FINE STRUCTURE OF THE ZEROS OF ORTHOGONAL POLYNOMIALS, I. A TALE OF TWO PICTURES

BARRY SIMON*

Abstract. Mhaskar-Saff found a kind of universal behavior for the bulk structure of the zeros of orthogonal polynomials for large n. Motivated by two plots, we look at the finer structure for the case of random Verblunsky coefficients and for what we call the BLS condition: α_{n} = Cb^{n} + O((bΔ)^{n}). In the former case, we describe results of Stoiciu. In the latter case, we prove asymptotically equal spacing for the bulk of zeros.

1. Prologue: A Theorem of Mhaskar and Saff

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A recurrent theme of Ed Saff’s work has been the study of zeros of orthogonal polynomials defined by measures in the complex plane. So I was happy that some thoughts I’ve had about zeros of orthogonal polynomials on the unit circle (OPUC) came to fruition just in time to present them as a birthday bouquet. To add to the appropriateness, the background for my questions was a theorem of Mhaskar-Saff [25] and the idea of drawing pictures of the zeros was something I learned from some of Ed’s papers [32, 21]. Moreover, ideas of Barrios-López-Saff [4] played a role in the further analysis.

Throughout, dμ will denote a probability measure on ∂𝔻 = {z ∈ ℂ | |z| = 1} which is nontrivial in that it is not supported on a finite set. Φ_{n}(z) (resp. φ_{n}(z)) will denote the monic orthogonal polynomials (resp. orthonormal polynomials φ_{n} = Φ_{n}/∥Φ_{n}∥). I will follow my book [34, 35] for notation and urge the reader to look there for further background.

A measure is described by its Verblunsky coefficients

α_{n} = −\overline{Φ_{n+1}(0)}  (1.1) 1.1a

Date: January 31, 2005.

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

Talk given at Constructive Functions Tech-04.

To appear in conference proceedings.

which enter in the Szegő recursion

Φ_{n+1}(z) = zΦ_{n}(z) − ᾱ_{n}Φ_{n}^{∗}(z)  (1.2) 1.1
Φ_{n+1}^{∗}(z) = Φ_{n}^{∗}(z) − α_{n}zΦ_{n}(z)  (1.3) 1.2

where

P_{n}^{∗}(z) = z^{n} \overline{P_{n}(1/z̄)}  (1.4) 1.3

Φ_{n} has all its zeros in 𝔻 [34, Theorem 1.7.1]. We let dν_{n} be the pure point measure on 𝔻 which gives weight k/n to each zero of Φ_{n} of multiplicity k. For simplicity, we will suppose there is a b ∈ [0, 1] so that

lim|α_{n}|^{1/n} = b  (1.5) 1.4

(root asymptotics). If b = 1, we also need

lim_{n→∞} \frac{1}{n} ∑_{j=0}^{n−1} |α_{j}| = 0  (1.6) 1.5

(which automatically holds if b < 1). Here is the theorem of Mhaskar-Saff [25]:

T1.1 Theorem 1.1 (Mhaskar-Saff [25]). If (1.5) and (1.6) hold, then dν_{n} converges weakly to the uniform measure on the circle of radius b.

We note that both this result and the one of Nevai-Totik [28] I will mention in a moment define b by lim sup|α_{n}|^{1/n} and the Mhaskar-Saff result holds for dν_{n(j)} where n(j) is a subsequence with lim_{j→∞}|α_{n(j)}|^{1/n(j)} = b.

I want to say a bit about the proof of Theorem 1.1 in part because I will need a slight refinement of the first part (which is from Nevai-Totik [28]) and in part because I want to make propaganda for a new proof [34] of the second part.

The proof starts with ideas from Nevai-Totik [28] that hold when b < 1:

(1) By (1.2) and |Φ_{n}(e^{iθ})| = |Φ_{n}^{∗}(e^{iθ})|, one sees inductively that

sup_{n,z∈∂𝔻}|Φ_{n}^{∗}(z)| ≤ ∏_{j=0}^{∞} (1 + |α_{j}|) < ∞  (1.7) 1.6

and so, by the maximum principle and (1.4),

C ≡ sup_{n,z∉𝔻}|z|^{−n}|Φ_{n}(z)| < ∞  (1.8) 1.7

(2) By (1.3),

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

if |z|b < 1 by (1.5). Thus, in the disk {z | |z| < b^{−1}}, Φ_{n}^{∗} has a limit. Since b < 1, the Szegő condition (see [34, Chapter 2]) holds, so

φ_{n}^{∗}(z) → D(z)^{−1}  (1.9) 1.8

on 𝔻 (see [34, Theorem 2.4.1]), we conclude that D(z)^{−1} has an analytic continuation to the disk of radius b^{−1} and (1.9) holds there. (Nevai-Totik also prove a converse: If the Szegő condition holds, dμ_{s} = 0 and D(z)^{−1} has an analytic continuation to the disk of radius b^{−1}, then lim sup|α_{n}|^{1/n} ≤ b.)

(3) When (1.5) holds, D(z) is analytic in 𝔻 and continuous on 𝔻̄ so D(z)^{−1} has no zeros in 𝔻̄. We define the Nevai-Totik points {z_{j}}_{j=1}^{N} (N in {0, 1, 2, …, } ∪ ∞) with 1 > |z_{1}| ≥ |z_{2}| ≥ ⋯ > b to be all the solutions of D(1/z̄)^{−1} = 0 in 𝔸_{b} = {z | b < |z| < 1}. Since (1.9) holds and φ_{n}^{∗}(z) = 0 ⇔ φ_{n}(1/z̄) = 0, Hurwitz’s theorem implies that the z_{j} are precisely the limit points of zeros of φ_{n} in the region 𝔸_{b}. If N = ∞, |z_{j}| → b so we conclude that for each ε > 0, the number of zeros of φ_{n} in {z | |z| > b + ε} is bounded as n → ∞.

That concludes our summary of some of the results from Nevai-Totik. The next step is from Mhaskar-Saff.

(4) By (1.1), if {z_{jn}}_{j=1}^{n} are the zeros of φ_{n}(z) counting multiplicity, then

|α_{n}| = ∏_{j=1}^{n} |z_{jn}|

so, by (1.5),

lim_{n→∞} \frac{1}{n} ∑_{j=1}^{n} log|z_{jn}| = log b

This together with (3) implies that the bulk of zeros must asymptotically lie on the circle of radius b and, in particular, any limit point of dν_{n} must be a measure on {z | |z| = b}.

Mhaskar-Saff complete the proof by using potential theory to analyze the limit points of the ν_{n}. Instead, I will sketch a different idea from [34, Section 8.2] that exploits the CMV matrix (see [5] and [34, Sections 4.2– 4.5]).

(5) If C^{(N)} is the N × N matrix in the upper left of C, then

Φ_{N}(z) = det(z − C^{(N)})  (1.10) 1.9a

and, in particular,

∫ z^{k} dν_{N}(z) = \frac{1}{N} Tr([C^{(N)}]^{k})  (1.11) 1.9b

It is not hard to see that (1.6) implies that on account of the structure of C, the right side of (1.11) goes to 0 as N → ∞ for each k > 0. Thus any limit point, dν, of dν_{N} has ∫ z^{k} dν(z) = 0 for k = 1, 2, . . .. That determines the measure dν uniquely since the only measure on b[∂𝔻] with zero moments is the uniform measure.

That completes the sketch of the proof of the Mhaskar-Saff theorem. Before going on, I have two remarks to make. It is easy to see ([34, Theorem 8.2.6]) that ⟨δ_{ℓ}, C^{k}δ_{ℓ}⟩ = ∫_{0}^{2π} e^{ikθ}|φ_{ℓ}(e^{iθ})|^{2} dμ(θ). Thus the moments of the limit points of the Ces̀aro average

dη_{N} = \frac{1}{N} ∑_{j=0}^{N−1} |φ_{j}(e^{iθ})|^{2} dμ(θ)

are the same as the moments of the limits of dν_{N} (so if dν_{N} has a limit that lives on ∂𝔻, dη_{N} has the same limit).

Second, a theorem like Mhaskar-Saff holds in many other situations. For example, if β_{n} is periodic and α_{n} − β_{n} → 0, then the dν_{n} converge to the equilibrium measure for the essential support of dμ, which is a finite number of intervals. (See Ed Saff’s book with Totik [33].)

One critical feature of the Mhaskar-Saff theorem is its universality. So long as we look at cases where (1.5) and (1.6) hold, the angular distribution is the same. Our main goal here is to go beyond the universal setup where the results will depend on more detailed assumptions on asymptotics. In particular, we will want to consider two stronger conditions than root asymptotics, (1.5), namely, ratio asymptotics

lim_{n→∞} \frac{α_{n+1}}{α_{n}} = b  (1.12) 1.8x

for some b ∈ (0, 1) and what we will call BLS asymptotics (or the BLS condition):

α_{n} = Cb^{n} + O((bΔ)^{n})  (1.13) 1.9

where b ∈ (0, 1), C ∈ ℂ, and Δ ∈ (0, 1).

The name BLS is for Barrios-López-Saff [4] who proved

T1.2 Theorem 1.2 (Barrios-López-Saff [4]). A set of Verblunsky coefficients obeys the BLS condition if and only if dμ_{s} = 0 and the numeric inverse of the Szegő function D(z)^{−1}, defined initially for z ∈ 𝔻, has a meromorphic continuation to a disk of radius (bΔ′)^{−1} for some Δ′ ∈ (0, 1) which is analytic except for a single pole at z = b^{−1}.

This is Theorem 2 in [4]. They allow b ∈ 𝔻\{0} but, by the rotation invariance of OPUC (see [34, Eqns. (1.6.62)–(1.6.67)]), any b = |b|e^{iθ} can be reduced to |b|. Another proof of Theorem 1.2 can be found in [34, Section 7.2] where it is also proven that

C = −[ lim_{z→b^{−1}} (z^{−1} − b)D(z)^{−1}] \overline{D(b)}  (1.14) 1.10

and that if Δ or Δ′ is in (\frac{1}{2}, 1), then Δ = Δ′.

We summarize the contents of the paper after the next and second preliminary section. I’d like to thank Mourad Ismail, Rowan Killip, Paul Nevai, and Mihai Stoiciu for useful conversations.

2. Two Pictures and an Overview of the Fine Structure

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Take a look at two figures (Figure 1 and Figure 2) from my book [34, Section 8.4]. The first shows the zeros of Φ_{n}(z) for α_{n} = (\frac{1}{2})^{n+1} and the second for α_{n} independent random variables uniformly distributed on [−\frac{1}{2}, \frac{1}{2}]. (Of course, the second is for a particular choice of the random variables made by Mathematica using its random number generator.) They are shown for n = 5, 10, 20, 50, 100, and 200.

n = 5  n = 10

n = 20  n = 50

n = 100  n = 200

Figure 1

Figure 1 beautifully illustrates Theorem 1.1 and the Nevai-Totik theorem. All the zeros but one clearly approach the circle |z| = \frac{1}{2}. There is one zero that approaches approximately 0.84 + 0i. It is the single

n = 5  n = 10

n = 20  n = 50

n = 100  n = 200

Figure 2

Nevai-Totik zero in this case. That there are no limiting zeros inside |z| = \frac{1}{2} is no accident; see Corollary 1 of [4] and Theorem 8.2 below. And the zeros certainly seem uniformly distributed — indeed, when I first ran the program that generated Figure 1, I was impressed by how uniform the distribution seemed to be, even for N = 10.

The conclusion of the Mhaskar-Saff theorem is not true for the example in Figure 2 (nor, of course, the hypotheses since (1.6) fails), although it would be if uniform density on [−\frac{1}{2}, \frac{1}{2}] were replaced by any rotation invariant density, dγ, with ∫ |log(z)| dγ(z) < ∞ (see [35, Theorem 10.5.19]). But, by Theorems 10.5.19 and 10.5.21 of [35], dν_{n} has a limit dν in the case of Figure 2, and this limit can be seen to be of the form f(θ)\frac{dθ}{2π} with f ∈ C^{∞} and f ≢ 1 but not too far from 1 (e.g., all odd moments vanish and ∫ z^{2} dν = (\frac{1}{24})^{2}). My initial thought was that the roughness was trying to emulate the pure point spectrum.

I now think I was wrong in both initial reactions.

Expectation 1: Poisson Behavior. For Figure 2, I should have made the connection with work of Molchanov [27] and Minami [26] who proved in the case of random Schrödinger operators that, locally, eigenvalues in a large box had Poisson distribution. This leads to a conjecture. First some notation:

We say a collection of intervals Δ_{1}^{(n)}, …, Δ_{k}^{(n)} in ∂𝔻 is canonical if Δ_{j}^{(n)} = \{e^{iθ} | θ ∈ [\frac{2πa_{j}}{n} + θ_{j}, \frac{2πb_{j}}{n} + θ_{j}]\} where 0 ≤ θ_{1} ≤ ⋯ ≤ θ_{k} ≤ 2π, and if θ_{j} = θ_{j+1}, then b_{j} < a_{j+1}.

Con2.1 Conjecture 2.1. Let {α_{ℓ}}_{ℓ=0}^{∞} be independent identically distributed random variables, each uniformly distributed in {z | |z| ≤ ρ} for some ρ < 1 and let z_{1}^{(n)}, …, z_{n}^{(n)} be the random variable describing the zeros of the Φ_{n} associated to α. Then for some C_{1}, C_{2}, (1)

𝔼(#{z_{j}^{(n)} | |z_{j}^{(n)}| < 1 − e^{−C_{1}n}}) ≤ C_{2}(log n)^{2}  (2.1) 2.1

(2) For any collection Δ_{1}^{(n)}, …, Δ_{k}^{(n)} of canonical intervals and any ℓ_{1}, …, ℓ_{k} in {0, 1, 2, … },

ℙ(#{j | arg(z_{j}^{(n)}) ∈ Δ_{m}} = ℓ_{m} for m = 1, …, k)
→ ∏_{m=1}^{k} [\frac{(b_{m} − a_{m})^{ℓ_{m}}}{ℓ_{m}!} e^{−(b_{m}−a_{m})}] (2.2) 2.2

Remarks. 1. This says that, asymptotically, the distribution of z’s is the same as n points picked independently, each uniformly distributed.

2. See the next section for a result towards this conjecture.

3. 𝔼 means expectation and ℙ probability.

4. I base the precise e^{−C_{1}n} and (log n)^{2} on the results of Stoiciu [39], but I would regard as very interesting any result that showed, except for a small fraction (even if not as small as (log n)^{2}/n), all zeros are very close (even if not as close as e^{−C_{1}n}) for ∂𝔻.

5. There is one aspect of this conjecture that is stronger than what is proven for the Schrödinger case. The results of Molchanov [27] and Minami [26] are the analog of Conjecture 2.1 if θ_{1} = θ_{2} = ⋯ = θ_{k}, which I would call a local Poisson structure. That there is independence of distant intervals is conjectured here but not proven in the Schrödinger case. That this is really an extra result can be seen by the fact that Figure 2 is likely showing local Poisson structure about any θ_{0} ≠ 0, π, but because the α’s are real, the set of zeros is invariant under complex conjugation, so intervals about, say, π/2 and 3π/2 are not independent.

As far as Figure 1 is concerned, it is remarkably regular so there is an extra phenomenon leading to

Expectation 2: Clock Behavior. If for b ∈ (0, 1) and C ∈ ℂ, α_{n}/b^{n} converges to C sufficiently fast, then the non-Nevai-Totik zeros approach |z| = b and the angular distance between nearby zeros in 2π/n.

Remarks. 1. Proving this expectation when “sufficiently fast” means BLS convergence is the main new result of this paper; see Section 4.

2. This is only claimed for local behavior. We will see that, typically, errors in the distance between the zeros are O(1/n^{2}) and will add up to shift zeros that are a finite distance from each other relative to a strict clock.

