WEAK CONVERGENCE OF CD KERNELS AND APPLICATIONS

BARRY SIMON^{∗}

Abstract. We prove a general result on equality of the weak limits of the zero counting measure, dν_{n}, of orthogonal polynomials (defined by a measure dμ) and \frac{1}{n}K_{n}(x, x) dμ(x). By combining this with Máté–Nevai and Totik upper bounds on nλ_{n}(x), we prove some general results on ∫_{I} \frac{1}{n}K_{n}(x, x) dμ_{s} → 0 for the singular part of dμ and ∫_{I}|ρ_{E}(x)− \frac{w(x)}{n}K_{n}(x, x)| dx → 0, where ρ_{E} is the density of the equilibrium measure and w(x) the density of dμ.

1. Introduction

We will discuss here orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) (see [30, 10, 9, 21, 22, 27]). dμ will denote a measure on ∂𝔻 = {z ∈ ℂ | |z| = 1} (positive but not necessarily normalized), Φ_{n}(z, dμ) and φ_{n}(z, dμ) its monic and normalized orthogonal polynomials, and {α_{n}}_{n=0}^{∞} its Verblunsky coefficients determined by (Φ_{n}^{∗}(z) ≡ z^{n} \overline{Φ_{n}(1/z̄)})

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

and

∥Φ_{n}∥_{L^{2}(∂𝔻,dμ)} = μ(∂𝔻)^{1/2} ∏_{j=0}^{n−1}(1 − |α_{j}|^{2})^{1/2}  (1.2)

dμ will also denote a measure on ℝ of compact support, P_{n}(x, dμ) and p_{n}(x, dμ) its monic and normalized orthogonal polynomials. {a_{n}, b_{n}}_{n=1}^{∞} are its Jacobi parameters defined by

xp_{n}(x) = a_{n+1}p_{n+1}(x) + b_{n}p_{n}(x) + a_{n}p_{n−1}(x)  (1.3)

Date: July 11, 2007. 2000 Mathematics Subject Classification. Primary: 33C45; Secondary: 60B10,

05E35.

Key words and phrases. CD kernel, orthogonal polynomials, weak convergence.

∗ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125. E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592 and U.S.–Israel Binational Science Foundation (BSF) Grant No. 2002068.

and

∥P_{n}∥_{L^{2}(ℝ,dμ)} = μ(ℝ)^{1/2}(a_{1} … a_{n})  (1.4)

The CD kernel is defined by (some authors sum only to n − 1)

K_{n}(z, w) = ∑_{j=0}^{n} \overline{φ_{n}(z)} φ_{n}(w)  (1.5)

K_{n}(x, y) = ∑_{j=0}^{n} p_{j}(x)p_{j}(y)  (1.6)

The Lebesgue decomposition

dμ(e^{iθ}) = w(θ) \frac{dθ}{2π} + dμ_{s}(e^{iθ})  (1.7)
dμ(x) = w(x) dx + dμ_{s}(x)  (1.8)

with dμ_{s} Lebesgue singular will enter.

To model the issues that concern us here, we recall two consequences of the Szegő condition for OPUC, namely,

∫ log(w(θ)) \frac{dθ}{2π} > −∞  (1.9)

Here are two central results:

Theorem 1.1 (Szegő, 1920 [29]). If the Szegő condition holds, then

lim_{n→∞} ∫ |φ_{n}(e^{iθ})|^{2} dμ_{s} = 0  (1.10)

Remark. In distinction, if w = 0, ∫ |φ_{n}(e^{iθ})|^{2} dμ_{s} ≡ 1.

Theorem 1.2 (Máté–Nevai–Totik [16]). If the Szegő condition holds, then for a.e. θ,

w(θ) \frac{1}{n + 1} K_{n}(e^{iθ}, e^{iθ}) → 1  (1.11)

They state the result in an equivalent form involving the Christoffel function

λ_{n}(z_{0}) = inf{∫ |Q_{n}(e^{iθ})|^{2} dμ(θ) | deg Q_{n} ≤ n; Q_{n}(z_{0}) = 1} (1.12)

The minimizer is

Q_{n}(e^{iθ}) = \frac{K_{n}(z_{0}, e^{iθ})}{K_{n}(z_{0}, z_{0})}  (1.13)
λ_{n}(z_{0}) = K_{n}(z_{0}, z_{0})^{−1}  (1.14)

and since (1.9) implies w(θ) > 0 for a.e. θ, (1.11) is equivalent to

nλ_{n−1}(e^{iθ}) → w(θ)  (1.15)

