z_{0}). Since deg P = n − 1, P ⊥ Φ_{n}. Thus

∥P ∥^{2} = ∥zP ∥^{2} = ∥z_{0}P + Φ_{n}∥^{2} = |z_{0}|^{2}∥P ∥^{2} + ∥Φ_{n}∥^{2}  (4.1)

so ∥Φ_{n}∥^{2} = (1 − |z_{0}|^{2})∥P ∥^{2}, implying |z_{0}| < 1. Since Φ_{n}^{∗}(z_{0}) = 0 ⇔ Φ_{n}(1/z̄) = 0, the result for Φ_{n} implies the result for Φ_{n}^{∗}. □

Next, we will identify measures with α_{j}(dμ) = 0 for j ≥ n_{0}. The key is a calculation that goes back to Erdélyi et al. [22].

Proposition 4.2. Let P_{n} be a polynomial of degree n with all zeros in 𝔻. Let

dμ = \frac{c dθ}{2π|P_{n}(e^{iθ})|^{2}}  (4.2)

where c is picked to make dμ a probability measure. Then for all integral j < n (including j < 0),

⟨z^{j}, P ⟩_{L^{2}(∂𝔻,dμ)} = 0  (4.3)

Proof.

⟨z^{j}, P ⟩_{L^{2}(∂𝔻,dμ)} = ∫ e^{−ijθ}P (e^{iθ}) \frac{dθ}{2πz^{−n}P^{∗}(z)P (z)|_{z=e^{iθ}}}

= \frac{1}{2πi} ∮ z^{n−j−1} \frac{dz}{P^{∗}(z)}

is zero for n − j − 1 ≥ 0 since P^{∗}(z) is nonvanishing on \overline{𝔻}. □

Theorem 4.3. Let dμ be a nontrivial probability measure on ∂𝔻. Let

dμ_{n} = \frac{dθ}{2π|φ_{n}(e^{iθ}, dμ)|^{2}}  (4.4)

Then dμ_{n} is a probability measure with

α_{j}(dμ_{n}) = \begin{cases}α_{j}(dμ) & j ≤ n − 1 \\ 0 & j ≥ n \end{cases}  (4.5)

Proof. Let dν = c dμ_{n} where c is picked so that ∫ dν = 1 (eventually, we will prove c = 1). By Proposition 4.2, ⟨z^{j}, Φ_{n}( ·; dμ)⟩_{L^{2}(∂𝔻,dν)} = 0 for j = 0, 1, …, n − 1, so Φ_{n}(z; dν) = Φ_{n}(z; dμ). It follows from Theorem 2.3 that α_{j}(dν) = α_{j}(dμ) for j = 0, …, n − 1 and φ_{n}(z; dμ) = φ_{n}(z; dν). Therefore, 1 = ∫ |φ_{n}|^{2} dν = c, so dν = dμ_{n}.

By Proposition 4.3, for any k ≥ 0,

⟨z^{j}, z^{k}Φ_{n}⟩_{L^{2}(∂𝔻,dμ_{n})} = 0  j = 0, …, n + k − 1  (4.6)

It follows that Φ_{n+k}(z; dμ_{n}) = z^{k}Φ_{n}(z; dμ_{n}) and thus, Φ_{n+k}(0) = 0 for k ≥ 1. Therefore, by (2.18), α_{j}(dμ_{n}) = 0 for j ≥ n. □

Even though Theorem 4.3 was proven by Verblunsky [103] and rediscovered by Geronimus [31] (to whom it is often credited), dμ_{n} are called Bernstein-Szegő approximations since Szegő [94] first considered measures of this form (3.2) and Bernstein [11] their OPRL analog. Since, for each fixed j, α_{j}(dμ_{n}) → α_{j}(dμ) (indeed, they are equal for n > j), dμ_{n} → dμ weakly since S is a homeomorphism.

Some thought about the form of dμ_{n} suggests its Carathéodory function should be a rational function whose denominator is φ_{n}^{∗}. We will prove this by identifying the numerator. The second kind polynomials, ψ_{n}, are the OPUC for the measure dμ_{−1} with α_{j}(dμ_{−1}) = −α_{j}(dμ). Notice that in terms of the matrix A of (2.29),

\binom{ψ_{n+1}}{−ψ_{n+1}^{∗}} = A(z, α_{n}(dμ))\binom{ψ_{n}}{−ψ_{n}^{∗}}  (4.7)

(note α_{n}(dμ), not α_{n}(dμ_{−1})). Thus

( \begin{matrix}ψ_{n} & φ_{n} \\ −ψ_{n}^{∗} & φ_{n}^{∗} \end{matrix} ) = A(z, α_{n−1}) … A(z, α_{0}) ( \begin{matrix}1 & 1 \\ −1 & 1 \end{matrix} )  (4.8)

Taking determinants, using det(A) = z,

φ_{n}^{∗}ψ_{n} + φ_{n}ψ_{n}^{∗} = 2z^{n}  (4.9)

Theorem 4.4 (Verblunsky [103]). Let dμ_{n} be given by (4.4). Then

F (z, dμ_{n}) = \frac{ψ_{n}^{∗}(z; dμ)}{φ_{n}^{∗}(z; dμ)}  (4.10)

Proof. For z = e^{iθ}, (4.9) can be rewritten as Re( \overline{φ_{n}(e^{iθ})} ψ_{n}(e^{iθ})) = 1. Thus, if G(z) is the right side of (4.10),

Re(G(e^{iθ})) = \frac{1}{|φ_{n}(e^{iθ})|^{2}}  (4.11)

Since G is analytic in a neighborhood of \overline{𝔻}, Re G > 0 on 𝔻. Since G(0) = 1, the complex Poisson representation (see Rudin [83, pg. 235]) and (4.10) imply that G(z) is the Carathéodory function of dμ_{n}. □

It is useful to think of dμ and dμ_{−1} as embedded in a family dμ_{λ} for λ ∈ ∂𝔻. The Aleksandrov family associated to dμ is defined by

α_{j}(dμ_{λ}) = λα_{j}(dμ)  (4.12)

Given Geronimus’ theorem (Theorem 3.1), it is easy to see that

f(z, dμ_{λ}) = λf(z, dμ)  (4.13)

(for γ_{0}(λf) = λγ_{0}(f) and (λf)_{1} = λ(f_{1})). So, by (2.4) and its inverse, zf(z) = (F (z) − 1)/(F (z) + 1),

F (z, dμ_{λ}) = \frac{(1 − λ) + (1 + λ)F (z, dμ)}{(1 + λ) + (1 − λ)F (z, dμ)}  (4.14)

which is the original definition of Aleksandrov [2]; it is Golinskii-Nevai [39] who realized its relevance to OPUC and boundary conditions. If φ_{n}^{(λ)}(z) = φ_{n}(z; dμ_{λ}), then

\binom{φ_{n+1}^{(λ)}}{λ̄φ_{n+1}^{(λ)}} = A(z, α_{n})\binom{φ_{n}^{(λ)}}{λ̄φ_{n}^{(λ)}}  (4.15)

so φ_{n} and φ_{n}^{(λ)} obey the same difference equation, but the n = 0 boundary values change from \binom{1}{1} to \binom{1}{λ̄}. The Aleksandrov family is the analog of variation of boundary conditions in second-order ODE’s.