3. Clock behavior has been discussed for OPRL. Szegő [40] has C/n upper and lower bounds (different C’s) in many cases and Erdös-Turan [11] prove local clock behavior under hypotheses on the measure, but their results do not cover all Jacobi polynomials. In Section 6, we will prove a clock result for a class of OPRL in terms of their Jacobi matrix parameters (∑_{n=1}^{∞} n(|b_{n}| + |a_{n} − 1|) < ∞), and in Section 7, a simple analysis that proves local clock behavior for Jacobi polynomials. I suppose this is not new, but I have not located a reference.

4. A closer look at Figure 1 suggests that this conjecture might not be true near z = b. In fact, the angular gap there is 2(2π/n) + o(1/n), as we will see.

I should emphasize that the two structures we suspect here are very different from what is found in the theory of random matrices. This is most easily seen by looking at the distribution function for distance between nearest zeros scaled to the local density. For the Poisson case, there is a constant density, while for clock, it is a point mass at a point θ_{0} ≠ 0 depending on normalization. For the standard random matrix (GUE, GOE, CUE), the distribution is continuous but vanishing at 0 (see [24]).

Since any unitary with distinguished cyclic vector can be represented by a CMV matrix, CUE has a realization connected with OPUC, just not either the totally random or BLS case. Indeed, Killip-Nenciu [19] have shown that CUE is given by independent α_{j}’s but not identically distributed.

In Section 3, we describe a new result of Stoiciu [39] on the random case. In Section 4, we overview our various clock results: paraorthogonal OPUC in Section 5, OPRL proven in Sections 6 and 7, and BLS in Sections 8–13. We mention some examples in Section 13.

3. Stoiciu’s Results on the Random Case

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Recall that given β ∈ ∂𝔻 and {Φ_{n}}_{n=0}^{∞}, a set of orthogonal polynomials, the paraorthogonal polynomials (POPs) [18, 16] are defined by

Φ_{n}(z; β) = zΦ_{n−1}(z) − β̄Φ_{n−1}^{∗}(z)  (3.1) 3.1x

They have all their zeros on ∂𝔻 (see, e.g., [34, Theorem 2.2.12]). Stoiciu [39] has proven the following result:

T3.1 Theorem 3.1 (Stoiciu [39]). Let {α_{j}}_{j=0}^{∞} be independent identically distributed random variables with common distribution uniform in {z | |z| ≤ σ} for some σ < 1. Let {β_{j}}_{j=0}^{∞} be independent identically distributed random variables uniformly distributed on ∂𝔻. Let z_{j}^{(n)} be the zeros of Φ_{n}(z; β_{n−1}). Let Δ_{1}^{(n)}, …, Δ_{k}^{(n)} be canonical intervals with the same θ, that is, θ_{1} = θ_{2} = ⋯ = θ_{k}. Then (2.2) holds.

This differs from Conjecture 2.1 in two ways: The zeros are of the POPs, not the OPUC, and the result is only local (i.e., all θ’s are equal). While the proof has some elements in common with the earlier work on OPRL of Molchanov [27] and Minami [26], there are many differences forced, for example, by the fact that rank one perturbations of selfadjoint operators differ in many ways from rank two perturbations of unitaries. Since the proof is involved and the earlier papers have a reputation of being difficult, it seems useful to summarize here the strategy of Stoiciu’s proof.

Following Minami, a key step is the proof of what are sometimes called fractional moment bounds and which I like to call Aizenman-Molchanov bounds after their first appearance in [2]. A key object in these bounds is the Green’s function which has two natural analogs for OPUC:

G_{nm}(z) = ⟨δ_{n}, (C − z)^{−1}δ_{m}⟩  (3.2) 3.1

F_{nm}(z) = ⟨δ_{n}, (C + z)(C − z)^{−1}δ_{m}⟩  (3.3) 3.2

These are related by

F_{nm}(z) = δ_{nm} + 2zG_{nm}(z)  (3.4) 3.3

so controlling one on ∂𝔻 is the same as controlling the other.

As we will see below, F and G lie in the Hardy space H^{p} for any p < 1, so we can define

G_{nm}(e^{iθ}) = lim_{r↑1} G_{nm}(re^{iθ})  (3.5) 3.4

for a.e. e^{iθ}. In the random case, rotation invariance will then imply that for any e^{iθ} ∈ ∂𝔻, (3.5) holds for a.e. α. In treatments of Aizenman-Molchanov bounds for Schrödinger operators, it is traditional to prove bounds on the analog of G_{ij}(z) for z = re^{iθ} with r < 1 uniform in r in (\frac{1}{2}, 1). Instead, the Stoiciu proof deals directly with r = 1, requiring some results on boundary values of H^{p} functions to replace a uniform estimate.

Given N, we define Ĉ^{(N)} to be the random CMV matrix ([5] and [34, Chapter 4]) obtained by setting α_{N} to the random value β_{N} ∈ ∂𝔻. Ĉ^{(N)} decouples into a direct sum of an N × N matrix, C^{(N)}, and an infinite matrix which is identically distributed to the random C if N is even and C̃, the random alternate CMV matrix, if N is odd. (This is a slight oversimplification. Only if β_{N} = −1 is the infinite piece of Ĉ^{(N)} a CMV matrix since the 1 × 1 piece in the M half of the LM factorization has −β_{N} in place of 1. As explained in [34, Theorem 4.2.9], there is a diagonal unitary equivalence taking such a matrix to a CMV matrix with Verblunsky coefficients −β_{N}^{−1}α_{j+N+1} and the distribution of these is identical to the distribution of the α_{j+N+1}. We will ignore this subtlety in this sketch.)

We define F_{nm}^{(N)}(z) and G_{nm}^{(N)}(z) for n, m ∈ ({0, 1, …, N − 1} by replacing C in (3.2)/(3.3) by C^{(N)}.

C −Ĉ^{(N)} is a rank two matrix with (C −Ĉ^{(N)})_{nm} ≠ 0 only if |n−N| ≤ 2, |m − N| ≤ 2. Moreover, any matrix element of the difference is bounded in absolute value by 2. If n ∈ {0, …, N − 1}, m ≥ N, then (Ĉ^{(N)} − z)_{nm}^{−1} = 0, so the second resolvent formula implies

\begin{aligned}n & ≤ N − 1, m ≥ N ⇒ |G_{nm}(z)| \\ & ≤ 2( ∑_{k=N−1,N−2,N−3} |G_{nk}^{(N)}(z)|)( ∑_{|k−N+ \frac{1}{2} |≤ \frac{5}{2}} |G_{km}(z)|) \end{aligned}  (3.6) 3.4a

which we will call the decoupling formula.

Similarly, we have

\begin{aligned}n, m & ≤ N ⇒ |G_{nm}(z) − G_{nm}^{(N)}(z)| \\ & ≤ 2( ∑_{k=N−1,N−2,N−3} |G_{nk}^{(N)}(z)|)( ∑_{|k−N+ \frac{1}{2} |≤ \frac{5}{2}} |G_{km}(z)|) \end{aligned} (3.7) 3.4b

and

\begin{aligned}n, m & ≥ N ⇒ |G_{nm}(z) − Ĝ_{nm}^{(N)}(z)| \\ & ≤ 2( ∑_{k=N,N+1,N+2} |Ĝ_{nk}^{(N)}(z)|)( ∑_{|k−N+ \frac{1}{2} |≤ \frac{5}{2}} |G_{km}(z)|) \end{aligned} (3.8) 3.4c

where we recall that if n, m ≥ N and N is even, then Ĝ_{nm}^{(N)}(z) = G_{n−N,m−N}(z, {α_{j+N+1}}_{j=0}^{∞}) and if N is odd, then Ĝ_{nm}^{(N)}(z) = G_{m−N,n−N} (z, {α_{j+N+1}}_{j=0}^{∞}).

Stoiciu’s argument has five parts, each with substeps: Part 1: Some preliminaries concerning H^{p} properties of F_{ij}, positivity

of the Lyapunov exponent, and exponential decay of F_{ij} for a.e. α. Part 2: Proof of the Aizenman-Molchanov estimates. Part 3: Using Aizenman-Molchanov estimates to prove that eigenval-

ues of C^{(N)} are, except for O(log N) of them, very close to eigenvalues of log N independent copies of C^{(N/ log N)}.

Part 4: A proof that the probability of C^{(N)} having two eigenvalues in

an interval of size 2πx/N is O(x^{2}). Part 5: Putting everything together to get the Poisson behavior.

Part 2 uses Simon’s formula for G_{ij} (see [34, Section 4.4]) and ideas of Aizenman, Schenker, Friedrich, and Hundertmark [3], but the details are specific to OPUC and exploit the rotation invariance of the distribution in an essential way. Part 3 uses the strategy of Molchanov-Minami with some ideas of Aizenman [1], del Rio et al. [9], and Simon [36]. But again, there are OPUC-specific details that actually make the argument simpler than for OPRL. Part 4 is a new and, I feel, more intuitive argument than that used by Molchanov [27] or Minami [26]. It depends on rotation invariance. Part 5, following Molchanov and Minami, is fairly standard probability theory. Here are some of the details.

In the arguments below, we will act as if log N and N/ log N are integers rather than doing what a true proof does: use integral parts and wiggle blocks of size [N/ log N] by 1 to get [log N] of them that add to N.

\underline{Step 1.1} (H^{p} properties of Carathéodory functions). A Carathéodory function is an analytic function on 𝔻 with F (0) = 1 and Re F (z) > 0. By Kolmogorov’s theorem (see [10, Section 4.2]), such an F is in H^{p}, 0 < p < 1 with an a priori bound (0 < p < 1),

sup_{0<r<1} ∫ |F (re^{iθ})|^{p} \frac{dθ}{2π} ≤ cos(\frac{pπ}{2} )^{−1}  (3.9) 3.5

For any unit vector η, ⟨η, \frac{C+z}{C−z} η⟩ is a Carathéodory function so, by polarization, we have the a priori bounds

sup_{0<r<1} ∫ |F_{nm}(re^{iθ})|^{p} \frac{dθ}{2π} ≤ 2^{2−p} cos(\frac{pπ}{2} )  (3.10) 3.6

0 < p < 1, all m, n. The same bound holds for F^{(N)}. \underline{Step 1.2} (Pointwise estimates on expectations). H^{p} functions have boundary values and the mean converges (see [10, Theorem 2.6]), so (3.10) holds for r = 1 and the sup dropped. If one averages over the random α (or random α and β for F^{(N)}) and uses the rotation invariance to see that the expectation is θ-independent, we find

𝔼(|F_{nm}(e^{iθ})|^{p}) ≤ 2^{2−p} cos(\frac{pπ}{2} )  (3.11) 3.7

all n, m, all θ ∈ [0, 2π), and for all F^{(N)}. \underline{Step 1.3} (Positive Lyapunov exponent). By the rotation invariance and the Thouless formula (see [35, Theorems 10.5.8 and 10.5.26]), the density of zeros is dθ/2π, the Lyapunov exponent exists for all z and is given by (see [35, Theorem 12.6.2])

γ(z) = −\frac{1}{2} ∫_{|z|≤σ} log(1 − |z|^{2}) \frac{d^{2}z}{πσ^{2}} + log(max(1, |z|))  (3.12) 3.8

and, in particular, γ(e^{iθ}) > 0. \underline{Step 1.4} (Pointwise decay of G). Let z_{0} ∈ ∂𝔻 and let α be a random sequence of α’s for which \frac{1}{n} log ∥T_{n}(z; α)∥ → γ and |F_{00}(z_{0})| < ∞. Since γ(z_{0}) > 0, the Ruelle-Osceledec theorem (see [35, Theorem 10.5.29]) implies there is a λ ≠ 1 for which the OPUC with boundary condition λ (see [34, Section 3.2]) obeys |φ_{n}^{λ}(z_{0})| → 0. It follows from Theorem 10.9.3 of [35] that |G_{0 n}(z_{0})| → 0. Thus, for a.e. α,

lim_{n→∞} |G_{0 n}(z_{0})| → 0  (3.13) 3.9

By Theorem 10.9.2 of [35], F_{0 0} ∈ iℝ and this implies that the solutions π and ρ of Section 4.4 of [34] obey |π_{k}(z_{0})| = |ρ_{k}(z_{0})| so |(C−z)_{mn}^{−1}| is symmetric in m and n. Thus (3.13) implies

lim_{n→∞} |G_{n 0}(z_{0})| → 0

\underline{Step 1.5} (Decay of 𝔼(|G_{0 n}(z_{0})|^{p})). The proof of (3.13) shows, for fixed α, |G_{0n}(z_{0})| decays exponentially, but since the estimates are not uniform in α, one cannot use this alone to conclude exponential decay of the expectation. But a simple functional analytic argument shows that if h_{n} are functions on a probability measure space, sup_{n} 𝔼(|h_{n}|^{p}) < ∞ and |h_{n}(ω)| → 0 for a.e. ω, then 𝔼(|h_{n}|^{r}) → 0 for any r < p. It follows from (3.11) and (3.13) that for any z_{0} ∈ ∂𝔻 and 0 < p < 1,

lim_{n→∞} 𝔼(|G_{0 n}(z_{0})|^{p}) = 0  (3.14) 3.10

\underline{Step 1.6} (General decay of 𝔼(|G|^{p})). By the Schwartz inequality and repeated use of (3.6), (3.7), and (3.8), one sees first for p < \frac{1}{2} and then by Hölder’s inequality that

lim_{n→∞} sup_{|m−k|≥n} 𝔼(|G_{mk}(z_{0})|^{p}) = 0  (3.15) 3.11

and

lim_{n→∞} sup_{\begin{matrix}|m−k|≥n \\ 0≤m,k≤N−1 \\ all N \end{matrix}} 𝔼(|G_{mk}^{(N)}(z_{0})|^{p}) = 0  (3.16) 3.12

That completes Part 1.

\underline{Step 2.1} (Conditional expectation bounds on diagonal matrix elements). Let

H(α, γ) = \frac{α + γ}{1 + ᾱγ}  (3.17) 3.13

Then a simple argument shows that for 0 < p < 1,

sup_{β,γ} ∫_{|α|≤1} |\frac{1}{1 − βH(α, γ)}|^{p} d^{2}α < ∞

because, up to a constant, the worst case is β = γ = 1, and in that case, the denominator vanishes only on Re α = 0. Applying this to Khrushchev’s formula (see [35, Theorem 9.2.4]) provides an a priori bound on the conditional expectation

𝔼(|F_{kk}(z)|^{p} | {α_{j}}_{j≠k}) ≤ C  (3.18) 3.14

where C is a universal constant depending only on σ (the radius of the support of the distribution of α) and a similar result for conditioning on {α_{j}}_{j≠k−1}, that is, averaging over α_{k−1}.