It is also known under a global w(θ) > 0 condition that there are similar results that come from what is called Rakhmanov theory (see [22, Ch. 9] and references therein):

Theorem 1.3. If w(θ) > 0 for a.e. θ, then

(i) (1.10) holds. (ii)

lim_{n→∞} ∫ |w(θ)|φ_{n}(e^{iθ})|^{2} − 1| \frac{dθ}{2π} = 0  (1.16)

Remark. (i) is due to Rakhmanov [18, 19]; (ii) is due to Máté, Nevai, and Totik [15].

Theorems 1.1 and 1.2 are known to hold under a local Szegő condition together with regularity in the sense of Stahl–Totik [28] (see also [24] and below).

One of our goals here is to prove the first results of these genres with neither a local Szegő condition nor a global a.c. condition. While we focus on the OPRL case, for comparison with the above theorems, here are our new results for OPUC:

Theorem 1.4. If I is an interval on ∂𝔻 so that

(a) w(θ) > 1 a.e. on I

(b) dμ is regular for ∂𝔻, that is,

(ρ_{1} … ρ_{n})^{1/n} → 1  (1.17)

then

(i)  ∫_{I} \frac{1}{n + 1} K_{n}(e^{iθ}, e^{iθ}) dμ_{s}(θ) → 0  (1.18)
(ii)  ∫_{I} |1 − w(θ) \frac{1}{n + 1}K_{n}(e^{iθ}, e^{iθ})| \frac{dθ}{2π} → 0  (1.19)

Remarks. 1. We do not have pointwise convergence (1.11), but do have “one-half” of it, namely, for e^{iθ} ∈ I,

lim inf w(θ) \frac{1}{n + 1} K_{n}(e^{iθ}, e^{iθ}) → 1  (1.20)

2. One associates existence of limits of |φ_{n}|^{2} dμ with Rakhmanov which we only expect when dμ has support on all of ∂𝔻 or a single interval of ℝ. At best, with multiple intervals, one expects almost periodicity of |φ_{n}|^{2} dμ rather than existence of the limits. For this reason, the Cesàro averages of Theorem 1.4 are quite natural.

There is nothing sacred about ∂𝔻—regularity is defined for any set, and for both OPRL and OPUC, all we need is regularity plus w(θ) > 0 on an interval. We also have interesting new bounds on the density of zeros when regularity fails—these generalize a theorem of Totik–Ullman [33].

These new theorems do not involve tweaking the methods used to prove Theorems 1.1–1.3, but a genuinely new technique (plus one general method of Máté–Nevai used in the proof of Theorem 1.2). Our point here is as much to emphasize this new technique as to prove the results. The new technique is the following:

Theorem 1.5. Let dμ be a measure on ℝ with bounded support or a measure on ∂𝔻. For ℝ, let dν_{n} be the normalized zero counting measure for the OPRL and for ∂𝔻 for the zeros of the paraorthogonal polynomials (POPUC). Let n(j) be a subsequence n(1) < n(2) < . . .. Then

dν_{n(j)+1} ⟶^{w} dν_{∞} ⟺ \frac{1}{n(j) + 1} K_{n(j)}(x, x) dμ(x) ⟶^{w} dν_{∞} (1.21)

Remarks. 1. We will discuss POPUC and related objects on ∂𝔻 in Section 2.

2. As we will discuss, dν_{n(j)} and dν_{n(j)+1} have the same limits.

In one sense, this result is more than twenty-five years old! It is a restatement of the invariance of the density of states under change of boundary conditions proven in this context first by Avron–Simon [2]. But this invariance is certainly not usually stated in these terms. We also note that for OPUC, I noted this result in my book (see [22, Thm. 8.2]) but did not appreciate its importance.

We note that for OPUC, limits of \frac{1}{n+1}K_{n}(e^{iθ}, e^{iθ}) dμ(θ) have been studied by Golinskii–Khrushchev [12] without explicitly noting the connection to CD kernels. Their interesting results are limited to the case of OPUC and mainly to situations where the support is all of ∂𝔻.