A direct calculation (via contour integrals) shows that if Re(a) > 0, then

∫_{0}^{2π} \frac{(1 − e^{iθ}) + (1 + e^{iθ})a}{(1 + e^{iθ}) + (1 − e^{iθ})a} \frac{dθ}{2π} = 1  (4.16)

Since 1 is the Carathéodory function of dθ/2π, (4.16) and (4.14) imply

Theorem 4.5 (Aleksandrov [2], Golinskii [38]). For the Aleksandrov family, we have

∫_{θ}[dμ_{e^{iθ}}(φ)] \frac{dθ}{2π} = \frac{dφ}{2π}  (4.17)

This is the OPUC analog of the Javrjan [42]-Wegner [104] averaging for Schrödinger operators, which is the basis of the localization proof of Simon-Wolff [93]. It can be used [89] to prove localization for suitable random OPUC.

5. The CMV Matrix

Perturbation theory involves looking at similarities of measures when their Verblunsky coefficients are close in some suitable sense. In the analogous OPRL situation, the Jacobi matrices, (1.4), are an invaluable tool. If one defines the essential support of a measure to be the support with isolated points removed, and if ρ and γ are measures on [c, d] ⊂ ℝ with Jacobi parameters a_{n}, b_{n} and ã_{n}, b̃_{n}, then ρ and γ have the same essential support if |a_{n} − ã_{n}| + |b_{n} − b̃_{n}| → 0. This can be seen by noting that the difference of the Jacobi matrices is compact and then appealing to Weyl’s theorem on the invariance of essential spectrum.

In this section, we discuss a suitable matrix representation for multiplication by z in L^{2}(∂𝔻, dμ). There is an obvious choice, namely, ⟨φ_{n}, zφ_{m}⟩, but this is not the “right” one. It has two problems. If ∑|α_{j}|^{2} < ∞, {φ_{n}}_{n=0}^{∞} is not a basis, and so this matrix is not unitary. Even worse, this matrix has finite columns (⟨φ_{n}, zφ_{m}⟩ = 0 if n > m + 1) but, in general, it does not have finite rows.

The right basis, as discovered by Cantero, Moral, and Velázquez [14] is the one, χ_{0}, χ_{1}, χ_{2}, …, obtained by orthonormalizing 1, z, z^{−1}, z^{2}, z^{−2}, . . .. We will also want to consider the basis, x_{0}, x_{1}, x_{2}, … obtained by orthonormalizing 1, z^{−1}, z, z^{−2}, . . .. Remarkably, the χ’s can be expressed in terms of φ’s and φ^{∗}’s, and the matrix elements in terms of α’s and ρ’s.

Proposition 5.1.

(a)  χ_{2n}(z) = z^{−n}φ_{2n}^{∗}(z)  χ_{2n−1}(z) = z^{−n+1}φ_{2n−1}(z)  (5.1)
(b)  x_{2n}(z) = z^{−n}φ_{2n}(z)  x_{2n−1}(z) = z^{−n}φ_{2n−1}^{∗}(z)  (5.2)

Proof. In terms of the projections P_{[k,ℓ]} of (2.22), we have

φ_{m} = \frac{(1 − P_{[0,m−1]})z^{m}}{∥ … ∥}  φ_{m}^{∗} = \frac{(1 − P_{[1,m]})1}{∥ … ∥}  (5.3)

where ∥ … ∥ is the norm of the numerator. Since multiplication by z^{ℓ} is unitary,

z^{−n}φ_{2n} = \frac{(1 − P_{[−n,n−1]})z^{n}}{∥ … ∥} = x_{2n}

proving the first half of (5.2). The others are similar. □

We define four matrices (C = CMV matrix) by:

C_{kℓ} = ⟨χ_{k}, zχ_{ℓ}⟩ C̃_{kℓ} = ⟨x_{k}, zx_{ℓ}⟩ L_{kℓ} = ⟨χ_{k}, zx_{ℓ}⟩ M_{kℓ} = ⟨x_{k}, χ_{ℓ}⟩  (5.4)

Clearly,

C = LM  C̃ = ML  C̃ = C^{t}  (5.5)

where the last comes from the fact that the explicit formulae below show L and M are (complex) symmetric. Define, for α ∈ \overline{𝔻}, the 2 × 2 symmetric matrix:

Θ(α) = ( \begin{matrix}ᾱ & ρ \\ ρ & −α \end{matrix} )  (5.6)

Theorem 5.2. Let 1 be the 1 × 1 unit matrix. Then

M = 1 ⊕ Θ(α_{1}) ⊕ Θ(α_{3}) ⊕ ⋯ L = Θ(α_{0}) ⊕ Θ(α_{2}) ⊕ Θ(α_{4}) ⊕ ⋯  (5.7)

Proof. This is an expression of the Szegő recursion formula. For example, the 2n row (labelling rows 0, 1, 2, …) of L says that zx_{2n} = ᾱ_{2n}χ_{2n} + ρ_{2n}χ_{2n+1} which, by Proposition 5.1, is equivalent to zφ_{2n} = ᾱ_{2n}φ_{2n}^{∗} + ρ_{2n}φ_{2n+1}, which is the top row of (2.28). □

While L and M have direct sum structures, in general (i.e., if all |α_{j}| < 1), C does not. Indeed, by (5.5) and (5.7),

C = ( \begin{matrix}ᾱ_{0} & ᾱ_{1}ρ_{0} & ρ_{1}ρ_{0} & 0 & 0 & … \\ ρ_{0} & −ᾱ_{1}α_{0} & −ρ_{1}α_{0} & 0 & 0 & … \\ 0 & ᾱ_{2}ρ_{1} & −ᾱ_{2}α_{1} & ᾱ_{3}ρ_{2} & ρ_{3}ρ_{2} & … \\ 0 & ρ_{2}ρ_{1} & −ρ_{2}α_{1} & −ᾱ_{3}α_{2} & −ρ_{3}α_{2} & … \\ 0 & 0 & 0 & ᾱ_{4}ρ_{3} & −ᾱ_{4}α_{3} & … \\ … & … & … & … & … & … \end{matrix} )  (5.8)

Thus C has a 4 × 2 block structure and is generally five-diagonal. It is the simplest unitary matrix with a cyclic vector; for example [13], any tridiagonal semi-infinite unitary is a direct sum of 1 × 1 and 2 × 2 matrices.