\underline{Step 2.2} (Conditional expectation bounds on near diagonal matrix elements). Since ρ^{−1} ≤ (1 − σ^{2})^{−1/2} ≡ Q for all α with |α| ≤ σ, we have, by equation (1.5.30) of [34], that

|\frac{φ_{k}}{φ_{m}} | ≤ (2Q)^{|k−m|}

on ∂𝔻. A similar estimate for the solutions π and ρ of Section 4.4 of

[34] (using |π_{k}| = |ρ_{k}|; see the end of Step 1.4) proves

|\frac{π_{k}}{π_{m}} | ≤ (2Q)^{|k−m|}

This implies, by Theorem 4.4.1 of [34], that

|\frac{G_{kl}(z)}{G_{mn}(z)}| ≤ (2Q)^{|k−m|+|ℓ−n|}  (3.19) 3.15

something clearly special to OPUC. This together with (3.18) and (3.4) implies

𝔼(|G_{mn}(z)|^{p} | {α_{j}}_{j≠k}) ≤ (2Q)^{|m−k|+|n−k|}(1 + 2C)  (3.20) 3.16

\underline{Step 2.3} (Double decoupling). This step uses an idea of Aizenman, Schenker, Friedrich, and Hundertmark [3]. Given n, we look at N < n − 3 and decouple first at N and then at N + 3 to get, using (3.6) and (3.8), that

\begin{aligned}|G_{0 n}(z)| ≤ 4( & ∑_{k=N−1,N−2,N−3} |G_{0 k}^{(N)}(z)|) \\ & ( ∑_{\begin{matrix}|k−N+ \frac{1}{2} |≤ \frac{5}{2} \\ |ℓ−N+ \frac{7}{2} |≤ \frac{5}{2} \end{matrix}} |G_{kℓ}|)( ∑_{ℓ=N+3,N+4,N+5} |Ĝ_{ℓn}^{(N+3)}(z)|) \end{aligned} ; (3.21) 3.17

Using (3.19) and generously overestimating the number of terms, we find

|G_{0 n}(z)| ≤ 4 · 3 · 6 · 6 · 3(2Q)^{10}|G_{0 N−1}^{(N)}(z)| |G_{N+1 N+1}(z)| |Ĝ_{N+3 n}^{(N+3)}(z)|

(3.22) 3.18

Raise this to the p-th power and average over α_{N+1} with {α_{k}}_{k≠N+1} fixed. Since G_{0 N−1}^{(N)} and Ĝ_{N+3 N}^{(N+3)} are independent of α_{N+1}, they come out of the conditional expectation which can be bounded by (3.18).

After that replacement has been made, the other two factors are independent. Thus, if we integrate over the remaining α’s and use the structure of Ĝ, we get

𝔼(|G_{0 n}(z)|^{p}) ≤ C_{p}𝔼(|G_{0 N−1}^{(N)}(z)|^{p})𝔼(|G_{0 n−N−3}^{(N)}|^{p})  (3.23) 3.19

where C_{p} is p-dependent but N-independent. \underline{Step 2.4} (Aizenman-Molchanov bounds). By (3.16), for p fixed, we can pick N so large that in (3.23), we have C_{p}𝔼(|G_{0 N−1}^{(N)}(z)|^{p}) < 1. If we iterate, we then get exponential decay, that is, we get the Aizenman-Molchanov bound; for any p ∈ (0, 1), there is D_{p} and κ_{p} so that

𝔼(|G_{nm}(z)|^{p}) ≤ D_{p} e^{−κ_{p}|n−m|}  (3.24) 3.20

and n, m ∈ [0, N − 1],

𝔼(|G_{nm}^{(N)}(z)|^{p}) ≤ D_{p} e^{−κ_{p}|n−m|}  (3.25) 3.21

One gets (3.25) from (3.24) by repeating Step 1.6.

That completes Part 2.

\underline{Step 3.1} (Dynamic localization). In the Schrödinger case, Aizenman [1] shows (3.25) bounds imply bounds on sup_{t}|(e^{−ith})_{nm}|. The analog of this has been proven by Simon [36]. Thus, (3.25) implies

𝔼( sup_{ℓ}|(C^{ℓ})_{nm}|^{p}) ≤ D_{p} e^{−κ_{p}|n−m|}  (3.26) 3.23

and similarly with C replaced by C^{(N)}. \underline{Step 3.2} (Pointwise a.e. bounds). For a.e. α, there is D(α) so

[(C^{(N)})^{ℓ}]_{nm} ≤ D(α)(N + 1)^{8} e^{−κ|n−m|}  (3.27) 3.34

for, by (3.26) and its N analog with p = \frac{1}{2},

𝔼( ∑_{\begin{matrix}n,m,N \\ n,m≤N \end{matrix}} sup_{ℓ}|(C^{ℓ})_{nm}|^{1/2}(N + 1)^{−4}e^{+ \frac{1}{2} κ_{1/2}|n−m|}) < ∞  (3.28) 3.35

\underline{Step 3.3} (SULE for OPUC). Following del Rio, Jitomirskaya, Last, and Simon [9], we can now prove SULE in the following form. For each eigenvalue ω_{k} of C^{(N)}, define m_{k} to maximize the component of the corresponding eigenvector u_{k} (the eigenvalues are simple), that is,

|u_{k,m_{k}}| = max_{ℓ=1,...,N} |u_{k,ℓ}|  (3.29) 3.36

Since

\frac{1}{L} ∑_{ℓ=0}^{L−1} ω̄_{k}^{ℓ}[(C^{(N)})^{ℓ}]_{nm} → u_{k,n}ū_{k,m}

and max|u_{k,l}| ≥ N^{−1/2} (since ∑_{ℓ=0}^{N−1}|u_{k,ℓ}|^{2} = 1), (3.27) implies

|u_{k,n}| ≤ D(α)(N + 1)^{8.5} e^{−κ|n−m_{k}|}  (3.30) 3.37

which is what del Rio et al. call semi-uniform localization (SULE). \underline{Step 3.4} (Bound on the distribution of u_{k}). If |n−m_{k}| ≥ κ^{−1}[(9.5) log(N+

1) + log D(α)], |u_{k,n}| ≤ (N + 1)^{−1} so ∑_{such n}|u_{k,n}|^{2} ≤ \frac{1}{2}, and thus,

|u_{k,m_{k}}|^{2} ≥ \frac{1}{C[log(N) + log D(α)]}  (3.31) 3.38

Since ∑_{k=1}^{N}|u_{k,ℓ}|^{2} = 1 for each ℓ, (3.31) implies for each m,

#{k | m_{k} = m} ≤ C[log(N) + log D(α)]

\underline{Step 3.5} (Decoupling except for bad eigenvalues). Let (C^{♯})^{(N)} be the matrix obtained from C^{(N)} by decoupling in log(N) blocks of size N/ log(N) where decoupling is done with random values of β_{jN/ log(N)} in ∂𝔻. Call an eigenvalue of C^{(N)} bad if its m_{k} lies within C_{1}[log(N) + log D(α)] and good if not. A good eigenvector is centered at an m_{k} well within a single block and, by taking C_{1} large, is of order at most O(N^{−2}) at the decoupling points. It follows, by using trial functions, that good eigenvalues move by at most C_{2}N^{−2} if C^{(N)} is replaced by (C^{♯})^{(N)}. \underline{Step 3.6} (Decoupling of probabilities). Fix the k intervals of Theorem 3.1. We claim if z_{j}^{(N)} are the eigenvalues of C^{(N)} and z_{j}^{♯(N)} of C^{♯(N)}, then

\begin{matrix}ℙ(#(\{j | & z_{j}^{(N)} ∈ Δ_{m}^{(N)}\} = ℓ_{m}, m = 1, …, k) \\ − ℙ(#(j | z_{j}^{♯(N)} ∈ Δ_{m}^{(N)}) = ℓ_{m}, m = 1, …, k) → 0 \end{matrix} (3.32) 3.39

This follows if we also condition on the set where D(α) ≤ D because the distribution of bad eigenvalues conditioned on D(α) ≤ D is rotation invariant, and so the conditional probability is rotation invariant. Thus, with probability approaching 1, no bad eigenvalues lie in the Δ_{m}^{(N)}. Also, since the conditional distribution of good eigenvalues is dθ/2π, they will lie within O(N^{−2}) of the edge with probability N^{−1}. Thus (3.32) holds with the conditioning. Since lim_{D→∞}(D(α) > D) → 0, (3.32) holds.

That concludes Part 3 of the proof. For the fourth part, we note that the POP

Φ_{n} = zΦ_{n−1} − β̄_{n}Φ_{n−1}^{∗} = 0 ⇔ \frac{β_{n}zΦ_{n−1}}{Φ_{n}^{∗}} = 1  (3.33) 3.33x

\underline{Step 4.1} (Definition and properties of η_{n}). Define η_{n}(θ; α_{0}, …, α_{n−1}, β_{n}) for θ ∈ [0, 2π]

\frac{β_{n}zΦ_{n−1}}{Φ_{n−1}^{∗}} = exp(iη_{n})|_{z=e^{iθ}}  (3.34) 3.34x

The ambiguity in η_{n} is settled by usually thinking of it as only defined mod 2π, that is, in ℝ/2πℤ. η_{n} is then real analytic and has a derivative dη_{n}/dθ lying in ℝ. We first claim

\frac{dη_{n}}{dθ} > 0  (3.35) 3.35x

for if η̃(θ) is defined by e^{iη̃} = (z − z_{0})/(1 − zz̄_{0}) for z_{0} ∈ ℝ, and z = e^{iθ}, then

\frac{dη̃}{dθ} = \frac{1 − |z_{0}|^{2}}{|e^{iθ} − z_{0}|^{2}} > 0  (3.36) 3.36x

from which (3.35) follows by writing Φ_{n−1} as the product of its zeros, all of which lie in 𝔻. We also have

∫_{0}^{2π} \frac{dη_{n}}{dθ} dθ = 2πn  (3.37) 3.37x

This follows from the argument principle if we note β_{n}zΦ_{n−1}/Φ_{n−1}^{∗} is analytic in 𝔻 with n zeros there. Alternatively, since the Poisson kernel maps 1 to 1, (3.36) implies ∫ \frac{dη̃}{2θ} \frac{dθ}{2π} = 1, which also yields (3.37).

\underline{Step 4.2} (Independence of η_{n}(e^{iθ}) and \frac{dη_{n}}{dθ} (e^{iθ_{1}})). β_{n} drops out of dη_{n}/dθ at all points. On the other hand, β_{n} is independent of zΦ_{n−1}/Φ_{n−1}^{∗} and uniformly distributed. It follows that η_{n}(e^{iθ_{0}}) and \frac{dη_{n}}{dθ} (e^{iθ_{1}}) at any θ_{0} and θ_{1} are independent. Moreover, η_{n}(e^{iθ}) is uniformly distributed. \underline{Step 4.3} (Calculation of 𝔼(dη_{n}/dθ)). As noted, dη_{n}/dθ is only dependent on {α_{j}}_{j=0}^{n−2} and, by rotation covariance of the α’s (see [34, Eqns. (1.6.62)–(1.6.68)]),

\frac{dη_{n}}{dθ} (θ_{0}; e^{−(j+1)φ}α_{j}) = \frac{dη_{n}}{dθ} (θ_{0} − φ; α_{j})

It follows that since the α_{j}’s are rotation invariant that 𝔼(\frac{dη_{n}}{dθ} (θ_{0})) is independent of θ_{0} and so, by applying 𝔼 to (3.37),

𝔼(\frac{dη}{dθ} (θ_{0})) = n  (3.38) 3.38x

\underline{Step 4.4} (Bound on the conditional expectation). Let I_{n} be an interval on ∂𝔻 of size 2πy/n. Let λ_{0} ∈ I_{n} and consider the conditional probability

ℙ(I_{n} has 2 or more eigenvalues | λ_{0} is an eigenvalue)  (3.39) 3.39x

(where we use “eigenvalue” to refer to zeros of the POP since they are eigenvalues of a C^{(N)}). If that holds, η_{n}(λ_{0}) = 1 and, for some λ_{1} in I, η(λ_{1}) = 1, so ∫_{I_{n}} \frac{dη_{n}}{dθ} dθ ≥ 2π. Thus the conditional probability (3.39) is bounded by the conditional probability

ℙ(∫_{I_{n}} \frac{dη_{n}}{dθ} dθ ≥ 2π | η(λ_{0}) = 1)  (3.40) 3.40

While (3.39) is highly dependent on the value of η(λ_{0}), (3.40) is not since dη_{n}/dθ is independent of η(λ_{0}). Thus, by Chebyshev’s inequality,

(3.39) ≤ (3.40) ≤ ℙ(∫_{I_{n}} \frac{dη_{n}}{dθ} dθ ≥ 2π)

≤ (2π)^{−1}𝔼(∫_{I_{n}} \frac{dη_{n}}{dθ} dθ)

= (\frac{2πy}{n} )(2π)^{−1}n = y  (3.41) 3.41

by (3.38). \underline{Step 4.5} (Two eigenvalue estimate). By a counting argument,

ℙ(I_{n} has exactly m eigenvalues)

= \frac{1}{m} ∫_{θ∈I_{n}} ℙ(I_{n} has exactly m eigenvalues | η_{n}(θ) = 1) dκ(θ)

where dκ(θ) is the density of eigenvalues which is \frac{n}{2π}dθ by rotation invariance. Since m ≥ 2 ⇒ \frac{1}{m} ≤ \frac{1}{2}, we see

ℙ(I_{n} has 2 or more eigenvalues) ≤ \frac{1}{2} ∫_{I_{n}} (3.39) dκ(θ)
≤ \frac{y}{2} (\frac{n}{2π} \frac{2πy}{n} ) = \frac{y^{2}}{2}  (3.42) 3.42

The key is that for y small, (3.42) is small compared to the probability that I_{n} has at least one eigenvalue which is order y. This completes Part 4.

\underline{Step 5.1} (Completion of the proof). It is essentially standard theory of Poisson processes that an estimate like (3.42) for a sum of a large number of independent point processes implies the limit is Poisson. The argument specialized to this case goes as follows. Use Step 3.6 to consider log N independent of POPs of degree N/ log N. The union of the Δ_{m}^{(N)} lies in a single interval, Δ̃^{(N)}, of size C/N (here is where the θ_{0} = ⋯ = θ_{k} condition is used) which is y_{N} \frac{2π}{(N/ log N)} with y_{N} = C/ log N. Thus the probability of any single POP having two zeros in Δ̃^{(N)} is O((log N)^{−2}). The probability of any of the log N POPs having two zeros is O((log N)^{−1}) → 0.

The probability of any single eigenvalue in a Δ_{m}^{(N)} is O(1/ log N), so each interval is described by precisely the kind of limit where the Poisson distribution results. Since, except for a vanishing probability, no interval has eigenvalues from a POP with an eigenvalue in another, and the POPs are independent, we get independence of intervals. This completes our sketch of Stoiciu’s proof of his result.

4. Overview of Clock Theorems

s4

The rest of this paper is devoted to proving various theorems about equal spacings of zeros under suitable hypotheses. In this section, we will state the main results and discuss the main themes in the proofs. It is easiest to begin with the case of POPs for OPUC:

T4.1 Theorem 4.1. Let {α_{j}}_{j=0}^{∞} be a sequence of Verblunsky coefficients so that

∑_{j=0}^{∞} |α_{j}| < ∞  (4.1) 4.1

and let {β_{j}}_{j=1}^{∞} be a sequence of points on ∂𝔻. Let {θ_{j}^{(n)}}_{j=1}^{n} be defined

so 0 ≤ θ_{1}^{(n)} ≤ ⋯ ≤ θ_{n}^{(n)} < 2π and so that e^{iθ_{j}^{(n)}} are the zeros of the POPs

Φ_{n}^{(β)}(z) = zΦ_{n−1}(z) − β̄_{n}Φ_{n−1}^{∗}(z)  (4.2) 4.2

Then (with θ_{n+1}^{(n)} ≡ θ_{1}^{(n)} + 2π)

sup_{j=1,...,n} n|θ_{j+1}^{(n)} − θ_{j}^{(n)} − \frac{2π}{n} | → 0  (4.3) 4.3

as n → ∞.

The intuition behind the theorem is very simple. Szegő’s theorem and Baxter’s theorem imply on ∂𝔻 that (with φ_{n}^{(β)} = Φ_{n}^{(β)}/∥Φ_{n−1}∥)

φ_{n}^{(β)}(e^{iθ}) ∼ e^{inθ} \overline{D(e^{iθ})^{−1}} − β̄_{n}D(e^{iθ})^{−1}  (4.4) 4.4

and the zeros of the right side of (4.4) obey (4.3)! (4.4) holds only on ∂𝔻 and does not extend to complex θ without much stronger hypotheses on α. That works since we know by other means that φ_{n}^{(β)} has all its zeros on ∂𝔻. But when one looks at true OPUC, we will not have this a priori information and will need stronger hypotheses on the α’s.