By itself, Theorem 1.4 is interesting (e.g., it could be used to streamline the proof of a slightly weaker version of Corollary 2 of Totik [31]), but it is really powerful when used with the following collection of results:

Theorem 1.6 (Máté–Nevai [14]). For any measure on ∂𝔻,

lim sup nλ_{n−1}(e^{iθ}) ≤ w(θ)  (1.22)

Máté–Nevai–Totik [16] noted that this applies to OPRL on [−1, 1] using the Szegő mapping, and Totik, in a brilliant paper [31], shows how to extend it to any measure of bounded support, E, on ℝ so long as E contains an interval:

Theorem 1.7 (Totik [31]). Let I ⊂ E ⊂ ℝ where I = (a, b) is an interval and E is compact. Suppose dμ is a measure with support contained in E so that

dμ(x) = w(x) dx + dμ_{s}(x)  (1.23)

Let dρ_{E}(x) be the potential theoretic equilibrium measure for E (so it is known dρ_{E} ↾ I = ρ_{E}(x) dx for some strictly positive, real analytic weight ρ_{E}). Then for Lebesgue a.e. x ∈ I,

lim sup nλ_{n−1}(x) ≤ \frac{w(x)}{ρ_{E}(x)}  (1.24)

Remarks. 1. Totik concentrated on the deeper and more subtle fact that if there is a local Szegő condition on I and μ is regular for E, then the limit exists and equals w(x)/ρ_{E}(x) for a.e. x. But along the way he proved (1.24).

2. We will actually prove equality in (1.24) (for lim sup) when E = supp(dμ) and μ is regular.

3. Later (see Section 8), we will prove the analog of Theorem 1.7 for closed subsets of ∂𝔻.

Some of our results assume regularity of μ so we briefly summarize the main results from that theory due largely to Stahl–Totik [28] in their book; see my recent paper [24] for an overview.

A measure μ on ℝ is called regular if E = supp(dμ) is compact and

lim_{n→∞} (a_{1} … a_{n})^{1/n} = C(E)  (1.25)

where C(E) is the (logarithmic) capacity of E. For OPUC, (1.25) is replaced by

lim_{n→∞} (ρ_{1} … ρ_{n})^{1/n} = C(E)  (1.26)

Some insight is gained if one knows in both cases, for any μ (supported on E compact in ℝ or ∂𝔻), the lim sup is bounded above by C(E). For this paper, regularity is important because of (this is from [28, Sect. 2.2] or [24, Thm. 2.5]).

Theorem 1.8. If dμ is regular, then with dν_{n}, the density of zeros of the OPRL or of the POPUC, we have

dν_{n} ⟶^{w} dρ_{E}  (1.27)

the equilibrium measure for E. Conversely, if (1.27) holds, either μ is regular or μ is supported on a set of capacity zero.

We should mention one criterion for regularity that goes back to Erdös–Turán [8] for [−1, 1] and Widom [35] for general E:

Theorem 1.9. If E = supp(dμ) and

dμ(x) = f(x) dρ_{E}(x) + dμ_{s}(x)

where μ_{s} is ρ_{E}-singular and f(x) > 0 for ρ_{E}-a.e. x, then μ is regular.

There is a proof of Van Assche [34] of the general case presented in [24] that is not difficult. But in the case where E^{int} (interior in the sense of ℝ) differs from E by a set of capacity zero (e.g., E = [−1, 1]), we will find a proof that uses only our Theorem 1.5/1.7 strategy. In particular, we will have a proof of the Erdös–Turán result that uses neither potential theory nor polynomial inequalities.

We can now describe the content of this paper. In Section 2, we prove the main weak convergence result, Theorem 1.5. In Section 3, we prove the analog of the Erdös–Turán result for ∂𝔻 and illustrate how these ideas are connected to regularity criterion of Stahl–Totik [28]. Section 4 proves Theorem 1.4 for μ regular on ∂𝔻. Sections 5 and 6 then parallel Sections 3 and 4 but for general compact sets E ⊂ ℝ. Section 7, motivated by work of Totik–Uhlmann [33], provides a comparison result about densities of zeros. Section 9 does the analog of Sections 5 and 6 for general E ⊂ ∂𝔻. To do this, we need Totik’s result (1.24) in that situation. This does not seem to be in the literature, so Section 8 fills that need.