Theorem 5.3. If C^{(N)} is the top left N × N block of C, then

det(z_{0}1 − C^{(N)}) = Φ_{N}(z_{0})  (5.9)

Sketch. If ζ is the operator of multiplication by z in L^{2}(∂𝔻, dμ), then C^{(N)} = P_{[−ℓ,N−ℓ−1]}ζP_{[−ℓ,N−ℓ−1]} restricted to ranP_{[−ℓ,N−ℓ−1]} where P_{[j,k]} is given by (2.22) and ℓ is either (N − 1)/2 or N/2. Since multiplication by z^{ℓ} is unitary, C^{(N)} is unitarily equivalent to P_{[0,N−1]}ζP_{[0,N−1]} on ranP_{[0,N−1]}.

z_{0} is an eigenvalue of P_{[0,N−1]}ζP_{[0,N−1]} if and only if there is Q of degree N − 1 so (z − z_{0})Q = Φ_{N}(z), that is, if and only if Φ_{N}(z_{0}) = 0. This proves (5.9) if Φ_{n} has distinct zeros. By a limiting argument (see Theorem 1.7.18 of [88]), (5.9) holds in general. □

Remark. This theorem sheds light on a result of Fejér [23] that for OP’s of general measures on ℂ, their zeros lie in the convex hull of supp(dμ). For (5.9) implies the zeros are in the numerical range of C^{(N)}, so the numerical range of C, which is the convex hull of supp(dμ) by the spectral theorem. In particular, Fejér’s theorem implies in the OPUC case that if ζ ∈ ∂𝔻 with d = dist(ζ, supp(dμ)) > 0 and Φ_{n}(z_{0}) = 0, then |z_{0} − ζ| ≥ \frac{1}{2}d^{2}.

Notice that if |α_{j}| = 1, Θ(α_{j}) = ( \begin{matrix}ᾱ_{j} & 0 \\ 0 & −α_{j} \end{matrix} ) is a direct sum, and so C = LM has a (j + 1) × (j + 1) unitary block in the upper corner. This implies that if β ∈ ∂𝔻, then Φ_{N}^{(β)} ≡ zΦ_{N−1} − βΦ_{N−1}^{∗} has all its zeros on ∂𝔻 since they are eigenvalues of a unitary matrix. The Φ_{N}^{(β)} are called paraorthogonal polynomials and studied in [31, 44, 37].

Dombrowski [20] proved that a Jacobi matrix with lim inf a_{n} = 0 has no a.c. spectrum by picking a subsequence with ∑_{j=0}^{∞} a_{n(j)} < ∞ and trace class perturbing to a decoupled direct sum of finite rank matrices. Unaware of this work, Simon-Spencer [92] proved a similar result if lim sup|b_{n}| = ∞. As noted by Golinskii-Simon [40], the same idea and CMV matrices prove the following, originally proven by other means [81].

Theorem 5.4 (Rakhmanov’s Lemma [81]). If μ is a probability measure on ∂𝔻 so lim sup|α_{n}(dμ)| = 1, then μ is singular with respect to dθ/2π.

Golinskii-Simon also use perturbations of CMV matrices to prove:

Theorem 5.5 ([40]). If μ, ν are two probability measures on ∂𝔻 so |α_{n}(dμ) − α_{n}(dν)| → 0, then ess sup(dμ) = ess sup(dν). If ∑_{n}|α_{n}(dμ) − α_{n}(dν)| < ∞, then the absolutely continuous parts of μ and ν are mutually absolutely continuous.

Aleksandrov families fit into CMV matrices with a twist. C({λα_{n}}) and C({α_{n}}) do not differ by a rank one perturbation — rather they do up to a unitary equivalence. Specifically:

Theorem 5.6. Let λ ∈ ∂𝔻 and {α_{n}} ∈ 𝔻^{∞}. Let D be the diagonal matrix with elements 1, λ^{−1}, 1, λ^{−1}, . . .. Then DC({λα_{n}})D^{−1} = L({α_{n}})M_{λ}({α_{n}}) where M_{λ} differs from M by having λ in the (1, 1) position instead of 1.

This is a restatement of Theorem 4.2.9 of [88]. A generalization to rank one perturbation in the n-th diagonal can be found in Simon [90].

CMV matrices have been generalized in two directions. First, OPUC can be thought of as an analog of half-line ODE. The whole-line analog is an extended CMV matrix, E, defined on ℓ_{2}(−∞, ∞) by a two-sided sequence {α_{n}}_{n=−∞}^{∞} as a product of ⋯ ⊕ Θ(α_{−2}) ⊕ Θ(α_{0}) ⊕ Θ(α_{2}) ⊕ ⋯ and ⋯ ⊕ Θ(α_{−1}) ⊕ Θ(α_{1}) ⊕ ⋯ where Θ(α_{j}) acts on the span of δ_{j} and δ_{j+1}. This is discussed in Sections 4.5, 10.5, and 10.16 of [88, 89]. It is useful in the study of ergodic (Section 6) and periodic (Section 10) OPUC. Gesztesy-Zinchenko [34, 35] have further results on E.

Second, if U is an n×n unitary matrix and φ is cyclic in that {U^{j}φ}_{j=0}^{n−1} is a basis, then the spectral measure for φ has n points, defines polynomials Φ_{0}, …, Φ_{n} and Verblunsky coefficients α_{0}, …, α_{n−2} ∈ 𝔻 and α_{n−1} ∈ ∂𝔻. U is unitarily equivalent to a finite CMV matrix, the upper block of an infinite matrix where α_{n−1} is taken in ∂𝔻.

Just as the theory of selfadjoint matrices with cyclic vector is identical to the theory of Jacobi matrices, the theory of unitary matrices with cyclic vector (i.e., {U^{j}φ}_{j=−∞}^{∞} spanning) is identical to the theory of CMV matrices. The Verblunsky coefficients are a complete set of unitary invariants.

In this regard, there is a natural question answered by Killip-Nenciu [50]. Let 𝕌(n) be the group of n×n unitary matrices and consider Haar measure on 𝕌(n). For a.e. U, (1 0 … 0)^{t} is cyclic, so there is induced a measure on Verblunsky coefficients α_{0}, …, α_{n−2} ∈ 𝔻 and α_{n−1} ∈ ∂𝔻. The measure is the same if (1 0 … 0)^{t} is replaced by any other vector or by a random choice (say, uniform distribution on the unit sphere in ℂ^{n}).

Theorem 5.7 ([50]). Under the measure induced by Haar measure on 𝕌(n), the α_{j} are independent (i.e., the induced measure is a product measure), α_{n−1} is uniformly distributed on ∂𝔻, and α_{j}, j = 1, …, n − 2, is distributed via

\frac{(n − j − 1)}{π} (1 − |α|^{2})^{(n−j−2)} d^{2}α  (5.10)

6. Transfer Matrices, Weyl Solutions, and Lyapunov Exponents

In this section, we present a potpourri of results connected with solutions of Szegő recursion (2.28) where the two components are freed of u_{2}^{∗} = u_{1} — indeed, we look at solutions for a fixed z. Thus, solutions have the form

u(z; n) = T_{n}(z)u(z; 0)  T_{n}(z) = A(z, α_{n−1}) … A(z, α_{0})  (6.1)

with A given by (2.29). T_{n} is called the transfer matrix. By (4.8), we have

\begin{aligned}T_{n}(z) & = \frac{1}{2} (\begin{matrix}φ_{n}(z) + ψ_{n}(z) & φ_{n}(z) − ψ_{n}(z) \\ φ_{n}^{∗}(z) − ψ_{n}^{∗}(z) & φ_{n}^{∗}(z) + ψ_{n}^{∗}(z) \end{matrix}) \\ & = ( ∏_{j=0}^{n−1} ρ_{j}^{−1/2}) (\begin{matrix}zB_{n−1}^{∗}(z) & −A_{n−1}^{∗}(z) \\ −zA_{n−1}(z) & B_{n−1}(z) \end{matrix}) \end{aligned}  (6.2)

where A_{n−1} and B_{n−1} are degree n − 1 polynomials and the ^{∗} term is ^{∗,n−1}. The degree count uses φ_{n}(0) = −ψ_{n}(0), φ_{n}^{∗}(0) = ψ_{n}^{∗}(z). A_{n} and B_{n} are the Wall polynomials which are related to the Schur approximants, f^{[n]}, of (3.9) by f^{[n]}(z) = A_{n}(z)/B_{n}(z).