There is a second issue connected with the ∼ in (4.4). It means uniform convergence of the difference to zero. If f_{n} and g are uniformly close, f_{n} must have zeros close to the zeros of g, and we will have enough control on the right side of (4.4) to be sure that φ_{n}^{(β)} has zeros near those of the right side of (4.4). So uniform convergence will be existence of zeros.

A function like f_{n}(x) = sin(x) − \frac{2}{n} sin(nx), which has three zeros near x = 0, shows uniqueness is a more difficult problem.

There are essentially two ways to get uniqueness. One involves control over derivatives and/or complex analyticity which will allow uniqueness via an appeal to an intermediate value theorem or a use of Rouché’s theorem. These will each require extra hypotheses on the Verblunsky coefficients or Jacobi parameters. In the case of genuine OPUC where we already have to make strong hypotheses for existence, we will use a Rouché argument.

There is a second way to get uniqueness, namely, by counting zeros. Existence will imply an odd number of zeros near certain points. If we have n such points and n zeros, we will get uniqueness. This will be how we will prove Theorem 4.1. Counting will be much more subtle for OPRL because the close zeros will lie in [−2, 2] (if a_{n} → 1 and b_{n} → 0) and there can be zeros outside. For counting to work, we will need only finitely many mass points outside [−2, 2]. This will be obtained via a Bargmann bound, which explains why our hypothesis in the next theorem is what it is:

T4.2 Theorem 4.2. Let {a_{n}}_{n=1}^{∞} and {b_{n}}_{n=1}^{∞} be Jacobi parameters that obey

∑_{n=1}^{∞} n(|a_{n} − 1| + |b_{n}|) < ∞  (4.5) 4.4x

Let {P_{n}}_{n=0}^{∞} be the monic orthogonal polynomials and let {E_{j}}_{j=1}^{∞} (J < ∞) be the mass points of the associated measure which lie outside [−2, 2]. Then for n sufficiently large, P_{n}(x) has precisely J zeros outside [−2, 2] and the other n − J in [−2, 2]. Define 0 < θ_{1}^{(n)} < ⋯ < θ_{n−J}^{(n)} < π so that the zeros of P_{n}(x) on [−2, 2] are exactly at {2 cos(θ_{ℓ}^{(n)})}_{ℓ=1}^{n−J}. Then

sup_{j=1,...,n−J−1} n|θ_{j+1}^{(n)} − θ_{j}^{(n)} − \frac{2π}{2n}| → 0  (4.6) 4.5

Remarks. 1. The Jacobi parameters are defined by the recursion relation

xP_{n}(x) = P_{n−1}(x) + b_{n+1}P_{n}(x) + a_{n}^{2}P_{n−1}(x)  (4.7) 4.6

(with P_{0}(x) = 1 and P_{−1}(x) = 0).

2. It is known for all Jacobi polynomials that the Jacobi parameters have |b_{n}|+|a_{n}−1| = O(n^{−2}). So (4.5) fails. In Section 7, we will provide a different argument that proves clock behavior for Jacobi polynomials.

3. We will also say something about |θ_{1}| and |π − θ_{n−J}|, but the result is a little involved so we put the details in Section 6.

Finally, we quote the result for OPUC obeying the BLS condition:

T4.3 Theorem 4.3. Suppose a set of Verblunsky coefficients obeys (1.13) for C ∈ ℂ, C ≠ 0, b ∈ (0, 1), and Δ ∈ (0, 1). Then the number, J, of Nevai-Totik points is finite, and for n large, Φ_{n}(z) has J zeros near these points. The other n − J zeros can be written {z_{j}^{(n)}}_{j=1}^{n−J} where

z_{j}^{(n)} ≡ |z_{j}^{(n)}|e^{iθ_{j}^{(n)}} with (for n large) θ_{0}^{(n)} ≡ 0 < θ_{1}^{(n)} < θ_{2}^{(n)} < ⋯ < θ_{n−J}^{(n)} < 2π = θ_{n−J+1}^{(n)}. We have that

sup_{j} | |z_{j}^{(n)}| − b | = O(\frac{log(n)}{n})  (4.8) 4.7
sup_{j=0,...,n−J} n|θ_{j+1}^{(n)} − θ_{j}^{(n)} − \frac{2π}{n} | → 0  (4.9) 4.8

and

\frac{|z_{j+1}^{(n)}|}{|z_{j}^{(n)}|} = 1 + O(\frac{1}{n log n})  (4.10) 4.9a

Remarks. 1. We will see that “usually,” the right side of (4.8) can be replaced by O(1/n) and the right side of (4.10) by 1 + O(1/n^{2}).

2. Since θ_{0} = 0 and θ_{n−J+1} = 2π are not zeros, the angular gap between z_{1}^{(n)} and z_{n−J}^{(n)} is 2(2π/n).

3. (4.9) and (4.10) imply that |z_{j+1}^{(n)} − z_{j}^{(n)}| → \frac{2π}{n}b.

The key to the proofs of Theorems 4.2 and 4.3 will be careful asymptotics for P_{n} and Φ_{n}. For P_{n}, we will use well-known Jost-Szegő asymptotics. For Φ_{n}, our analysis seems to be new.

We will also prove two refined results on the Nevai-Totik zeros, one of which has a clock!

T4.4 Theorem 4.4. Suppose that (1.5) holds with b < 1. Let z_{0} obey |z_{0}| > b and D(1/z̄_{0})^{−1} = 0 (i.e., z_{0} is a Nevai-Totik zero). Let z_{n} be zeros of φ_{n}(z) near z_{0} for n large. Then for some ε > 0 and n large,

|z_{n} − z_{0}| ≤ e^{−εn}  (4.11) 4.11

Remark. In general, if z_{0} is a zero of order k of \overline{D(1/z̄)}^{−1}, then there are k choices of z_{n} and all obey (4.11).

T4.5 Theorem 4.5. Suppose that (1.13) holds for C ∈ ℂ, C ≠ 0, b ∈ (0, 1), and Δ ∈ (0, 1). There exists Δ_{2}, Δ_{3} ∈ (0, 1) so that if b < z_{0} < bΔ_{2} is a zero of order k of \overline{D(1/z̄)}^{−1}, then for large n, the k zeros of φ_{n}(z) near z_{0} have a clock pattern:

z_{n}^{(j)} = z_{0} + C_{1}(\frac{b}{|z_{0}|})^{n/k} exp(−2πi \frac{n}{k} arg(z_{0}))e^{2πij/k} + O((\frac{bΔ_{3}}{|z_{0}|} )^{n/k})

(4.12) 4.12

for j = 0, 1, …, k − 1 (so the k zeros form a k-fold clock).

This completes the description of clock theorems we will prove in this article, but I want to mention three other situations where the pictures in [34] suggest there are clock theorems plus a fourth situation: (A) Periodic Verblunsky Coefficients. As Figures 8.8 and 8.9 of

[34] suggest, if the Verblunsky coefficients are periodic (or converge sufficiently rapidly to the periodic case), the zeros are locally equally spaced, but are spaced inversely proportional to a local density of states. We will prove this in a future paper [38]. For earlier related results, see [22, 29, 30].

(B) Barrios-López-Saff [4] consider α_{n}’s which are decaying as b^{n} with a periodic modulation. A strong version of their consideration is that there is a period p sequence, c_{1}, c_{2}, …, c_{p}, all nonzero with

α_{n} = b^{n}c_{n} + O((bΔ)^{n})  (4.13) 4.10a

with 0 < Δ < 1. In [37], we will prove clock behavior for such α_{n}’s. In general, there are p missing points in the clock at bω^{j} with ω = exp(2πi/p) a p-th root of unity. Indeed, we will treat the more general

α_{n} = ∑_{ℓ=1}^{m} c_{ℓ}e^{inθ_{ℓ}}b^{n} + O((bΔ)^{n})  (4.14) 4.10b

for any θ_{1}, …, θ_{m} ∈ [0, 2π). (C) Power Regular Baxter Weights. Figure 8.5 of [34] (which

shows zeros for α_{n} = (n+2)^{−2}) suggests that if β > 1 and n^{β}α_{n} → C sufficiently fast, then one has a strictly clock result. By [4], all zeros approach ∂𝔻, and we believe that if their phases are 0 ≤ θ_{1} ≤ θ_{2} ≤ ⋯ ≤ θ_{n} < 2π and θ_{n+1} = θ_{1}+2π, then sup_{j}|n[θ_{j+1}− θ_{j}] − 2π| → C.

(D) Slow Power Decay. Figures 8.6 and 8.7 of [34] (which show α_{n} = (n + 2)^{−1/2} and (n + 2)^{−1/8}) were shown by Ed Saff at a conference as a warning that pictures can be misleading because they suggest there is a gap in the spectrum while we know that the Mhaskar-Saff theorem applies! In fact, I take their prediction of the gap seriously and suggest if (n+2)^{β}α_{n} → C fast enough for β < 1, then we have clock behavior away from θ = 0, that is, if θ_{0} is fixed and θ_{j}, θ_{j+1} are the nearest zeros to θ_{0}, then n(θ_{j+1} − θ_{j}) → 2π, but that there is a single zero near θ = 0 with the next nearby zero θ′ obeying n|θ′| → ∞.

(C) and (D) present interesting open problems.

5. Clock Theorems for POPs in Baxter’s Class

s5

In this section, we will prove Theorem 4.1.

L5.1 Lemma 5.1. If (4.1) holds, then

sup_{e^{iθ}∈𝔻} |\frac{e^{iθ}φ_{n−1}(e^{iθ})}{φ_{n−1}^{∗}(e^{iθ})} − \frac{e^{inθ} \overline{D(e^{iθ})}^{−1}}{D(e^{iθ})^{−1}} | → 0  (5.1) 5.1

as n → ∞.

Remarks. 1. Recall φ_{n}^{(β)} = Φ_{n}^{(β)}/∥Φ_{n−1}∥, that is,

φ_{n}^{(β)} = zφ_{n−1} − β̄φ_{n−1}^{∗}  (5.2) 5.1a

2. Implicit here is the fact that D(z) defined initially on 𝔻 has a continuous extension to 𝔻̄.

Proof. Baxter’s theorem (see [34, Theorem 5.2.2]) says that D(z) lies in the Wiener algebra and, in particular, has a (unique) continuous extension to 𝔻̄, and that φ_{n−1}^{∗}(z) → D(z)^{−1} uniformly on 𝔻̄ and, in particular, uniformly on ∂𝔻. (5.2) plus φ_{n−1}(e^{iθ}) = e^{i(n−1)θ} \overline{φ_{n−1}^{∗}(e^{iθ})} completes the proof. □

Since D is nonvanishing on 𝔻̄ (see [34, Theorem 5.2.2]), the argument principle implies D(e^{iθ}) = |D(e^{iθ})|e^{iψ(θ)} with ψ continuous and ψ(2π) = ψ(0). We will suppose ψ(0) ∈ (−π, π].

L5.2 Lemma 5.2. For each n and each η ∈ [2ψ(0), 2ψ(0) + 2π), there are

solutions, θ̃_{j,η̃}^{(n)}, of

nθ̃ + 2ψ(θ̃) = 2πj + η̃  (5.3) 5.2

for j = 0, 1, …, n − 1. We have that

sup_{η̃,j=0,1,2,...,n−1} n| θ̃_{j+1,η̃}^{(n)} − θ̃_{j,η̃}^{(n)} − \frac{2π}{n} | → 0  (5.4) 5.2a

where θ̃_{n,η̃}^{(n)} ≡ 2π+θ̃_{0,η̃}^{(n)}. Moreover, for any ε, there is an N so for n > N,

sup_{|η−η′|< \frac{ε}{2}} n|θ̃_{j,η̃}^{(n)} − θ̃_{j,η̃′}^{(n)}| ≤ ε  (5.5) 5.3

Proof. As θ̃ runs from 0 to 2π, the LHS of (5.3) runs from 2ψ(0) to 2πn + 2ψ(0). By continuity, (5.3) has solutions. If there are multiple ones, pick the one with θ̃_{j,η̃}^{(n)} as small as possible.

Since ψ is bounded, there is C so

|θ̃_{j,η̃}^{(n)} − \frac{2πj}{n} | ≤ \frac{C}{n}  (5.6) 5.4

so subtracting (5.3) for j + 1 from (5.3) for j,

| θ̃_{j+1,η̃}^{(n)} − θ̃_{j,η̃}^{(n)} − \frac{2π}{n} | ≤ \frac{2}{n} max_{|θ−θ′|≤ \frac{C+1}{n}} |ψ(θ) − ψ(θ′)|  (5.7) 5.5

Since ψ is continuous on [0, 2π], it is uniformly continuous, and thus the max in (5.7) goes to zero, which implies (5.4).

To prove (5.5), we note that subtracting (5.3) for η̃ from (5.3) for η̃′,

n|θ̃_{j,η̃}^{(n)} − θ̃_{j,η̃′}^{(n)}| ≤ |η̃ − η̃′| + 2|ψ(θ̃_{j,η̃}^{(n)}) − ψ(θ̃_{j,η̃′}^{(n)})|  (5.8) 5.6

(5.8) first implies

|θ̃_{j,η̃}^{(n)} − θ̃_{j,η̃′}^{(n)}| ≤ \frac{C}{n}

and then implies (5.5) picking N so

sup_{|θ−θ′|≤ \frac{C}{n}} |ψ(θ) − ψ(θ′)| < \frac{ε}{4}  □

Remark. The proof shows that (5.5) continues to hold for any solutions of (5.3).

Proof of Theorem 4.1. The phase, ζ_{n}(θ), of zφ_{n−1}/φ_{n−1}^{∗}|_{z=e^{iθ}} is monotone increasing and runs from ζ_{n}(0) to 2πn = ζ_{n}(0) at θ = 2π (see Step 4.1 in Section 3 for the monotonicity), so for any fixed β_{n} = e^{−iη_{n}} with η_{n} ∈ [ζ_{n}(0), ζ_{n}(0) + 2π), there are exactly n solutions, θ_{j}^{(n)}, j = 0, 1, …, n − 1 of

ζ_{n}(θ_{j}^{(n)}) = 2πj + η_{n}  (5.9) 5.7

By (5.1), θ_{j}^{(n)} solves (5.3) with

η′ − η_{n} → 0

as n → ∞. By the remark after Lemma 5.2, that shows that (4.3) holds. □

I believe this result has an extension to a borderline where α_{n} = C_{0}n^{−1} + error, where the error goes to zero sufficiently fast (ℓ^{1} error may suffice; for applications, |error| ≤ Cn^{−2} is all that is needed). The extension is on zeros away from θ = 0. I believe in this case that D has an extension to ∂𝔻\{1} (see [35, Section 12.1]) and one has convergence there. Replacing uniformity should be some control of derivatives of φ_{n}^{∗} (an O(n) bound). By the Szegő mapping (see [35, Section 13.1]), this would provide another approach to Jacobi polynomials.