It is a pleasure to thank Jonathan Breuer, Yoram Last, and especially Vilmos Totik for useful conversations. I would also like to thank Ehud de Shalit and Yoram Last for the hospitality of the Einstein Institute of Mathematics of the Hebrew University during part of the preparation of this paper.

2. Weak Convergence

Our main goal here is to prove a generalization of Theorem 1.5 (and so also that theorem). We let μ be a measure of compact support in ℂ and

N(μ) = sup{|z| | z ∈ supp(dμ)}  (2.1)

We let M_{z} be multiplication by z on L^{2}(ℂ, dμ), so

∥M_{z}∥ = N(μ)  (2.2)

and let Q_{n} be the (n+1)-dimensional orthogonal projection onto polynomials of degree n or less. All estimates here depend on Proposition 2.1. Fix ℓ = 1, 2, . . .. Then

Q_{n}M_{z}^{ℓ}Q_{n} − (Q_{n}M_{z}Q_{n})^{ℓ}  (2.3)

is an operator of rank at most ℓ and norm at most 2N(μ)^{ℓ}.

Proof. Let X be the operator in (2.3). Clearly, X = 0 on ran(1−Q_{n}) = ker(Q_{n}). Since M_{z} maps ran(Q_{j}) to ran(Q_{j+1}), X = 0 on ran(Q_{n−ℓ}). This shows that X has rank at most ℓ. The norm estimate is immediate from (2.2). □

Here is the link to K_{n} and to dν_{n+1}. Let X_{j}(z, dμ) be the monic OPs for μ and let x_{n} = X_{n}/∥X_{n}∥.

Proposition 2.2. (i) We have for all w ∈ ℂ,

det_{Q_{n}}(w − Q_{n}M_{z}Q_{n}) = X_{n+1}(w, dμ)  (2.4)

In particular, if dν_{n+1} is the zero counting measure for X_{n+1}, then

\frac{1}{n + 1}Tr((Q_{n}M_{z}Q_{n})^{ℓ}) = ∫ z^{ℓ} dν_{n+1}(z)  (2.5)

(ii) Let

K_{n}(z, w) = ∑_{j=0}^{n} \overline{x_{j}(z)} x_{j}(w)  (2.6)

Then

Tr(Q_{n}M_{z}^{ℓ}Q_{n}) = ∫ z^{ℓ}K_{n}(z, z) dμ(z)  (2.7)

Remark. The proof of (i) is due to Davies and Simon [7].

Proof. (i) Let z_{0} be a zero of X_{n+1} of order ℓ. Let φ(z) = X_{n+1}(z)/(z − z_{0})^{ℓ}. Since Q_{n}[X_{n+1}] = 0, we see, with M_{z}^{(n)} = Q_{n}M_{z}Q_{n},

(M_{z}^{(n)} − z_{0})^{ℓ}φ = 0 (M_{z}^{(n)} − z_{0})^{ℓ−1}φ ≠ 0  (2.8)

showing that z_{0} is a zero of det_{Q_{n}}(w − M_{z}^{(n)}) of order at least ℓ. In this way, we see X_{n+1}(w) and the det have the same zeros. Since both are monic, we obtain (2.4).

In particular, this shows

Tr((M_{z}^{(n)})^{ℓ}) = ∑_{zeros z_{j} of multiplicity m_{j}} m_{j}z_{j}^{ℓ} = (n + 1) ∫ z^{ℓ} dν_{n+1}(z)

proving (2.5)

(ii) In L^{2}(ℂ, dμ), {x_{j}}_{j=0}^{n} span ran(Q_{n}) and are an orthonormal basis, so

Tr(Q_{n}M_{z}^{ℓ}Q_{n}) = ∑_{j=0}^{n}⟨x_{j}, z^{ℓ}x_{j}⟩

= ∫ ∑_{j=0}^{n} z^{ℓ}|x_{j}(z)|^{2} dμ(z)

proving (2.7). □

Proposition 2.3. Let dη_{n} be the probability measure

dη_{n}(z) = \frac{1}{n + 1} K_{n}(z, z) dμ(z)  (2.9)

Then for ℓ = 0, 1, 2, …,

|∫ z^{ℓ} dη_{n}(z) − ∫ z^{ℓ} dν_{n+1}(z)| ≤ \frac{2ℓN(μ)^{ℓ}}{n + 1}  (2.10)

Suppose there is a compact set, K ⊂ ℂ, containing the supports of all dν_{n} and the support of dμ so that {z^{ℓ}}_{ℓ=0}^{∞} ∪ {z̄^{ℓ}}_{ℓ=0}^{∞} are ∥ · ∥_{∞}- total in the continuous function on K. Then for any subsequence n(j), dη_{n(j)} → dν_{∞} if and only if dν_{n(j)+1} → dν_{∞}.