For z ∈ ∂𝔻, T_{n} lies in the group 𝕌(1, 1) of matrices obeying M^{∗} ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) M = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). Features of this group play a role in advanced aspects of the theory; see [89], especially Section 10.4.

The solutions u_{φ} = (φ_{n}, φ_{n}^{∗}) and u_{ψ} = (ψ_{n}, −ψ_{n}^{∗}) of (6.1) can be combined into an ℓ_{2} solution for |z| < 1:

Theorem 6.1 (Geronimo [27]; Golinskii-Nevai [39]). Fix z ∈ 𝔻. Then u_{ψ}(z; n) + ru_{φ}(z; n) → 0 as n → ∞ for fixed r ∈ ℂ if and only if r = F (z). Moreover, u_{ψ} + F (z)u_{φ} is in ℓ^{2}.

Remark. In analogy to ODE theory, u_{ψ} + F (z)u_{φ} is called the Weyl solution.

Sketch ([39]) . Looking at the second component, we see that if u_{ψ} +ru_{φ} → 0, then −ψ_{n}^{∗} + rφ_{n}^{∗} → 0. By (2.33), r − ψ_{n}^{∗}/φ_{n}^{∗} → 0, so by (4.10), r = F (z). That the first components go to zero for r = F (z) will follow from the ℓ^{2} proof.

By using the CD formula (2.32) for φ and ψ plus a mixed CD formula obtained from (2.31) by using \binom{1}{1} and \binom{1}{−1}, one finds that

(1−|z|^{2}) ∑_{j=0}^{n−1} |ψ_{j}(z)+rφ_{j}(z)|^{2} = 4 Re(r)+|ψ_{n}^{∗}(z)−rφ_{n}^{∗}|^{2} −|ψ_{n}(z)+rφ_{n}(z)|^{2} (6.3)

Taking r = ψ_{n}^{∗}(z)/φ_{n}^{∗}(z), one finds

k ≤ n − 1 ⇒ ∑_{j=0}^{k} |ψ_{j} + \frac{ψ_{n}^{∗}}{φ_{n}^{∗}} φ_{j}|^{2} ≤ 4 Re(\frac{ψ_{n}^{∗}}{φ_{n}^{∗}} )  (6.4)

Taking n → ∞ and then k → ∞ shows

∑_{j=0}^{∞} |ψ_{j} + F φ_{j}|^{2} ≤ 4 Re(F)  (6.5)

The inequality in (6.5) plus equality in (6.3) imply that |ψ_{j}^{∗} − F φ_{j}^{∗}| ≤ |ψ_{j} + F φ_{j}|, so (6.5) implies u_{ψ} + F u_{φ} ∈ ℓ^{2}. □

Another way of proving the ℓ^{2} result, from [27], is illuminating. It starts from a formula which was Geronimus’ original definition of the second kind polynomials,

ψ_{n}(z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} [φ_{n}(e^{iθ}) − φ_{n}(z)] dμ(θ)  (6.6)

This and its image under the map ^{∗} imply

\begin{aligned}F (z)φ_{n}(z) + ψ_{n}(z) & = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} φ_{n}(e^{iθ}) dμ(θ) \\ F (z)φ_{n}^{∗}(z) − ψ_{n}^{∗}(z) & = z^{n} ∫ \frac{e^{iθ} + z}{e^{iθ} − z} \overline{φ_{n}(e^{iθ})} dμ(θ) \end{aligned}  (6.7)

Using ∫ |φ_{n}| dμ ≤ 1 and (e^{iθ} + z)(e^{iθ} − z)^{−1} = 1 + ∑_{n=1}^{∞} 2(e^{−iθ}z)^{n}, we see the Taylor coefficients of each expression in (6.7) are bounded by 2. Since ∫ e^{−ikθ}φ_{n}(e^{iθ}) dμ(θ) = 0 for k = 0, …, n − 1, we see |F φ_{n} + ψ_{n}| ≤ 2|z|^{n}(1 − |z|)^{−1}, while |F φ_{n}^{∗} − ψ_{n}^{∗}| ≤ 2|z|^{n+1}(1 − |z|)^{−1}. This proves not only an ℓ^{2} property but exponential decay.

The next issue we want to discuss is Lyapunov exponents. To understand them, it pays to also discuss the density of zeros, an object of independent interest. Given dμ, a nontrivial probability measure in ∂𝔻, define the measure dν_{n} on \overline{𝔻} to be the point measure which gives weight k/n to a zero of Φ_{n} of multiplicity k. On account of (5.9) for ℓ = 0, 1, 2, …,

∫ z^{ℓ} dν_{n}(z) = \frac{1}{n} Tr([C^{(n)}]^{ℓ})  (6.8)

which can help show that dν_{n} sometimes has a weak limit; if it does, we say the limit is the density of zeros. The limit may not exist; there even exist examples (see Example 1.1.17 of [88]) where the set of limit points of dν_{n} is all measures on \overline{𝔻}! Here is how (6.8) can be used:

Theorem 6.2 (Mhaskar-Saff [63]). If

lim_{n→∞} |α_{n}|^{1/n} = r  and  \frac{1}{n} ∑_{j=0}^{n−1} |α_{j}| → 0  (6.9)

(automatic if r < 1), then dν_{n} converges to the uniform measure on the circle of radius r.

Sketch. (See Sections 8.1 and 8.2 of [88] for details.) An argument that exploits Theorem 9.1 below and the fact that |α_{n−1}| = |Φ_{n}(0)| is the product of zeros shows that when the first equation in (6.9) holds, then all limits of dν_{n} are concentrated on the circle of radius r. The second equation and (6.8) show that any such limit, dν, has ∫ z^{ℓ} dν = δ_{ℓ0} for ℓ ≥ 0. □

The other case where we know ν_{n} has a limit is ergodic families of Verblunsky coefficients. Let (Ω, dβ) be a probability measure space, T : Ω → Ω, an invertible ergodic transformation, and V : Ω → 𝔻. For each ω ∈ Ω, define a measure dμ_{ω} by

α_{j}(dμ_{ω}) = V (T^{j}ω)  (6.10)

An argument using the ergodic theorem, (6.8), and control of lim|α_{n}(dμ_{ω})|^{1/n} show that so long as ∫ [− log V (ω)] dβ(ω) < ∞, then dμ_{ω} has for a.e. ω a limit supported on ∂𝔻 and ω-independent. The most important examples of ergodic families are random, periodic, almost periodic, and subshifts (see Chapters 10–12 of [89]).