6. Clock Theorems for OPRL With Bargmann Bounds

s6new

Our goal here is to prove Theorem 4.2 which shows clock behavior for OPUC when (4.5) holds. It is illuminating to consider two simple examples first:

En6.1 Example 6.1. Take a_{n} ≡ 1, b_{n} ≡ 0. Then

P_{n}(2 cos θ) = c_{n} \frac{sin((n + 1)θ)}{sin(θ)}  (6.1) n6.1a

the Chebyshev polynomials of the second kind. The zeros are precisely at

θ_{j} = \frac{πj}{n + 1} j = 1, 2, …, n

Note

θ_{j+1} − θ_{j} = \frac{π}{n + 1} = \frac{π}{n} + O(\frac{1}{n^{2}} )  (6.2) n6.1b

for j = 1, …, n − 1. In addition,

θ_{1} = \frac{π}{n} + O(\frac{1}{n^{2}} ) π − θ_{n} = \frac{π}{n} + O(\frac{1}{n^{2}} )  (6.3) n6.1c
□

En6.2 Example 6.2. Take a_{1} = \sqrt{2}, a_{n} = 1 (n ≥ 2), b_{n} = 0. Then

P_{n}(2 cos θ) = c_{n} cos(nθ)  (6.4) n6.1d

the Chebyshev polynomials of the first kind. The zeros are precisely at

θ_{j} = \frac{π(j − \frac{1}{2})}{n}  j = 1, …, n  (6.5) n6.1e

(6.2) holds in this case also but instead of (6.3), we have

θ_{1} = \frac{π}{2n} + O(\frac{1}{n^{2}} ) π − θ_{n} = \frac{π}{2n} + O(\frac{1}{n^{2}} )  (6.6) n6.1f
□

We will make heavy use of the construction of the Jost function in this case. For Jacobi matrices, Jost functions go back to a variety of papers of Case and collaborators; see, for example, [6, 15]. I will follow ideas of Killip-Simon [20] and Damanik-Simon [7, 8], as discussed in [35, Chapter 13].

First, we note some basic facts about the zeros of P_{n}(x), some only true when (4.5) holds.

Pn6.1 Proposition 6.3. Let dμ be a measure on ℝ whose Jacobi parameters obey (4.5). Then

(a) supp(dμ) = [−2, 2] ∪ {E_{j}^{+}}_{j=1}^{N_{+}} ∪ {E_{j}^{−}}_{j=1}^{N_{−}} with E_{1}^{−} < ⋯ < E_{N_{−}}^{−} < −2 and E_{1}^{+} > E_{2}^{+} > ⋯ > E_{N_{+}}^{+} > 2.

(b) N_{+} < ∞ and N_{−} < ∞.

(c) For any n, P_{n}(x) has at most one zero in each (E_{j}^{−}, E_{j+1}^{−}) (j = 1, …, N_{−}), in each (E_{j+1}^{+}, E_{j}^{+}) (j = 1, …, N_{+}), and in (E_{N_{−}}^{−}, −2) and (2, E_{N_{+}}^{+}).

(d) For some N_{0} and n > N_{0}, P_{n}(x) has exactly one zero in each of the above intervals and all other zeros lie in (−2, 2).

Proof. (a) holds because if J_{0} is the free Jacobi matrix (the one with a_{n} ≡ 1, b_{n} ≡ 0), then J − J_{0} is compact.

(b) This follows from the Bargmann bound for Jacobi matrices as proven by Geronimo [13, 14] and Hundertmark-Simon [17].

(c) That there is at most one zero in any interval disjoint from supp(dμ) is a standard fact [12].

(d) By a simple variational argument, using the trial functions in (1.2.61) of [34], each E_{j}^{±} is a limit point of zeros. This and (c) imply that each interval has a zero for large N. By a comparison argument, the E_{j}’s cannot be zeros for such n and also shows that ±2 are not zeros. Since all zeros lie in [E_{1}^{−}, E_{1}^{+}] (see [34, Subsection 1.2.5]), the other zeros lie in [−2, 2]. □

Remark. It is possible N_{+} and/or N_{−} are zero, in which case the above proof changes slightly, for example, E_{1}^{−} is replaced by −2.

Next, we use the fact (see [35, Theorem 13.6.5]) that when (4.5) holds, there is a Jost function, u(z), described most simply in the variable z with E = z + z^{−1} (z ∈ 𝔻 maps to ℂ\[−2, 2] and ∂𝔻 is a twofold cover on [−2, 2] with e^{iθ} → 2 cos θ).

Pn6.2 Proposition 6.4. Let dμ be a measure on ℝ whose Jacobi parameters obey (4.5). There exists a function u(z) on 𝔻̄, analytic on 𝔻, continuous on 𝔻̄, and real on 𝔻̄ ∩ ℝ, so that

(a) Uniformly on [0, 2π],

(sin θ)p_{n−1}(2 cos θ) − Im( \overline{u(e^{iθ})} e^{inθ}) → 0  (6.7) n6.1

(b) The only zeros u has in 𝔻 are at those points β_{j}^{±} ∈ 𝔻 with β_{j}^{±} + (β_{j}^{±})^{−1} = E_{j}^{±}.

(c) The only possible zeros of u and ∂𝔻 are at z = ±1, and if u(±1) = 0, then lim_{θ→0} θ^{−1}u(±e^{iθ}) = c exists and is nonzero.

Proof. This is part of Theorems 13.6.4 and 13.6.5 of [35]. □

Proof of Theorem 4.2. Write

u(e^{iθ}) = |u(e^{iθ})|e^{iη(θ)}  (6.8) n6.2

Mod 2π, η is uniquely defined on (0, π) and (π, 2π) since u is nonvanishing there.

Suppose first that u(±1) ≠ 0. Then η can be chosen continuously at ±1 and so η can be chosen continuously on [0, 2π] with

η(2π) − η(0) = 2π(N_{+} + N_{−})

by the argument principle and the fact that the number of zeros of u in 𝔻 is N_{+} + N_{−}.

It follows that if

g_{n}(θ) = nθ − η(θ)  (6.9) n6.2a

that

g_{n}(2π) = 2π(n − N_{+} − N_{−})

and, in particular,

Im( \overline{u(e^{iθ})} e^{inθ})

has at least 2(n − N_{+} − N_{−}) zeros at points where θ̃_{j}

g_{n}(θ̃_{j}) = πj  j = 0, 1, …, 2(n − N_{+} − N_{−}) − 1  (6.10) n6.3

Since u is real,

θ̃_{2(n−N_{+}−N_{−})−j} = 2π − θ̃_{j}  (6.11) n6.4

and θ̃_{0} = 0, θ̃_{n−N_{+}−N_{−}} = π.

By the boundedness of η, (6.9), and (6.10),

θ̃_{j+1} − θ̃_{j} = \frac{π}{n} + o(\frac{1}{n})  (6.12) n6.5

uniformly in j.

By (6.7), sin(θ)p_{n−1}(2 cos θ), θ ∈ [0, 2π] has at least n − N_{+} − N_{−} + 1 zeros on [0, π] at points θ_{j} with θ_{j} − θ̃_{j} = o(1/n). Since 0 and π are zeros of sin θ and for large n, p_{n−1}(2 cos θ) only has n − N_{+} − N_{−} − 1 zeros on (0, π), {θ_{j}} are the only zeros. (6.12) implies (4.6) and also

θ_{1} = \frac{π}{n} + O(\frac{1}{n}) θ_{n−N_{+}−N_{−}−1} = π − \frac{π}{n} + O(\frac{1}{n})  (6.13) n6.6

Next, we turn to the case where u vanishes as +1 and/or −1. Suppose that u(1) = 0, u(−1) ≠ 0. By (a) of Proposition 6.4 and the reality of u (i.e., \overline{u(e^{iθ})} = u(e^{−iθ})), we have that u(e^{iθ}) = icθ + o(θ) where c ≠ 0, so

η(0) = ±\frac{π}{2}  η(2π) = ∓\frac{π}{2}  (mod 2π)

By the argument principle taking into account the zero at z = 1,

η(2π) − η(0) = 2π(N_{+} + N_{−} + \frac{1}{2})

As in the regular case, (6.12) holds, but in place of (6.7),

θ_{1} = \frac{π}{2n} + O(\frac{1}{n}) θ_{n−N_{+}−N_{−}−1} = π − \frac{π}{n} + O(\frac{1}{n})  (6.14) n6.7

The other cases are similar. □

To summarize, we use a definition. Definition. We say J has a resonance at +2 if and only if u(1) = 0 and a resonance at −2 if u(−1) = 0.

Tn6.3 Theorem 6.5. Let (4.5) hold. Let θ_{j}^{(n)} be the points where P_{n}(2 cos θ) = 0, 0 < θ_{1}^{(n)} < ⋯ < θ_{n−N_{+}−N_{−}}^{(n)} < π. Then

θ_{1}^{(n)} = \frac{π}{n} + O(\frac{1}{n^{2}} )

if +2 is not a resonance and

θ_{1}^{(n)} = \frac{π}{2n} + O(\frac{1}{n^{2}} )

if +2 is a resonance. Similar results hold for θ_{n−N_{+}−N_{−}}^{(n)} with regard to a resonance at −2.

7. Clock Theorems for Jacobi Polynomials

s6

The Jacobi polynomials, P_{n}^{(α,β)}(x), are defined [31, 40] to be orthogonal for the weight

w(x) = (1 − x)^{α}(1 + x)^{β}  (7.1) n7.1

on [−1, 1] where α, β > −1 (to insure integrability of the weight). The P_{n}’s are normalized by

P_{n}^{(α,β)}(1) = \frac{[∏_{j=0}^{n−1}(1 + α − j)]}{n!}  (7.2) n7.2

They are neither monic nor orthonormal, but it is known ([40, Eqns. (4.3.2) and (4.3.3)]) that P_{n}^{(α,β)}(x) differ from the normalized polynomials by (n + 1)^{1/2}c_{n}^{(α,β)} with 0 < inf_{n} c_{n}^{(α,β)} ≤ sup_{n} c_{n}^{(α,β)} < ∞ (not uniform in α, β), and 2^{n}P_{n}^{(α,β)}(x) have leading term d_{n}^{(α,β)} with a similar estimate to c_{n}.

Our goal in this section is to prove

Tn7.1 Theorem 7.1. Fix α, β. Let θ_{j}^{(n)}, j = 1, 2, …, n, be defined by

0 < θ_{1}^{(n)} < ⋯ < θ_{n}^{(n)} < π  (7.3) n7.3

and x_{j}^{(n)} = cos(θ_{j}^{(n)}) are all the zeros of P_{n}^{(α,β)}(x). Then for each ε > 0,

sup_{j; θ_{j}∈[ε,π−ε]} n|θ_{j+1}^{(n)} − θ_{j}^{(n)} − \frac{π}{n}| → 0  (7.4) n7.4

as n → ∞.

Remarks. 1. It is not hard to see that θ_{j} ∈ [ε, π − ε] can be replaced by δn < j < (1 − δ)n for each δ > 0.

2. For restricted values of α, β, this result is a special case of results of Erdös-Turan [11]. Szegő [40] has bounds of the form

Cn^{−1} < θ_{j+1}^{(n)} − θ_{j}^{(n)} < Dn^{−1}

with C, D ε-dependent. I have not found (7.4), but the proof depends on such well-known results in such a simple way that I’m sure it must be known!

Proof. We will depend on two classical results. The first is Darboux’s formula ([40, Theorem 8.21.8]) for the large n asymptotics of P_{n}^{(α,β)}:

P_{n}(cos θ) = n^{−1/2}k(θ)^{−1/2} cos(nθ + γ(θ)) + O(n^{−3/2})  (7.5) n7.5

where

k(θ) = π^{− \frac{1}{2}} sin(\frac{θ}{2})^{−α− \frac{1}{2}} cos(\frac{θ}{2})^{−β− \frac{1}{2}}  (7.6) n7.6
γ(θ) = \frac{1}{2} (α + β + 1)θ − (α + \frac{1}{2}) \frac{π}{2}  (7.7) n7.7

and where the O(n^{−3/2}) is uniform in θ ∈ [ε, π − ε] for each fixed ε > 0 and fixed α, β. (7.5) is just pointwise Szegő-Jost asymptotics on [−1, 1] with explicit phase (k is determined by the requirement that k(θ)w(cos θ)d(cos θ) must be a multiple of dθ).

The second formula we need ([31, Eqn. (13.8.4)]) is

\frac{d}{dx} P_{n}^{(α,β)}(x) = \frac{1}{2} (1 + α + β + n)P_{n−1}^{(α+1,β+1)}(x)  (7.8) n7.8

which is a simple consequence of the Rodrigues formula.

Fix θ_{0}. Define θ^{(n)} by

θ^{(n)} = \frac{2π}{n} [\frac{nθ_{0}}{2π} ]  (7.9) n7.9

where [y] = integral part of y. Then θ^{(n)} ≤ θ_{0} < θ^{(n)} + \frac{2π}{n} and nθ^{(n)} ∈ 2πℤ. Define

f_{n}(y) = n^{1/2}P_{n}^{(α,β)}(cos(θ^{(n)} + \frac{y}{n}))  (7.10) n7.10

Then (7.5) implies that uniformly in θ_{0} ∈ [ε, π − ε] and y ∈ [−Y, Y] (any ε > 0, Y < ∞), we have as n → ∞,

f_{n}(y) → k(θ_{0})^{−1/2} cos(y + γ(θ_{0}))  (7.11) n7.11

Moreover, by (7.8), f′_{n}(y) converges uniformly to a continuous limit. Standard functional analysis (essentially the fundamental theorem of calculus!) says that if f_{n} → f_{∞} and f′_{n} → g_{∞} uniformly for C^{1} functions f_{n}, then g_{∞} = f′_{∞}. Thus

f′_{n}(y) → −k(θ_{0})^{−1/2} sin(y + γ(θ_{0}))  (7.12) n7.12

In particular, since k(θ_{0}) is bounded above and below on [−ε, π − ε], we see that for α, β, ε, Y fixed, there are C_{1}, C_{2} > 0 and N, so for all θ_{0} ∈ [ε, π − ε], |y_{j}| < Y for j = 1, 2, and n > N, we have

|y_{1} − y_{2}| < π, |f_{n}(y_{j})| < C_{1} ⇒ |f_{n}(y_{1}) − f_{n}(y_{2})| ≥ C_{2}(y_{1} − y_{2}) (7.13) n7.13

In the usual way (7.13) implies that for n large, there is at most one solution of f_{n}(y) = 0 within π of another solution. Since (7.5) implies existence, we can pinpoint the zeros of P_{n} in [ε, π − ε] as single points near \{\frac{π}{2n} + \frac{jπ}{n} − γ(\frac{π}{2n} + \frac{jπ}{n}) + o(\frac{1}{n})\}. From this, (7.4) follows. □

8. Asymptotics Away From the Critical Region

s7

This is the first of several sections which focus on proving Theorem 4.3. The key will be asymptotics of φ_{n}(z) in the region near |z| = b. In this section, for background, we discuss asymptotics away from |z| = b. We start with |z| > b. The first part of the following is a translation of results of Nevai-Totik [28] from asymptotics of φ_{n}^{∗} to φ_{n}:

T7.1 Theorem 8.1. If (1.5) holds for 0 < b < 1, then D is analytic in {z | |z| < b^{−1}} and for |z| > b,

lim_{n→∞} z^{−n}φ_{n}(z) = \overline{D(1/z̄)}^{−1}  (8.1) 7.1

and (8.1) holds uniformly in any region |z| ≥ b + ε with ε > 0. Indeed, on any region {z | b + ε < |z| ≤ 1},

|φ_{n}(z) − \overline{D(1/z̄)}^{−1}z^{n}| ≤ C_{ε}(b + \frac{ε}{2})^{n}  (8.2) 7.2

Remark. The point of (8.2) is that the error in (8.1) is approximately O(b^{n}/|z|^{n}), which is exponentially small if |z| > b + ε. It is remarkable that we get exponentially small errors with only (1.5).