Before leaving the subject of zeros, we note:

Theorem 6.3 (Widom [106]). If supp(dμ) is not all of ∂𝔻, then for any r < 1, sup_{n}(# of zeros of Φ_{n} in |z| < r) < ∞. In particular, any limit of dν_{n} is supported on ∂𝔻.

Theorem 6.4 (see Theorem 8.1.11 of [88]). If z_{0} in ∂𝔻 is an isolated point of supp(dμ), there is precisely a single zero of Φ_{n} near z_{0} for n large and it approaches z_{0} exponentially fast.

Finally, we discuss the Lyapunov exponent and Thouless formula:

Theorem 6.5 (see Theorem 10.5.8 of [89]). If the density of zeros measure, dν, exists and is supported on ∂𝔻, and if

ρ_{∞} = lim_{n→∞} ( ∏_{j=0}^{n−1} ρ_{j})^{1/n}  (6.11)

exists, then for z ∉ ∂𝔻, the following limit exists and is given by

γ(z) ≡ lim_{n→∞} \frac{1}{n} log ∥T_{n}(z)∥ = − log ρ_{∞} − ∫ log|e^{iθ} − z|^{−1} dν(θ)  (6.12)

γ is called the Lyapunov exponent. (6.12) is called the Thouless formula. For |z| > 1, |φ_{n}| > |φ_{n}^{∗}| and |ψ_{n}| > |ψ_{n}^{∗}|, we need only control the growth of |φ_{n}| and |ψ_{n}|. By (6.8), φ_{n} and ψ_{n} have the same density of zeros. Writing φ_{n} = ∏_{j=0}^{n−1} ρ_{j}^{−1} ∏_{zeros}(z − z_{ℓ}) easily yields (6.12).

See [89] for discussion of when (6.12) holds on ∂𝔻 and for further study of ergodic OPUC.

7. Khrushchev’s Formula, CMV Resolvents, and Rakhmanov’s Theorem

In two remarkable papers [48, 49], Khrushchev found deep connections between Schur iterates and the structure of OPUC. A key input for the theory is:

Theorem 7.1 (Khrushchev’s Formula). The Schur function for the measure |φ_{n}(e^{iθ}, dμ)|^{2} dμ(θ) is given by b_{n}(z)f_{n}(z), where f_{n} is the n-th Schur iterate (by Geronimus’ theorem, this is the Schur function of the measure with Verblunsky coefficients {α_{n+j}}_{j=0}^{∞}) and b_{n} is the Blaschke product,

b_{n}(z; dμ) = \frac{φ_{n}(z; dμ)}{φ_{n}^{∗}(z; dμ)}  (7.1)

Remark. Khrushchev’s formula illuminates (2.9). In this trivial measure case, {z_{j}}_{j=1}^{n−1} are the zeros of Φ_{n−1} and e^{iθ_{0}} is the Schur parameter, γ_{n−1}.

In terms of the CMV matrix, this gives a formula for ⟨δ_{n}, (C + z)(C − z)^{−1}δ_{n}⟩, and so when n = m for

G_{nm}(z) = ⟨δ_{n}, (C − z)^{−1}δ_{m}⟩  (7.2)

the analog of the Green’s function in ODE’s. Half-line Green’s functions for ODE’s have the form f_{−}(min(x, y))f_{+}(max(x, y)) where f_{−} (resp. f_{+}) obeys boundary conditions at x = 0 (resp. x = ∞). There is an analogous formula, due to Simon (even if n ≠ m), for G_{nm} in terms of the OPUC and Weyl solutions. It can be found in Section 4.4 of [88] and generalizes Theorem 7.1. Other proofs of Theorem 7.1 appear in Theorem 4.5.10 of [88] and Theorem 9.2.4 of [89]. The most important consequence of Khrushchev’s formula is:

Theorem 7.2 (Khrushchev [48]). The essential support of the a.c. part of dμ is all of ∂𝔻 if and only if

lim_{n→∞} ∫_{0}^{2π}|f_{n}(e^{iθ}, dμ)|^{2} \frac{dθ}{2π} = 0  (7.3)

Since

∫_{0}^{2π} f_{n}(e^{iθ}, dμ) \frac{dθ}{2π} = f_{n}(0) = α_{n}  (7.4)

an immediate corollary is

Theorem 7.3 (Rakhmanov’s Theorem). If the essential support of the a.c. part of dμ is all of ∂𝔻, then

lim_{n→∞} α_{n} = 0  (7.5)

This result is originally due to Rakhmanov [80, 81, 82] with important further developments by Máté-Nevai-Totik [59, 60, 65, 66]. Bello-López [10] extended this result to arcs, and Denisov [19] to OPRL. Here are some other important results of Khrushchev’s theory:

Theorem 7.4.

w-lim|φ_{n}(e^{iθ})|^{2} dμ = \frac{dθ}{2π} ⇔ (∀j) lim_{n→∞} α_{n+j}α_{n} = 0

Remark. (6.8) can be reinterpreted as saying weak Cesàro limits of |φ_{n}|^{2} dμ are the density of zeros when the latter is supported on ∂𝔻; see Section 8.2 of [88].

Theorem 7.5. Let f^{[n]} be the Schur approximates (given by (3.9)). Then

∫ |f^{[n]}(e^{iθ}) − f(e^{iθ})|^{2} \frac{dθ}{2π} → 0  (7.6)

if and only if either

(i) dμ_{ac} = 0, that is, μ is purely singular, or (ii) α_{n}(dμ) → 0. Moreover, if w-lim|φ_{n}(e^{iθ})|^{2} dμ = dθ/2π, then (7.6) holds.

As a consequence of these theorems, we get a result for sparse α’s:

Corollary 7.6. If lim_{n→∞} α_{n+j}α_{n} = 0 for all j, but lim sup_{n}|α_{n}| ≠ 0, then μ is purely singular continuous.

Theorem 7.7. Suppose that uniformly on compacts of ∂𝔻,

lim_{n→∞} \frac{Φ_{n+1}^{∗}(z)}{Φ_{n}^{∗}(z)} = G(z)  (7.7)

then either G(z) ≡ 1 or else for some a ∈ (0, 1] and λ ∈ ∂𝔻,

G(z) = \frac{1}{2} [(1 + λz) + \sqrt{(1 − λz)^{2} + 4a^{2}λz} ]  (7.8)

Note we have that G ≡ 1 if and only if lim_{n→∞} α_{n+j}α_{n} = 0 for all j and that Barrios-López have proven that (7.8) holds if and only if lim_{n→∞}|α_{n}| = a and lim_{n→∞} α_{n+1}α_{n}^{−1} = λ.

Khrushchev has also described all possible dν’s that can occur as w-lim|φ_{n}|^{2} dμ (i.e., for which the limit exists) and when they can occur (essentially, asymptotically period 1 or 2). The analogs of these w-limit and ratio asymptotic results for OPRL were found by Simon [87].