Proof. By step (2) in the proof of Theorem 1.1,

|φ_{n+1}^{∗}(z) − φ_{n}^{∗}(z)| ≤ C̃_{ε}[max(1, |z|)]^{n}|b + \frac{ε}{2} |^{n}  (8.3) 7.3

As noted there, this implies (1.9) which, using

φ_{n}(z) = z^{n} \overline{φ_{n}^{∗}(1/z̄)}  (8.4) 7.3a

implies (8.1). (8.3) then implies

|φ_{n}^{∗}(z) − D(z)^{−1}| ≤ C_{ε}[max(1, |z|)]^{n} |b + \frac{ε}{2}|^{n}  (8.5) 7.4

if |z|(b + \frac{ε}{3}) < 1, and this yields (8.2) after using (8.4). □

Remark. The restriction |z| ≤ 1 for (8.2) comes from |z| ≤ 1 in (1.8). But, by Theorem 8.1, (1.8) holds if |z| > b, and so we can conclude (8.2) in any region {z | b + ε < |z| < b^{−1} − ε}. By the maximum principle, we have that

sup_{|z|≤b+ε} |b + ε|^{−n}|Φ_{n}(z)| < ∞

which, plugged into the machine in (8.2), implies for |z| > b^{−1} − ε, we have

|φ_{n}(z) − \overline{D(1/z̄)}^{−1}z^{n}| ≤ C_{ε}|z|^{n}(b + \frac{ε}{2})^{n}

which is exponentially small compared to |z|^{n}.

Barrios-López-Saff [4] proved that ratio asymptotics (1.12) implies that φ_{n+1}(z)/φ_{n}(z) → b for |z| < b, thereby also proving there are no zeros of φ_{n} in each disk {z | |z| < b − ε} if n is large (see also [35, Section 9.1]). Here we will get a stronger result from a stronger hypothesis:

T7.2 Theorem 8.2. Suppose that

b^{−n}α_{n} → C ≠ 0  (8.6) 7.5

as n → ∞. Then for any |z| < b,

b^{−n}φ_{n}(z) → C̄(z − b)^{−1}D(z)^{−1}  (8.7) 7.6

Moreover, if BLS asymptotics (1.13) holds, then in each region {z | |z| < b − ε},

|b^{−n}φ_{n}(z) − C̄(z − b)^{−1}D(z)^{−1}| ≤ C_{1}Δ̃^{n}  (8.8) 7.6a

for some Δ̃ < 1.

Remark. In [37], we will prove a variant of this result that only needs ratio asymptotics as an assumption.

Proof. Define

u_{n}(z) = φ_{n}(z)b^{−n} A_{n} = −ᾱ_{n}b^{−n−1}  (8.9) 7.7

Then, Szegő recursion says

u_{n+1} = (\frac{z}{b} )u_{n} + A_{n}φ_{n}^{∗}(z)  (8.10) 7.8

Iterating, we see that

u_{n} = ∑_{j=1}^{n} A_{n−j}φ_{n−j}^{∗}(z)(\frac{z}{b} )^{j−1} + (\frac{z}{b} )^{n}u_{0}  (8.11) 7.9

Since A_{m} → −C̄b^{−1}, φ_{m}^{∗} → D^{−1}, and |z|/b < 1, (8.11) implies u_{n} has a limit u_{∞}. (8.10) then implies

bu_{∞} = zu_{∞} − C̄D(z)^{−1}  (8.12) 7.10

which implies (8.7).

If (1.13) holds, then

A_{n} − A_{∞} = O(Δ^{n})  (8.13) 7.11

Moreover, Szegő recursion for Φ_{n}^{∗} implies if |z| < 1,

|Φ_{n+1}^{∗} − Φ_{n}^{∗}| ≤ C_{1}b^{n}|ρ_{n}^{−1} − 1|

and then since |ρ_{n}^{−1} − 1| ≤ C_{1}|α_{n}| if |α_{n}| < \frac{1}{4}, |φ_{n+1}^{∗} − φ_{n}^{∗}| ≤ C_{2}b^{n}, and so

|z| ≤ 1 ⇒ |φ_{n}^{∗}(z) − D(z)^{−1}| ≤ C_{3}b^{n}  (8.14) 7.12

(8.11), (8.13), and (8.14) imply (8.8) with Δ̃ = max(Δ, b). □

9. Asymptotics in the Critical Region

s8

The key result in controlling the zeros when the BLS condition holds is

T8.1 Theorem 9.1. Let the BLS condition (1.13) hold for a sequence, {α_{n}}_{n=0}^{∞}, of Verblunsky coefficients and some b ∈ (0, 1) and C ∈ ℂ. Then there exist D, Δ_{1}, and Δ_{2} with 0 < Δ_{1} < Δ_{2} < 1 so that if

bΔ_{2} < |z| < bΔ_{2}^{−1}  (9.1) 8.1

then

|φ_{n}(z) − \overline{D(1/z̄)}^{−1}z^{n} − C̄(z − b)^{−1}D(z)^{−1}b^{n}| ≤ D(bΔ_{1})^{n}  (9.2) 8.2

Remarks. 1. Implicit in (9.2) is that \overline{D(1/z̄)}^{−1} has an analytic continuation (except at z = b; see Remark 2) to the region (9.1), that is, D(z)^{−1} has an analytic continuation to {z | |z| < b^{−1}Δ_{2}^{−1}} except for z = b^{−1}.

2. Since φ_{n}(z) is analytic at z = b and D(b)^{−1} ≠ 0, the poles in \overline{D(1/z̄)}^{−1} and D(z)/(z − b)^{−1} must cancel, that is, (1.4) must hold.

3. In this way, Theorem 9.1 includes a new proof of one direction of Theorem 1.2, that is, that the BLS condition implies that D(z)^{−1} is meromorphic in {z | |z| < b^{−1}Δ_{2}^{−1}} with a pole only at z = b^{−1} with (1.4).

4. The condition Δ_{1} < Δ_{2} implies that the error O((bΔ_{1})^{n}) is exponentially smaller than both z^{n} and b^{n} in the region where (9.1) holds.

We will prove (9.2) by considering the second-order equation obeyed by φ_{n} for n so large that α_{n−1} ≠ 0 (see [34, Eqn. (1.5.47)]):

φ_{n+1} = ρ_{n}^{−1}(z + \frac{ᾱ_{n}}{ᾱ_{n−1}} )φ_{n} − \frac{ᾱ_{n}}{ᾱ_{n−1}} \frac{ρ_{n−1}}{ρ_{n}} zφ_{n−1}  (9.3) 8.3

(the only other applications I know of this formula are in [4] and Mazel et al. [23]). By (1.13) which implies ρ_{n} = 1 + O(b^{2n}) and ᾱ_{n}/ᾱ_{n−1} = b + O(Δ^{n}), we have

ρ_{n}^{−1}(z + \frac{ᾱ_{n}}{ᾱ_{n−1}} ) = z + b + O(b^{2n} + Δ^{n})  (9.4) 8.4
(\frac{ᾱ_{n}}{ᾱ_{n−1}} )\frac{ρ_{n−1}}{ρ_{n}} z = bz + O(b^{2n} + Δ^{n})  (9.5) 8.5

In [37], we will analyze this critical region by an alternate method that, instead of analyzing (9.3) as a second-order homogeneous difference equations, analyzes the more usual Szegő recursion as a first-order inhomogeneous equation.

We thus study the pair of difference equations:

u_{n+1} = ρ_{n}^{−1}(z + \frac{ᾱ_{n}}{ᾱ_{n−1}} )u_{n} − \frac{ᾱ_{n}}{ᾱ_{n−1}} \frac{ρ_{n−1}}{ρ_{n}} zu_{n−1}  (9.6) 8.6

and

u_{n+1}^{(0)} = (z + b)u_{n}^{(0)} − bzu_{n−1}^{(0)}  (9.7) 8.7

expanding solutions of u_{n} in terms of solutions of u_{n}^{(0)}.

Two solutions of (9.7) are b^{n} and z^{n} (since x^{2} −(z +b)x+bz is solved by x = b, z). These are linearly independent if b ≠ z but not at b = z, so it is better to define

x_{n} = z^{n} y_{n} = \frac{z^{n} − b^{n}}{z − b}  (9.8) 8.8

with y_{n} interpreted as nb^{n−1} at z = b.

We rewrite (9.7) as

\binom{u_{n+1}^{(0)}}{u_{n}^{(0)}} = M^{(0)}\binom{u_{n}^{(0)}}{u_{n−1}^{(0)}}  (9.9) 8.9

with

M^{(0)} = (\begin{matrix}z + b & −bz \\ 1 & 0 \end{matrix})  (9.10) 8.10

and (9.6) as

\binom{u_{n+1}}{u_{n}} = (M^{(0)} + δM_{n})\binom{u_{n}}{u_{n−1}}  (9.11) 8.11

where δM_{n} is affine in z and, by (9.4)/(9.5), obeys

∥δM_{n}∥ ≤ C(1 + |z|)(b^{2n} + Δ^{n})  (9.12) 8.12

Now we use variation of parameters, that is, we define c_{n}, d_{n} by

u_{n} = c_{n}x^{n} + d_{n}y^{n}  (9.13) 8.13
u_{n−1} = c_{n}x^{n−1} + d_{n}y^{n−1}  (9.14) 8.14

or (u_{n}u_{n−1})^{t} = Q_{n}(c_{n}d_{n})^{t} where

Q_{n} = (\begin{matrix}x^{n} & y^{n} \\ x^{n−1} & y^{n−1} \end{matrix})  (9.15) 8.15

Since det(Q_{n}) = −z^{n−1}b^{n−1}, we have

Q_{n+1}^{−1} = −z^{−n}b^{−n} (\begin{matrix}y^{n} & −y^{n+1} \\ −x^{n} & x^{n+1} \end{matrix})  (9.16) 8.16

Since x_{n} and y_{n} solve (9.7),

Q_{n+1}^{−1}M^{(0)}Q_{n} = 1  (9.17) 8.17

Moreover, since

|x_{n}| ≤ |z|^{n} |y_{n}| ≤ n max(|z|, |b|)^{n}  (9.18) 8.18

(9.12), (9.15), and (9.16) imply that

δM̃_{n} ≡ Q_{n+1}^{−1}δM_{n}Q_{n}  (9.19) 8.19

obeys

∥δM̃_{n}∥ ≤ C_{n}[\frac{max(|z|, |b|)}{min(|z|, |b|)}]^{n}(1 + |z|)(b^{2n} + Δ^{n})  (9.20) 8.20

In particular, in the region (9.1),

∥δM̃_{n}∥ ≤ CΔ_{1}^{2n}  (9.21) 8.21

if we take Δ_{1} = Δ_{2}^{2} and Δ_{2} < 1 is picked so that max(b^{2}, Δ) < Δ_{2}. Since, in the region (9.1),

|z^{n}| ≤ |b|^{n}Δ_{2}^{−n}  (9.22) 8.22

we have that

n[|z|^{n} + |b|^{n}]∥δM̃_{n}∥ ≤ C(bΔ_{1})^{n}  (9.23) 8.23

We thus have the tools to prove the main input needed for Theorem 9.1:

P8.2 Proposition 9.2. There exist 0 < Δ_{1} < Δ_{2} < 1 and N so that for z in the region (9.1), there are two solutions u_{n}^{+}(z) and u_{n}^{−}(z) of (9.6) for n ≥ N with (i)

|u_{n}^{+} − x^{n}| − |u_{n}^{−} − y^{n}| ≤ D_{1}(bΔ_{1})^{n}  (9.24) 8.24

(ii) u_{n}^{±}(z) are analytic in the region (9.1). (iii) (u_{n+1}^{±}, u_{n}^{±})^{t} are independent for ^{+} and ^{−} for n ≥ N.

Proof. If u_{n} is related to c_{n}, d_{n} by (9.13)/(9.14), then (9.6) is equivalent to (9.11) and then to

\binom{c_{n+1}}{d_{n+1}} = (1 + δM̃_{n})\binom{c_{n}}{d_{n}}  (9.25) 8.25

By (9.23) and the fact that δM̃_{n} is analytic in the region (9.2), we see

L_{n}(z) = ∏_{j=n}^{∞}(1 + δM̃_{n}(z))  (9.26) 8.26

exists, is analytic in z (in the region (9.1)), and invertible for all z in the region and n ≥ N sufficiently large.

Define

\binom{c_{n}^{+}}{d_{n}^{+}} = L_{n}(z)^{−1}\binom{1}{0}  (9.27) 8.27

and c_{n}^{−}, d_{n}^{−} by the same formula with \binom{1}{0} replaced by \binom{0}{1} and then u_{n}^{±} by (9.13). Analyticity of u is immediate from the analyticity of L_{n}(z) and (9.24) follows from (9.23).

Independence follows from the invertibility of L_{n}(z)^{−1}. □

Proof of Theorem 9.1. By the independence, we can write

(φ_{N+1}(z), φ_{N}(z)) = f_{1}(z)(u_{N+1}^{+}(z), u_{N}^{+}(z)) + f_{2}(z)(u_{N+1}^{−}(z), u_{N}^{−}(z))

(9.28) 8.28

where f_{1}, f_{2} are analytic in the region (9.1) since φ and u^{±} are. By the fact that φ, u^{±} obey (9.6), we have for all n ≥ N that

φ_{n}(z) = f_{1}(z)u_{n}^{+}(z) + f_{2}(z)u_{n}^{−}(z)  (9.29) 8.29

Suppose first |z| < b in the region (9.1). Then, since b^{−n}|z|^{n} → 0, (9.24) implies that as n → ∞,

b^{−n}u_{n}^{+} → 0 b^{−n}u_{n}^{−} → −\frac{1}{z − b}  (9.30) 8.30

We conclude, by (8.7), that in that region,

f_{2}(z) = −C̄D(z)^{−1}  (9.31) 8.31

so, by analyticity, this holds in all of the region (9.1).

Next, suppose |z| > b, so |z|^{−n}b^{n} → 0. (9.24) implies that as n → ∞,

z^{−n}u_{n}^{+} → 1 z^{−n}u_{n}^{−} → \frac{1}{z − b}  (9.32) 8.32

By (8.1), we conclude that

f_{1}(z) + f_{2}(z)(z − b)^{−1} = \overline{D(1/z̄)}^{−1}  (9.33) 8.33

Again, by analyticity, this holds in all of the region (9.1) except z = b. It follows that D(z)^{−1} has a pole at b^{−1} and otherwise is analytic in {z | |z| < b^{−1}Δ_{2}^{−1}}. (9.33) also determines the residue to be given by (1.14).

(9.29) becomes

φ_{n}(z) = [ \overline{D(1/z̄)}^{−1} +C̄(z −b)^{−1}D(z)^{−1}]u_{n}^{+}(z)−C̄D(z)^{−1}u_{n}^{−}(z) (9.34) 8.34

(9.2) follows from this result, boundedness of f_{1}, f_{2}, and (9.24). □

10. Asymptotics of the Nevai-Totik Zeros

s10new

In this section, we prove Theorems 4.4 and 4.5. They will be simple consequences of Theorems 8.1 and 9.1.

Proof of Theorem 4.4. If \overline{D(1/z̄)}^{−1} has a zero of order exactly k, at z_{0} there is some C_{1} > 0 and δ > 0 so that

|z − z_{0}| < δ ⇒ |D(1/z̄)| ≥ C_{1}|z − z_{0}|^{k}  (10.1) n10.1

For n large, Hurwitz’s theorem implies |z_{n} − z_{0}| < δ. By (8.2) and (10.1), we have

C_{1}|z_{n} − z_{0}|^{k} ≤ C_{d}z_{n}^{−n}(b + \frac{d}{2})^{n}

for small d and n large. Picking d so z_{0}^{−1}(b + \frac{d}{2}) < 1, we see

|z_{n} − z_{0}|^{k} ≤ C_{2}e^{−2kεn}

for some ε > 0. This implies (4.11) for n large. □

Proof of Theorem 4.5. Pick Δ_{2} to be given by Proposition 9.2. Define Q(z) near z_{0} by

(z_{n} − z_{0})^{k}Q(z) = \overline{D(1/z̄)}^{−1}  (10.2) n10.2

so Q(z_{0}) ≠ 0 and Q is analytic near z_{0}. φ_{n}(z_{n}) = 0 and (9.2) implies

(z_{n} − z_{0})^{k}Q(z_{n}) = C(z_{n} − b)^{−1}D(z_{n})^{−1} \frac{b^{n}}{z_{n}^{n}} + O(\frac{(bΔ_{1})^{n}}{z_{n}^{n}} ) (10.3) n10.3

By Theorem 4.4, z_{n} − z_{0} = O(e^{−εn}) so (10.3) becomes

(z_{n} − z_{0})^{k} = C_{1} \frac{b^{n}}{z_{0}^{n}} + O(\frac{(bΔ_{1})^{n}}{z_{0}^{n}} ) + O(\frac{b^{n}}{z_{0}^{n}} e^{−εn})  (10.4) n10.4

since C̄(z_{n} − b)^{−1}D(z_{n})^{−1}Q(z_{n})^{−1} can be replaced by its value at z_{0} plus an O(e^{−εn}) error. (10.4) implies (4.12). □

11. Zeros Near Regular Points

s9

We call a point z with |z| = b singular if either z = b or D(1/z̄)^{−1} = 0. Regular points are all not singular points on {z | |z| = b}. There are at most a finite number of singular points. In this section, we will analyze zeros of φ_{n}(z) near regular points. In the next section, we will analyze the neighborhood of singular points.