8. Szegő’s and Baxter’s Theorems

Szegő’s theorems may well be the most celebrated in OPUC. While they have expressions purely in terms of OPUC objects, for historical reasons, one should state them in terms of Toeplitz determinants, D_{n}(dμ). This is defined as the determinant of the (n + 1) × (n + 1) matrix {c_{k−ℓ}}_{0≤k,ℓ≤n} with c given by (2.8). D_{n} is the Gram determinant of {z^{k}}_{k=0}^{n} since ⟨z^{k}, z^{ℓ}⟩_{L^{2}(dμ)} = c_{k−ℓ}. The invariance of such determinants under triangular change of basis implies (using also (2.20))

D_{n}(dμ) = ∏_{j=0}^{n} ∥Φ_{j}∥^{2} = ∏_{j=0}^{n−1}(1 − |α_{j}|^{2})^{n−j}  (8.1)

which immediately implies

F (dμ) ≡ lim_{n→∞} D_{n}(dμ)^{1/n} = ∏_{j=0}^{∞}(1 − |α_{j}|^{2}) = lim_{n→∞} ∥Φ_{n}∥^{2}  (8.2)

G(dμ) ≡ lim_{n→∞} \frac{D_{n}(dμ)}{F (dμ)^{n+1}} = ∏_{j=0}^{∞}(1 − |α_{j}|^{2})^{−j−1}  (8.3)

F is always defined, although it may be 0. G is defined so long as F > 0, that is, so long as ∑_{j=0}^{∞}|α_{j}|^{2} < ∞. G may be infinite and is finite if and only if ∑_{j=0}^{∞} j|α_{j}|^{2} < ∞.

Szegő’s theorems express F and G in terms of the a.c. weight, w, of dμ:

dμ = w(θ) \frac{dθ}{2π} + dμ_{s}  (8.4)

where w ∈ L^{1}(∂𝔻, \frac{dθ}{2π}) and dμ_{s} is singular with respect to dθ/2π.

Theorem 8.1 (Szegő’s Theorem).

F (dμ) = ∏_{j=0}^{∞}(1 − |α_{j}|^{2}) = exp(∫ log(w(θ)) \frac{dθ}{2π} )  (8.5)

Remark. Szegő proved this when dμ_{s} = 0 in 1915; the proof below is basically his proof in [95]. The result does not depend on dμ_{s} — this was shown first by Verblunsky [103]. [88, 89] have five proofs of Theorem 8.1

Sketch when dμ_{s} = 0. Since Φ_{n}^{∗} is nonvanishing on 𝔻 and Φ_{n}^{∗}(0) = 1, ∫ log|Φ_{n}^{∗}(e^{iθ})|\frac{dθ}{2π} = 1. Thus, by Jensen’s inequality,

∥Φ_{n}∥^{2} ≡ ∫ |Φ_{n}^{∗}(e^{iθ})|^{2}w(θ) \frac{dθ}{2π} ≥ exp(∫ log(w(θ)) \frac{dθ}{2π} )

so F (dμ) ≥ RHS of (8.5). On the other hand, since Φ_{n}^{∗} is the projection of 1 to the complement of [z, …, z^{n}] in [1, …, z^{n}], we have

∥Φ_{n}^{∗}∥^{2} = min{∥P ∥_{L^{2}(dμ)}^{2} | deg P ≤ n, P (0) = 1}  (8.6)

Using (8.2) and a limit argument,

F (dμ) = min{∥f∥_{L^{2}(dμ)}^{2} | f ∈ H^{∞}, f(0) = 1}  (8.7)

Pick the trial functions f_{ε}(z) = g_{ε}(z)/g_{ε}(0) where

g_{ε}(z) = exp(− ∫ \frac{e^{iθ} + z}{e^{iθ} − z} log(w(θ) + ε)\frac{dθ}{4π} )

and take ε ↓ 0 to get F (dμ) ≤ RHS of (8.5). □

Because their singular continuous part is arbitrary, once an ℓ^{2} condition is dropped, dμ can be arbitrarily “bad”:

Theorem 8.2. Let dρ be a measure on ∂𝔻 with support all of ∂𝔻. Then there exist dμ, a probability measure on ∂𝔻 mutually equivalent to dρ, so that for all p > 2,

∑_{n=0}^{∞} |α_{n}(dμ)|^{p} < ∞  (8.8)

This is Theorem 2.10.1 of [88], proven using ideas of Totik [100] and the bounds in (8.5).

By (8.5), we get one of the gems of spectral theory, equivalences between some recursion coefficient property and some spectral measure property:

Corollary 8.3.

∑_{j=0}^{∞} |α_{j}|^{2} < ∞ ⇔ ∫ log(w(θ)) \frac{dθ}{2π} > −∞  (8.9)

The equivalent conditions (8.9) are called the Szegő condition. When they hold, Szegő defined the Szegő function by

D(z) = exp(∫ \frac{e^{iθ} + z}{e^{iθ} − z} log(w(θ))\frac{dθ}{4π} )  (8.10)

Standard boundary value theory for the Poisson kernel implies D(e^{iθ}) = lim_{r↑1} D(re^{iθ}) exists for dθ/2π-a.e. θ ∈ [0, 2π) and

|D(e^{iθ})|^{2} = w(θ)  (8.11)

Theorem 8.4 (Szegő [95]). Suppose (8.9) holds. Let D_{ac}(e^{iθ}) = D(e^{iθ}) for a.e. θ and = 0 on a supporting set for dμ_{s}. Then

(i)  ∫ |φ_{n}^{∗}(e^{iθ}) − D_{ac}(e^{iθ})^{−1}|^{2} dμ → 0  (8.12)
(ii)  ∫ |φ_{n}(e^{iθ})|^{2} dμ_{s} → 0  (8.13)
(iii)  φ_{n}^{∗}(z) → D(z)^{−1} uniform on compacts in 𝔻  (8.14)

Sketch. A short preliminary argument proves that D ∈ ℍ^{2}(𝔻). Thus the Cauchy formula holds for φ_{n}^{∗}D, so

∫ (φ_{n}^{∗}D)(e^{iθ}) \frac{dθ}{2π} = φ_{n}^{∗}(0)D(0) → 1  (8.15)

since, by (8.5), φ_{n}^{∗}(0)D(0) = ∏_{j=0}^{n−1}(1 − |α_{j}|^{2})^{−1/2} ∏_{j=0}^{∞}(1 − |α_{j}|^{2})^{1/2}. By (8.11),

∫ |φ_{n}^{∗} − D(e^{iθ})^{−1}|^{2}w(θ) \frac{dθ}{2π} + ∫ |φ_{n}^{∗}(e^{iθ})|^{2} dμ_{s}
= ∥φ_{n}^{∗}∥_{L^{2}(dμ)}^{2} + 1 − 2 Re (LHS of (8.15)) → 0

by (8.15). This implies (i) and (ii). This then implies that Dφ_{n}^{∗} → 1 in L^{2}(∂𝔻, \frac{dθ}{2π}), so by ℍ^{2} theory, (iii) holds. □

(8.14) and the related

z^{−n}φ_{n}(z) → \overline{D(1/z̄)} on ℂ\\overline{𝔻}  (8.16)

are called Szegő asymptotics.

Theorem 8.5 (Sharp Form of the Szegő Strong Theorem [98, 41, 36]). If dμ_{s} = 0, the Szegő condition holds, and L̂_{k} are the Fourier coefficients of log w, then

G(dμ) = ∏_{j=0}^{∞}(1 − |α_{j}|^{2})^{−j−1} = exp( ∑_{n=0}^{∞} n|L̂_{n}|^{2})  (8.17)

Remark. Szegő [98] proved this when dμ has certain regularity properties. The general result is due to Ibragimov [41]; see also [36].

Seeing when G(dμ) < ∞ leads to a second gem:

Corollary 8.6.

∑_{j=0}^{∞} j|α_{j}|^{2} < ∞ ⇔ dμ_{s} = 0 and ∑_{n=0}^{∞} n|L̂_{n}|^{2} < ∞  (8.18)

This corollary relies also on a theorem of Golinskii-Ibragimov [36] that the LHS of (8.18) ⇒ dμ_{s} = 0. This result plus five distinct proofs of Theorem 8.5 are found in Chapter 6 of [88]. A sixth proof is in Section 9.10 of [89].

A final gem we want to mention is:

Theorem 8.7 (Baxter’s Theorem [8, 9]). Fix ℓ ≥ 0. The following are equivalent:

(a)  ∑_{n=0}^{∞} n^{ℓ}|α_{n}| < ∞

(b)  dμ_{s} = 0, inf_{θ} w(θ) > 0, and ∑_{n=0}^{∞} n^{ℓ}|c_{n}| < ∞

In particular if dμ_{s} = 0 and inf_{θ} w(θ) > 0, then w is C^{∞} if and only if sup_{n} n^{ℓ}|α_{n}| < ∞ for all ℓ ≥ 0.

This is proven in Chapter 5 of [88].

9. Exponential Decay

Suppose for some R > 1, we have

|α_{n}| ≤ CR^{−n}  (9.1)

By (2.17) and induction,

sup_{n, |z|=1} |Φ_{n}(z)| ≤ ∏_{j=0}^{∞}(1 + |α_{j}|) < ∞  (9.2)

so, by the maximum principle and (2.15),

sup_{|z|≥1} |z|^{−n}|Φ_{n}(z)| = sup_{|z|≤1} |Φ_{n}^{∗}(z)| < ∞

Thus, by (2.27), if |z| < R,

∑_{n=0}^{∞} |Φ_{n+1}^{∗}(z) − Φ_{n}^{∗}(z)| ≤ ∑_{n=0}^{∞} |α_{n}| |z|^{n+1} < ∞  (9.3)

It follows that Φ_{n}^{∗}(z) and so φ_{n}^{∗}(z) converges uniformly on compacts of {z | |z| < R} and so, by (8.14), D(z)^{−1} has a continuation to this disk. We thus have one-half of:

Theorem 9.1 (Nevai-Totik [67]). Fix R > 1. The following are equivalent:

(a) dμ_{s} = 0, the Szegő condition holds, and D(z)^{−1} has an analytic continuation to {z | |z| < R}.

(b) lim sup_{n→∞} |α_{n}|^{1/n} = R^{−1}

The other direction uses the useful formula,

dμ_{s} = 0 ⇒ α_{n} = −D(0)^{−1} ∫ \overline{Φ_{n+1}(e^{iθ})} D(e^{iθ})^{−1} dμ(θ)  (9.4)

Section 7.1 of [88] has a complete proof. One can say more ([88, Section 7.2]) when this holds: α_{n}+ Taylor coefficients of D(z)^{−1} \overline{D(1/z̄)} decays as 0(R^{−2n+ε}). There is also a lot known about asymptotics of the zeros when there is exponential decay (see [88, Sections 8.1 and 8.2], [91, 58] and references therein).

10. Periodic OPUC

The theory of one-dimensional periodic Schrödinger operators (a.k.a. Hill’s equation) and of periodic Jacobi matrices has been extensively developed [21, 24, 54, 55, 61, 101]. In the 1940’s, Geronimus [30] found the earliest results on OPUC with periodic Verblunsky coefficients, that is, for some p ≥ 1 and j = 0, 1, 2, …,

α_{j+p} = α_{j}  (10.1)

In particular, the case α_{j} ≡ a ∈ 𝔻\{0} yields OPUC called Geronimus polynomials (see Example 1.6.12 of [88]). Many of the general features for OPUC obeying (10.1) were found by Peherstorfer and collaborators [68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78]. Geronimo-Johnson [28, 29] have studied almost periodic Verblunsky coefficients. A reworking with some new results is Chapter 11 of [89], which uses methods mimicking the periodic Hill-Jacobi theory.

We suppose henceforth that p is even. A basic object is the discriminant,

Δ(z) = Tr(z^{−p/2}T_{p}(z))  (10.2)

where T_{p}(z) is the transfer matrix given by (6.2). The z^{−p/2} is included since, by det(A) = z, det(z^{−p/2}T_{p}(z)) = 1, and so z^{−p/2}T_{p}(z) has eigenvalues

\frac{Δ}{2} ± i\sqrt{1 − (\frac{Δ}{2})^{2}}. In particular, these eigenvalues have magnitude 1, that is, sup_{m} ∥T_{mp}(z)∥ < ∞ exactly when Δ(z) ∈ [−2, 2]. This is part of:

Theorem 10.1. There exists {x_{j}}_{j=1}^{2p}, {y_{j}}_{j=1}^{2p} with

x_{1} < y_{1} ≤ x_{2} < y_{2} ≤ ⋯ ≤ x_{p} < y_{p} ≤ x_{1} + 2π ≡ x_{p+1}

so that the solutions of Δ(z) = 2 (resp. −2) are exactly e^{ix_{1}}, e^{iy_{2}}, e^{ix_{3}}, …, e^{ix_{p}} (resp. e^{iy_{1}}, e^{ix_{2}}, e^{iy_{3}}, …, e^{iy_{p}}) and Δ(z) ∈ [−2, 2] exactly on

B = ⋃_{j=1}^{p}{e^{iθ} | x_{j} ≤ θ_{j} ≤ y_{j}}  (10.3)

the bands. B is the essential support of dμ_{ac} and the only possible singular spectrum are mass points which can occur in open gaps (i.e., nonempty sets of the form {e^{iθ} | y_{j} < θ < x_{j+1}}) with one (or zero) mass point in each gap.

Theorem 10.2. Let dρ be the equilibrium measure for B (i.e., the minimizer for E(ρ) = ∫ log|z − w|^{−1} dρ(z) dρ(w) with supp(dρ) ⊂ B and ρ(B) = 1). Let C_{B} be the logarithmic capacity of B (i.e., exp(− minimizing value of E(p))) and Q(z) the logarithmic potential for B (i.e., Q(z) = ∫ log|z − ω|^{−1} dρ(ω)). Then

(i) dρ is the density of zeros for dμ, −[Q(z) + log C_{B}] is the Lyapunov exponent, and C_{B} = ∏_{j=0}^{p−1}(1 − |α_{j}|^{2})^{1/2p}.