We will use Rouché’s theorem to reduce zeros of φ_{n}(z) to zeros of (z/b)^{n} − g(z_{0}b) (g defined in (11.1)) for z − z_{0}b small. Suppose bz_{0} with z_{0} ∈ ∂𝔻 is a regular point. Define

g(z) = \frac{C̄ \overline{D(1/z̄)}}{D(z)(b − z)}  (11.1) 9.1

which is regular and nonvanishing at z_{0}, so

g(bz_{0}) = ae^{iψ}  (11.2) 9.2

with a > 0 and ψ ∈ [0, 2π). Pick δ < bΔ_{2}^{−1} so that

|z − bz_{0}| < δ ⇒ |g(z) − g(bz_{0})| < \frac{a}{4}  (11.3) 9.3

Define

h_{1}^{(n)}(z) = (\frac{z}{b} )^{n} − g(z_{0}b)  (11.4) 9.4
h_{2}^{(n)}(z) = (\frac{z}{b} )^{n} − g(z)  (11.5) 9.5

h_{3}^{(n)}(z) = \frac{φ_{n}(z) \overline{D(1/z̄)}}{b^{n}}

Theorem 9.1 implies that

|z − bz_{0}| < δ ⇒ |h_{2}^{(n)}(z) − h_{3}^{(n)}(z)| ≤ D_{2}Δ_{1}^{n}  (11.6) 9.6

T9.1 Theorem 11.1. Let z_{0} be a regular point and δ < bΔ_{2}^{−1} so that (11.3) holds. Let j_{1} ≤ j_{2} be integers with |j_{k}| < (n − 1)δ/2. Let

I_{n} = \{z | |\frac{|z|}{b} −1| < \frac{δ}{2b}, arg(\frac{z}{z_{0}} ) ∈ (\frac{ψ}{n} + \frac{2πj_{1}}{n} − \frac{π}{n}, \frac{ψ}{n} + \frac{2πj_{2}}{n} + \frac{π}{n})\}

(11.7) 9.7

Then for n large, I_{n} has exactly (j_{2} − j_{1}) + 1 zeros {z_{ℓ}^{(n)}}_{ℓ=1}^{j_{2}−j_{1}+1} of φ_{n}. Moreover, (a)

|z_{ℓ}^{(n)}| = b(1 + \frac{1}{n} log|g(z_{ℓ}^{(n)})| + O(\frac{1}{n^{2}} ))  (11.8) 9.8
= b(1 + \frac{1}{n} log a + O(\frac{δ}{n}) + O(\frac{1}{n^{2}} ))  (11.9) 9.9

(b)

arg z_{ℓ}^{(n)} = \frac{arg g(z_{k}^{(n)})}{n} + \frac{2πℓ}{n} + O(\frac{1}{n^{2}} )  (11.10) 9.10

= \frac{ψ}{n} + \frac{2πℓ}{n} + O(\frac{δ}{n}) + O(\frac{1}{n^{2}} )  (11.11) 9.11

(c) If |j_{k}| ≤ J, (11.9) and (11.11) hold without the O(δ/n) term.

Proof. Note first that in I_{n}, since ||z| − b| ≤ \frac{δ}{2} and |arg(z/z_{0})| ≤ \frac{δ}{2}, we have |z−z_{0}b| < δ. Consider the boundary of the region I_{n} which has two arcs at |z| = b(1 ± δ) and two straight edges are arg(\frac{z}{z_{0}} ) = \frac{ψ}{n} + \frac{2πj_{1}}{n} − \frac{π}{n} and arg(\frac{z}{z_{0}} ) = \frac{ψ}{n} + \frac{2πj_{2}}{n} + \frac{π}{n}.

We claim on ∂I_{n}, we have for n large that

|h_{j}(z) − h_{1}(z)| ≤ \frac{1}{2} |h_{1}(z)| j = 2, 3  (11.12) 9.12

Consider the 4 pieces of ∂I_{n}:

\underline{|z| = b(1 + δ)}. |h_{1}^{(n)}(z)| ≥ (1 + δ)^{n} − a > a/2 for n large so, by (11.3) in this region,

|h_{1}^{(n)} − h_{2}^{(n)}| ≤ \frac{a}{4} < \frac{1}{2} |h_{1}^{(n)}|

and clearly, by (11.6) for n large,

|h_{1}^{(n)} − h_{3}^{(n)}| < \frac{1}{2} |h_{1}^{(n)}|

\underline{|z| = b(1 − δ)}. |h_{1}^{(n)}(z)| ≥ a − (1 − δ)^{n} > a/2 for n large so, as above, (11.12) holds there.

\underline{|arg z − \frac{ψ}{n} − \frac{2πj_{ℓ}}{n} | = \frac{π}{n}}. z^{n} = −e^{iψ}|z|^{n} so

|h_{1}^{(n)}(z)| = |\frac{z}{b} |^{n} + a > a

and the argument follows the ones above to get (11.12). Thus, (11.12) holds.

By Rouché’s theorem, h_{1}^{(n)}, h_{2}^{(n)}, h_{3}^{(n)} have the same number of zeros in each I_{n}. By applying this to each region with j_{1} = j_{2} and noting that h_{1}^{(n)} has exactly one zero in such a region, we see that each h_{k}^{(n)} has j_{1} + j_{2} + 1 zeros in each I_{n}, and there is one each in each pie slice of angle 2π/n about angles ψ/n + 2πℓ/n.

At the zeros of h_{3}^{(n)}, we have |h_{2}^{(n)}(z)| ≤ D_{2}Δ_{1}^{n} so

|\frac{z_{ℓ}^{(n)}}{b} | = e^{log|g(z_{ℓ}^{(n)})|/n} + O(Δ_{2}^{n})

= 1 + \frac{log|g(z_{ℓ}^{(n)})|}{n} + O(\frac{1}{n^{2}} )

proving (11.8). The proof of (11.10) is similar. Since |g(z) − g(z_{0})| ≤ Cδ, we get (11.9) and (11.11). If |j_{k}| ≤ J, |g(z_{ℓ}^{(n)}) − g(z_{0})| ≤ O(1/n) so the O(δ/n) term is not needed, proving (c). □

T9.2 Theorem 11.2. In each region I_{n} of Theorem 11.1, the zeros z_{ℓ}^{(n)} obey

|z_{ℓ}^{(n)}| − |z_{ℓ+1}^{(n)}| = O(\frac{1}{n^{2}} )  (11.13) 9.14
| arg z_{ℓ+1}^{(n)} − arg z_{ℓ}^{(n)} − \frac{2π}{n} | = O(\frac{1}{n^{2}} )  (11.14) 9.15

Proof. By (11.9) and (11.10), we have

|z_{ℓ+1}^{(n)} − z_{ℓ}^{(n)}| ≤ \frac{C}{n}

(in fact, C = 2πb + O(δ)). Thus

|g(z_{ℓ+1}^{(n)}) − g(z_{ℓ}^{(n)})| ≤ \frac{C_{1}}{n}

so (11.8) and (11.10) imply (11.13) and (11.14). □

12. Zeros Near Singular Points

s10

We first consider the singular point z = b which is always present, and then turn to other singular points which are quite different:

T10.1 Theorem 12.1. Let {α_{j}}_{j=0}^{∞} be a sequence of Verblunsky coefficients obeying BLS asymptotics (1.13). Fix any positive integer, j, with j < (n − 1)δ/2 where δ < bΔ_{2}^{−1} is picked so that

|z − b| < δ ⇒ |g(z) − 1| < \frac{1}{4}  (12.1) 10.1

Then

I_{n} = \{z | |\frac{|z|}{b} − 1| < \frac{δ}{2b}, |arg(z)| < \frac{2πj}{n} + \frac{π}{n}\}

has exactly 2j zeros {z_{ℓ}^{(n)}}_{ℓ=1}^{j} ∪ {z_{ℓ}^{(n)}}_{ℓ=−j}^{−1} with

||z_{ℓ}^{(n)}| − b| = O(\frac{δ}{n}) + O(\frac{1}{n^{2}} )  (12.2) 10.2
arg z_{ℓ}^{(n)} = \frac{2πℓ}{n} + O(\frac{δ}{n}) + O(\frac{1}{n^{2}} )  (12.3) 10.3

Moreover, for each fixed ℓ = ±1, ±2, …,

|z_{ℓ}^{(n)} − be^{2πiℓ/n}| = O(\frac{1}{n^{2}} )  (12.4) 10.4

Remark. We emphasize again the zero at z = b is “missing,” that is, z_{ℓ}^{(n)}, |ℓ| ≥ 2, has its nearest zeros in each direction a distance b2π/n + O(1/n^{2}), while z_{±}^{(n)} has a zero on one side at this distance, but on the other at distance b4π/n + O(1/n^{2}).

Proof. This is just the same as Theorem 11.1! g(z) is regular at z = b. Indeed, g(b) = 1 since \overline{D(1/z̄)}^{−1} and C̄D(z)/(z − b) have to have precisely cancelling poles. D(1/z̄) vanishes at z = b, so φ_{n}(z)\overline{D(1/z̄)}/b^{n} which, by the argument of Theorem 11.1, has 2j + 1 zeros in I_{n}, has one at z = b from D(1/z̄) and 2j from φ_{n}(z), (12.2)–(12.4) follow as did (11.9), (11.11), and (11.13)/(11.14). □

By a compactness argument and Theorems 11.1 and 12.1, we have the following strong form of Theorem 4.3 when there are no singular points other than z = b:

T10.2 Theorem 12.2. Let {α_{j}}_{j=0}^{∞} be a sequence of Verblunsky coefficients obeying the BLS condition (1.13). Suppose z = b is the only singular point, that is, D(z)^{−1} is nonvanishing on |z| = b^{−1}. Let {w_{j}}_{j=1}^{J} be the Nevai-Totik zeros and m_{j} their multiplicities. Let δ be such that for all j ≠ k, |w_{j} − w_{k}| > 2δ and |w_{j}| > b + 2δ, and let M = ∑_{j=1}^{J} m_{j} the total multiplicity of the NT zeros. Let C > sup_{|z|=b}|log(|g(z)|)|. Then (1) For some large N and all n > N, the zeros of φ_{n}(z) are m_{j} zeros

in {z | |z − w_{j}| < δ} for j = 1, …, J and n − M in the annulus

| |z| − |b| | < \frac{C}{n}  (12.5) 10.5

(2) The n − M zeros in the annulus can be labelled by increasing arguments 0 < arg(z_{1}^{(n)}) < ⋯ < arg(z_{n−J}^{(n)}) < 2π, and with arg(z_{n−J+1}^{(n)}) = arg(z_{1}^{(n)}) + 2π and |z_{n−J+1}^{(n)}| = |z_{1}^{(n)}|, we have

arg(z_{k+1}^{(n)}) − arg(z_{k}^{(n)}) = \frac{2π}{n} + O(\frac{1}{n^{2}} )  (12.6) 10.6

and

\frac{|z_{k+1}^{(n)}|}{|z_{k}^{(n)}|} = 1 + O(\frac{1}{n^{2}} )  (12.7) 10.7

uniformly in k = 1, 2, …, n − J + 1.

Remarks. 1. By [23], if α_{n} = −b^{n}, all zeros have |z| = b, so it can happen that there are no O(1/n) terms in |z_{k}^{(n)}|. However, since there are n zeros in (\frac{2π}{n}, \frac{n−1}{n}2π) (i.e., the zero near z = b is missing), there are always either NT zeros or O(1/n^{2}) corrections in some arg(z_{n}^{(ℓ)}).

2. While it can happen that there is no O(1/n) term, its absence implies strong restrictions on α_{n}. For

|g(z)| = 1 on |z| = b  (12.8) 10.9a

implies \overline{g(b/z̄)} g(z) = 1 near |z| = b, and that equation allows analytic continuations of g, and so of D or D^{−1}. In fact, (12.8) implies that D(z)^{−1} is analytic in {z | |z| < b^{−3}} except for a pole at z = b^{−1} and that implies, by Theorem 7.2.1 of [34], that α_{n} = −Cb^{n}+O(b^{2n})(1−ε)^{n} for all ε. Thus, if the BLS condition holds but lim inf|α_{n}−Cb^{n}|^{1/n} > b^{2}, then there must be O(1/n) corrections to |z_{k}^{(n)}| = b. Note that for the Roger’s Szegő polynomials, the poles of D(z)^{−1} are precisely at z ∈ {b^{−2k−1}}_{k=0}^{∞} (see [34, Eqn. (1.6.59)]), consistent with the |z| < b^{−3} statement above.

We now turn to an analysis of the other singular points. As a warmup, we study zeros of

f_{n}(z) = z^{n} − K(1 − z)^{k}  (12.9) 10.8

where

K = ae^{iψ}

is nonzero, a > 0, k a fixed positive integer, and we take n → ∞. We begin by localizing |z| and |z − 1|.

P10.3 Proposition 12.3. (i) There are M > 0 and N_{0} so that if |z| ≥

1 + M/n and n ≥ N_{0}, then f_{n}(z) ≠ 0. (ii) There is N_{1} so if n ≥ N_{1} and

|z| ≤ 1 − 2k \frac{log n}{n}  (12.10) 10.9

then f_{n}(z) ≠ 0. (iii) There is N_{2} so if n ≥ N_{2} and

|z − 1| ≤ \frac{k}{2} \frac{log n}{n}  (12.11) 10.10

then f_{n}(z) ≠ 0.

Remark. If one proceeds formally and lets y = 1 − z, then f_{n}(z) = 0 is equivalent to (1 − y)^{n} = Ky^{k} and finds

y = \frac{k}{n} (− log y) + O(y^{2}) + \frac{log a}{n} + \frac{iψ}{n}

= \frac{k}{n} log n − \frac{k}{n} log k − \frac{k}{n} log(log n) + O((\frac{log n}{n})^{2})

It was this formal calculation that caused us to pick 2k \frac{log n}{n} in (12.10) and \frac{k}{2} \frac{log n}{n} in (12.11).

44 B. SIMON

Proof. (i) For n sufficiently large and |z| ≥ 2, |z|^{n} ≥ a(1 + |z|)^{k} and for n large, (1 + \frac{M}{n})^{n} ≥ e^{M/2} > a2^{k} if M is suitable, so for such M and n, |z|^{n} ≥ a(1 + |z|)^{k} for 1 + \frac{M}{n} ≤ |z| ≤ 2.

(ii) If (12.10) holds,

|1 − z|^{k} ≥ (1 − |z|)^{k} ≥ (2k)^{k}(\frac{log n}{n})^{k}

while log(1 + x) ≤ x implies

|z|^{n} = exp(n log(1 + (|z| − 1)))
≤ exp(−n(1 − |z|))
≤ exp(−2k log n) = n^{−2k}

so for n large, a|1 − z|^{k} > |z|^{n}.