(ii) dρ can be written in terms of the discriminant as

dρ(θ) ≡ \frac{1}{p} \frac{|∂Δ(e^{iθ})/∂θ|}{(4 − Δ^{2}(e^{iθ}))^{1/2}} \frac{dθ}{2π}  (10.4)

(iii) For each j = 1, 2, …, p,

ρ(\{e^{iθ} | x_{j} ≤ θ ≤ y_{j}\}) = \frac{1}{p}  (10.5)

The proof of this result (see Section 11.1 of [89]) depends on noting that, by the Thouless formula, γ(z) is harmonic on ℂ\B and γ(z) = 0 on B. (iii) is related to half of the following result of Peherstorfer motivated by an OPRL result of Aptekarev [3]:

Theorem 10.3 (Peherstorfer [68]). Let B be a union of ℓ disjoint, closed intervals, B_{1}, …, B_{ℓ}, in ∂𝔻. Then B is the set of bands of a period p set of α’s if and only if (1) If dρ is the equilibrium measure of B, then pρ(B_{j}) ∈ ℤ for j = 1, …, ℓ.

(2) Let z_{1}, z_{2}, …, z_{2p} be defined clockwise around the circle so that z_{1} is the lower edge of B_{1} and the 2p points are the 2ℓ band edges and those interior points in a band with pρ(B_{j}) ≥ 2, that divide B_{j} into pρ(B_{j}) sets with ρ-measure 1/p, each counted twice. Then z_{1}z_{4}z_{5}z_{8}z_{9} … z_{2p} = 1.

If some ρ(B_{j}) is irrational, then there is no periodic family of α’s with those bands, but there is an almost periodic set, as proven by Geronimo-Johnson [29] (see Section 11.8 of [89]).

Given a measure dμ on ∂𝔻 so that (10.1) holds, the Dirichlet data is defined partly as the p points where \binom{1}{1} is an eigenvector for T_{p}(z), that is, zeros of φ_{p}^{∗}(z)− φ_{p}(z). There is one such point in each gap, including closed gaps (i.e., e^{iy_{j}} when y_{j} = x_{j+1}). If the value is at a gap edge, the eigenvalue, λ, of z^{−p/2}T_{p}(z) for \binom{1}{1} is ±1. Otherwise, it is in ℝ\\{0, −1, 1\}. In that case, we add σ_{j} = ±1 to the j-th Dirichlet point with σ_{j} = +1 (resp. −1) of the eigenvalue, |λ_{j}| < 1 (resp. |λ_{−j}| > 1). The point masses of dμ are precisely those Dirichlet points inside gaps with σ_{j} = +1. The set of allowed Dirichlet data are single points for closed gaps and a circle ([y_{j}, z_{j+1}]×{−1, 1} glued at the ends) for open gaps. Thus, the totality is a torus of dimension ℓ = # of open gaps.

Theorem 10.4. If Δ has ℓ open gaps, then the subset of {α_{j}}_{j=0}^{p−1} ∈ 𝔻^{p} which, when periodized, have discriminant Δ is a torus of dimension ℓ. The map from these α’s to the possible Dirichlet date is a bijection.

Critical to at least one understanding of this result is that the Carathéodory function, F, has a minimal degree meromorphic continuation to the genus ℓ − 1 hyperelliptic Riemann surface associated to \sqrt{Δ^{2} − 4}.

There is a natural symplectic form on 𝔻^{p} so that the real and imaginary parts of the coefficients of Δ are the set of integrals of a completely integrable system; this is described in Section 11.11 of [89] and in [64]. The associated flows include the defocusing AKNS flow.

11. The Szegő Mapping and the Geronimus Relations

Finally, we discuss a deep connection between OPRL and OPUC found by Szegő [96]. The map z ↦ z + z^{−1} maps 𝔻 biholomorphically to ℂ ∪ {∞} with a cut [−2, 2] removed. The map on the boundary, e^{iθ} → 2 cos θ, is a two-to-one map of ∂𝔻 to [−2, 2] that induces a map from M_{+,1}([−2, 2]) to those measures on ∂𝔻 which are invariant under complex conjugation. It is easy to see μ ∈ M_{+,1}(∂𝔻) has such invariance if and only if its Verblunsky coefficients are real. Explicitly, ρ, a probability measure on [−2, 2], is associated to μ = Sz(ρ), an even probability measure on ∂𝔻, via

∫ f(x) dρ(x) = ∫ f(2 cos θ) dμ(θ)  (11.1)

Szegő found the OPRL, P_{n}, for ρ in terms of the OPUC, Φ_{n}, for μ:

P_{n}(z + \frac{1}{z}) = [1 − α_{2n−1}(dμ)]^{−1}z^{−n}[Φ_{2n}(z) + Φ_{2n}^{∗}(z)]  (11.2)

and used this to convert Szegő asymptotics for OPUC (see (8.16)) to asymptotics for suitable OPRL. This asymptotics is often called Jost asymptotics in the discrete Schrödinger literature.

Geronimus [31] found the relation between the Jacobi parameters {a_{n}, b_{n}}_{n=1}^{∞} for ρ and the Verblunsky coefficients {α_{n}}_{n=0}^{∞} for μ (with α_{−1} ≡ 1):

a_{n+1}^{2} = (1 − α_{2n−1})(1 − α_{2n}^{2})(1 + α_{2n+1})  (11.3)
b_{n+1} = (1 − α_{2n−1})α_{2n} − (1 + α_{2n−1})α_{2n−2}  (11.4)

The map from α to (a, b) is local, that is, changing a single α only changes a finite number of a’s and b’s. That is not true for the inverse. Scaled Chebyshev polynomials of the first kind has a_{1} = \sqrt{2}, a_{n} = 1 (n ≥ 2), b_{n} = 0, and the corresponding α_{n} ≡ 0. Scaled Chebyshev polynomials of the second kind have a_{n} ≡ 1, b_{n} = 0 (i.e., they differ at a single a_{n}), but have α_{2n} = 0 and α_{2n−1} = −1/(n + 1).

Still the inverse can be computed ([31]). Given {a_{n}, b_{n}}_{n=1}^{∞}, define φ_{n}^{±} by φ_{0} = 0, φ_{1} = 1, and for n ≥ 1,

φ_{n+1}^{±} + a_{n−1}^{2}φ_{n−1}^{±} + b_{n}φ_{n}^{±} = ±2φ_{n}^{±}  (11.5)

By a Sturm oscillation theorem, {a_{n}, b_{n}}_{n=1}^{∞} are the Jacobi parameters of a measure supported on [−2, 2] if and only if φ_{n}^{+} > 0 and (−1)^{n}φ_{n}^{−} > 0. The Verblunsky coefficients are given by

u_{n} = \frac{φ_{n+2}^{+}}{φ_{n+1}^{+}}  v_{n} = −\frac{φ_{n+2}^{−}}{φ_{n+1}^{−}}  (11.6)
α_{2n} = \frac{v_{n} − u_{n}}{v_{n} + u_{n}}  α_{2n−1} = 1 − \frac{1}{2} (u_{n} + v_{n})  (11.7)

Recently, these mappings have been used by Denisov [19] and Damanik-Killip [17, 86] as a powerful tool in the study of discrete Schrödinger operators and of OPRL. For proofs and references, see Chapter 13 of [89].

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[108] Wikipedia entry on Rodney Dangerfeld: For those in the international community who don’t

     know of this reference, see http://en.wikipedia.org/wiki/Rodney Dangerfield.