(iii) If (12.11) holds, then

|z| ≥ 1 − |z − 1| ≥ 1 − \frac{k}{2} \frac{log n}{n}

so for n large,

|z|^{n} ≥ exp(\frac{3}{4} k log n)

= n^{− \frac{3}{4} k}

> (\frac{k}{2} \frac{log n}{n})^{k}

≥ |1 − z|^{k}

so f_{n}(z) ≠ 0. □

With this, we are able to control ratios of lengths of nearby zeros.

P10.4 Proposition 12.4. (i) If z, w are two zeros of f_{n}(z) and

| arg(\frac{z}{w})| ≤ \frac{C_{0} log n}{n}  (12.12) 10.11

then, for n large,

| |\frac{z}{w}| − 1 | ≤ \frac{D_{0}}{n}  (12.13) 10.12

(ii) If z, w are two zeros of f_{n}(z) and

| arg(\frac{z}{w})| ≤ \frac{C_{1}}{n}  (12.14) 10.13

then

| |\frac{z}{w}| − 1 | ≤ \frac{D_{1}}{n log n}  (12.15) 10.14

Remark. In both cases, D_{j} is a function of C_{j}, K, and k.

Proof. Since z and w are both zeros,

(\frac{z}{w})^{n} = \frac{(z − 1)^{k}}{(w − 1)^{k}}

so, by (iii) of the last proposition, |w − 1| ≥ \frac{k}{2} \frac{log n}{n}. Thus

|\frac{z}{w}| ≤ (1 + \frac{|z − w|}{|w − 1|})^{k/n}

≤ (1 + \frac{2n|z − w|}{log n})^{k/n}

If (12.12) holds, |\frac{z}{w}| ≤ (1+ \frac{2C_{0}}{n})^{k/n} ≤ 1+ \frac{D̃_{0}}{n} for n large. Interchanging z and w yields (12.13). The argument from (12.14) to (12.15) is identical.

□

As the final step in studying zeros of (12.9), we analyze arguments of nearby zeros.

P10.5 Proposition 12.5. Let z_{0} be a zero of f_{n}(z). Fix C_{2}. Then (i) If w is also a zero and |w − z_{0}| ≤ C_{2}/n, then for some ℓ ∈ ℤ,

arg(\frac{w}{z}) = \frac{2πℓ}{n} + O(\frac{1}{n log n})  (12.16) 10.15

with the size of the error controlled by a C_{2}, K, and k. (ii) For each L and n large, there exists exactly one zero obeying

(12.16) for each ℓ = 0, ±1, ±2, …, ±L. (iii) For any ψ_{0} ∈ [0, 2π) and δ, there is N so for n ≥ N, there is a

zero of f_{n} with arg z ∈ (ψ_{0} − \frac{π+δ}{n}, ψ_{0} + \frac{π+δ}{n}).

Remarks. 1. These propositions imply that the two nearest zeros to z_{0} are z_{0}e^{±2πi/n} + O(1/n log n).

2. If k is odd, the argument of (z − 1)^{k} changes by πk as z moves through 1 along the circle. Thus there are O(1) shifts as zeros swing around the forbidden circle |z −1| ∼ \frac{k}{n} log n. Since there are log n zeros near that circle, the phases really do slip by O(1/n log n). This is also why we do not try to specify exact phases, only relative phases.

Proof. Since z_{0}^{n} = K(z_{0} − 1)^{k}, we have

\frac{f(z)}{z_{0}^{n}} = (\frac{z}{z_{0}} )^{n} − \frac{(z − 1)^{k}}{(z_{0} − 1)^{k}}  (12.17) 10.16

If |z−z_{0}| ≤ C_{2}/n, then since |z_{0}−1| ≥ \frac{1}{2}K \frac{log n}{n} (by Proposition 12.3(iii)),

|\frac{(z − 1)^{k}}{(z_{0} − 1)^{k}} − 1| ≤ \frac{D|z − z_{0}|}{|z_{0} − 1|} ≤ \frac{D_{2}}{log n}  (12.18) 10.17

Fix η and any z_{0} with |z_{0}−1| ≥ \frac{1}{2}K \frac{log n}{n}, we want to look at solutions of

(\frac{z}{z_{0}} )^{n} = e^{iη} \frac{(z − 1)^{k}}{(z_{0} − 1)^{k}}  (12.19) 10.18

with |z − z_{0}| ≤ C_{2}/n. If f(z_{0}) = 0 and η = 0, we have solutions of (12.19) are exactly zeros of f.

By (12.18), (12.19) implies

n(arg(z) − arg(z_{0})) = η + O(\frac{1}{log n})

whose solutions are precisely

arg(\frac{z}{z_{0}} ) = \frac{2πℓ}{n} + η + O(\frac{1}{log n})  (12.20) 10.19

which is (12.16) when η = 0. If we prove that for n large, there is exactly one solution of (12.20) for ℓ = 0, ±1, ±2, …, ±L, we have proven (ii). We also have (iii). For given ψ_{0}, let

h(r) = r^{n} − |K| |re^{iψ_{0}} − 1|^{k}  (12.21) 10.20

If n > k, h(0) < 0 and h(∞) > 0, so there is at least one r_{0} solving h(r) = 0. Let z_{0} = re^{iψ_{0}} and define η by

z_{0}^{n} = Ke^{−iη}(1 − z_{0})^{k}

Solutions of f_{n}(z) = 0 are then precisely solutions of (12.19), and we have the existence statement in (iii) if we prove existence and uniqueness of solutions of (12.19).

Fix M. Consider the following contour with four parts:

C_{1} = \{z | |z| = |z_{0}|(1 + \frac{M}{n} ); (η − \frac{π}{2} )\frac{1}{n} ≤ arg(\frac{z}{z_{0}} ) ≤ (η + \frac{π}{2} )\frac{1}{n}\}
C_{2} = \{z | arg(\frac{z}{z_{0}} ) = (η + \frac{π}{2} )\frac{1}{n}; |z_{0}|(1 − \frac{M}{n} ) ≤ |z| ≤ |z_{0}|(1 + \frac{M}{n} )\}
C_{3} = \{z | z = |z_{0}|(1 − \frac{M}{n} ); (η − \frac{π}{2} )\frac{1}{n} ≤ arg(\frac{z}{z_{0}} ) ≤ (η + \frac{π}{2} )\frac{M}{n} \}

C_{4} = \{z | arg(\frac{z}{z_{0}} ) = (η − \frac{π}{2} )\frac{1}{n}; |z_{0}|(1 − \frac{M}{n} ) ≤ |z| ≤ |z_{0}|(1 + \frac{M}{n} )\}

(π/2 can be replaced by any angle in (0, π)). For n large, q(z) ≡ (z/z_{0})^{n}e^{−iη} follows arbitrarily close to

q[C_{1}] ≅ \{w | |w| = e^{M}; arg(w) ∈ (−\frac{π}{2} , \frac{π}{2} )\}
q[C_{2}] ≅ \{w | e^{−M} ≤ |w| ≤ e^{M}; arg(w) = \frac{π}{2} \}
q[C_{3}] ≅ \{w | |w| = e^{−M}; arg(w) ∈ (−\frac{π}{2} , \frac{π}{2} )\}
q[C_{4}] ≅ \{w | e^{−M} ≤ |w| ≤ e^{M}; arg(w) = −\frac{π}{2} \}

which surrounds w = 1 once.

By (12.18), if q̃(z) = e^{−iη}(z/z_{0})^{n} − (z − 1)^{k}/(z_{0} − 1)^{k}, then q̃[C] surrounds w = 0 once. It follows that (12.19) has one solution inside C. This proves existence and uniqueness within C. By part (i), there are no other solutions with |z − z_{0}| < C/n. □

These ideas immediately imply

T10.6 Theorem 12.6. Let {α_{j}}_{j=0}^{∞} be a sequence of Verblunsky coefficients obeying BLS asymptotics (1.13). Let bz_{0} be a singular point so that \overline{D(1/z̄)}^{−1} has a zero of order k at bz_{0}. Then there is δ > 0 with δ < bΔ_{2}^{−1} and N large so that for n ≥ N_{0}, all zeros of φ_{n}(z) in

S ≡ \{z|Δ_{2}^{−1} < |\frac{z}{b} | < Δ_{2}; |arg(\frac{z}{z_{0}} )| < δ\}  (12.22) 10.21

obey

1 − \frac{D}{n} ≤ |\frac{z}{b} | ≤ 1 + 2k \frac{log N}{n}  (12.23) 10.22

for some D. Moreover, for each z_{1} ∈ S and n > N, there is a zero, z, in S with

|arg(\frac{z}{z_{1}} )| ≤ \frac{2π}{n}  (12.24) 10.23

If z_{ℓ} is a zero in S, then the two nearest zeros to z_{ℓ} obey

arg(\frac{z_{ℓ±1}}{z_{ℓ}} ) = ±\frac{2π}{n} + O(\frac{1}{n log n})  (12.25) 10.24
|\frac{z_{ℓ+1}}{z_{ℓ}} | = 1 + O(\frac{1}{n log n})  (12.26) 10.25

|bz_{0} − z_{ℓ}| ≥ \frac{k}{2} \frac{log n}{n}  (12.27) 10.26

Remark. 2k in (12.23) can be replaced by any number strictly bigger than k (if we take N_{0} large enough). Similarly in (12.24), 2π can be any number strictly bigger than π and k/2 in (12.27) can be any number strictly less than k.

Proof. This follows by combining the analysis of f_{n} above with the ideas used to prove Theorem 11.1, picking h_{1}^{(z)} = (b/z)^{n} − K(z − z_{0}b)^{k}, h_{2}^{(z)} = (b/z)^{n} − g(z)^{−1}, and h_{3}^{(z)} = (z − b)D(z)φ_{n}(z)/C̄z^{n}. □

By compactness of b[∂𝔻] and Theorems 11.1, 12.1, and 12.6, we get the following precise form of Theorem 4.3:

T10.7 Theorem 12.7. Let {α_{j}}_{j=0}^{∞} be a sequence of Verblunsky coefficients obeying BLS asymptotics (1.13). The total multiplicity J of Nevai-Totik zeros is finite. There exists K, D > 0, so that for all n large, the n − J zeros not near the Nevai-Totik points can be labelled z_{j}^{(n)} =

|z_{j}^{(n)}|e^{iθ_{j}^{(n)}}, j = 1, …, n − J, with 0 = θ_{0}^{(n)} < θ_{1}^{(n)} < ⋯ < θ_{n−J+1}^{(n)} = 2π so that (1)

1 − \frac{D}{n} ≤ inf_{j=1,...,n−J} \frac{|z_{j}^{(n)}|}{b} ≤ sup_{j=1,...,n−J} \frac{|z_{j}^{(n)}|}{b} ≤ 1 + K \frac{log n}{n}

(2)

sup_{j=0,...,n−J} n|θ_{j+1}^{(n)} − θ_{j}^{(n)} − \frac{2π}{n} | = O(\frac{1}{log n})

(3) Uniformly in j in 0, …, n − J,

\frac{|z_{j+1}^{(n)}|}{|z_{j}|} = 1 + O(\frac{1}{log n})

where |z_{0}^{(n)}| and |z_{n−J+1}^{(n)}| are symbols for b. Moreover, for L fixed, uniformly in ℓ = 1, …, L, n − J, n − J − 1, …, n − J − L + 1,

z_{ℓ}^{(n)} = \begin{cases}be^{2πiℓ/n} & ℓ = 1, …, L \\ be^{−2πi(n−J−ℓ+1)} & ℓ = n − J, n − J − 1, …, n − J − L + 1 \end{cases}

13. Comments

s11

One question the reader might have is whether singular points other than z = b ever occur and whether there might be a bound on k. In fact, there are no restrictions for P13.1 Proposition 13.1. If f(z) is nonvanishing and analytic in a neigh-

borhood of 𝔻̄ and

∫ |f(e^{iθ})|^{2} \frac{dθ}{2π} = 1  (13.1) 13.1

then there is a measure dμ in the Szegő class with D(z; dμ) = f(z).

Proof. Just take dμ = |f(e^{iθ})|^{2} \frac{dθ}{2π}. □

E13.2 Example 13.2. Pick disjoint points z_{1}, …, z_{ℓ} with arg(z_{j}) ≠ 0 and |z_{j}| = b^{−1} and positive integers k_{1}, …, k_{ℓ}. Let

f(z) = c[ ∏_{j=1}^{ℓ}(z − z_{j})^{−k_{j}}](z − b^{−1})

where c is chosen so that (13.1) holds. Then D^{−1} has a single pole at b^{−1} and so, by Theorem 1.2, α_{j}(dμ) obeys the BLS condition. By Proposition 13.1,

D(z)^{−1} = c^{−1} \frac{1}{z − b^{−1}} ∏_{j=1}^{ℓ}(z − z_{j})^{k_{j}}

and so has zeros at z_{j} of order k_{j}. Thus we have singular points at 1/z̄_{j} of order k_{j}.

Let me emphasize that this procedure also lets one move Nevai-Totik zeros without changing the leading asymptotics of the Verblunsky coefficients or the error estimate. □

Secondly, we want to prove that in a suitable generic sense, the only singular point is z = b. Fix b and Δ in (0, 1). Let V_{b,Δ} be the space of sequences α_{j} in × 𝔻 so that for some C ≠ 0,

|C| + ∑ |(α_{j} − Cb^{j})Δ^{−j}b^{−1}| ≡ ∥α∥ < ∞  (13.2) 13.2

Obviously, any α ∈ V_{b,Δ} obeys the BLS condition.

T13.3 Theorem 13.3. {α ∈ V_{b,Δ} | The only single point is z = b} is a dense open set of V_{b,Δ}.

Proof. The proof of Theorem 7.2.1 of [34] shows that the finiteness condition of (13.2) is equivalent to D(z)^{−1} − C(z − b^{−1})^{−1} lying in the Wiener class of the disk {z | |z| ≤ (Δb)^{−1}} and that ∥ · ∥ convergence is equivalent to Wiener convergence of (z − b^{−1})D(z)^{−1}.

That the set of no singular points other than z = b is open follows from the fact that the just mentioned Wiener convergence implies uniform convergence, and so D(z)^{−1} is nonvanishing on {z | |z| = b^{−1}} is an open set.

As noted in Example 13.2, it is easy to move zeros by multiplying D(z)^{−1} by \frac{z−z_{1}}{z−z_{2}} . This shows that any D(z)^{−1} in the Wiener space is a limit of such D’s nonvanishing on {z | |z| = b^{−1}} and shows that the set of nonsingular α is dense. □

It is a worthwhile and straightforward exercise to compute the change of arg(g) along {z | |z| = b} and so verify that the number of zeros on the critical circle is exactly n minus the total order of the Nevai-Totik zeros.

Added Notes:

1. As I stated after Theorem 7.1, I was sure the result must be known. This is correct. Precise asymptotics of zeros of Jacobi P_{n} to order O(n^{−2}) is found in Vértesi (Stud. Sci. Math. Hungar. 85 (1999), 97–130) for zeros away from ±1 (there are also asymptotic results near ±1). See also J. Szabados and P. Vértesi, Interpolation of Functions, World Scientific, 1990; and Vértesi, Ann. Num. Math. 4 (1997), 561– 577.

2. Independently and approximately simultaneous with my work on OPUC obeying the BLS condition, Martínez-Finkelshtein, McLauglin, and Saff (in preparation) produced a study of asymptotics of OPUC obeying the BLS condition using Riemann-Hilbert methods. Their results overlap ours in Sections 8–12.

3. Last-Simon (in preparation) have a different and generally stronger approach to the question of controlling the uniqueness question, that is, only one zero of P_{n} near the zeros of the Jost-Szegő asymptotic limit. For example, they can prove a clock behavior away from ±2 if (4.5) is replaced by ∑_{n=1}^{∞}|a_{n} − 1| + |b_{n}| < ∞.

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