Proof. (2.10) is immediate from (2.5) and (2.7) if we note that, by Proposition 2.1,

|Tr(Q_{n}M_{z}^{ℓ}Q_{n}) − Tr((Q_{n}M_{z}Q_{n})^{ℓ})| ≤ 2N(μ)^{ℓ}ℓ  (2.11)

(2.10) in turn implies that we have the weak convergence result. □

While we were careful to use n(j) and n(j) + 1, we note that since |x_{j}(z)|^{2} dμ is a probability measure, we have

∥η_{n} − η_{n+1}∥ ≤ \frac{1}{n + 1} + n(\frac{1}{n} − \frac{1}{n + 1}) ≤ \frac{2}{n + 1}  (2.12)

so we could just as well have discussed weak limits of η_{n(j)+1} and ν_{n(j)+1}. Here is Theorem 1.5 for OPRL:

Theorem 2.4. For OPRL, dη_{n(j)} converges weakly to dν_{∞} if and only if dν_{n(j)+1} converges weakly to dν_{∞}.

Proof. {x_{j}}_{j=0}^{∞} are total in C([α, β]) for any real interval [α, β], so Proposition 2.3 is applicable. □

For OPUC, we need a few preliminaries: Define P: C(∂𝔻) → C(𝔻) (with 𝔻 = {z | |z| ≤ 1}) by

(Pf)(re^{iθ}) = ∫ \frac{1 − r^{2}}{1 + r^{2} − 2r cos(θ − φ)} f(e^{iθ}) \frac{dφ}{2π}  (2.13)

for r < 1 and (Pf)(e^{iθ}) = f(e^{iθ}) and P^{∗}: M_{+,1}(𝔻) → M_{+,1}(∂𝔻), the balayage, by duality. Then the following is well known and elementary:

Proposition 2.5. P^{∗}(dν) is the unique measure, η, on ∂𝔻 with

∫ e^{iℓθ} dη(θ) = ∫ z^{ℓ} dν(z)  (2.14)

for ℓ = 0, 1, 2, . . ..

Widom [35] proved that if μ is supported on a strict subset of ∂𝔻, then the zero counting measure, dν_{n}, has weak limits supported on ∂𝔻,

P^{∗}(dν_{n}) − dν_{n} ⟶^{w} 0  (2.15)

Finally, we note the following about paraorthogonal polynomials (POPUC) defined for β ∈ ∂𝔻 by

P_{n+1}(z, β) = zΦ_{n}(z) − β̄Φ_{n}^{∗}(z)  (2.16)

defined in [13] and studied further in [4, 5, 6, 11, 25, 36].

Proposition 2.6. Let β_{n} be an arbitrary sequence ∂𝔻 and let dν_{n+1}^{(β_{n})} be the zero counting measure for P_{n+1}(z, β_{n}) (known to live on ∂𝔻; see, e.g. [25]). Then for any ℓ ≥ 0,

|∫ z^{ℓ} dν_{n+1} − ∫ z^{ℓ} dν_{n+1}^{(β_{n})}| → 0  (2.17)

Proof. Let C_{n+1,F} be the truncated CMV matrix of size n + 1 whose eigenvalues are the zeros of Φ_{n+1}(z) (see [21, Ch. 4]) and let C_{n+1,F}^{(β_{n})} be the unitary dialation whose eigenvalues are the zeros of P_{n+1}(z, β_{n}) (see [6, 25, 23]). Then C_{n+1,F} − C_{n+1,F}^{(β_{n})} is rank one with norm bounded by 2, so C_{n+1,F}^{ℓ} − [C_{n+1,F}^{(β)} ]^{ℓ} is rank at most ℓ with norm 2. Thus

|Tr(C_{n+1,F}^{ℓ} − (C_{n+1,F}^{(β)} )^{ℓ})| ≤ 2ℓ  (2.18)

and

|LHS of (2.17)| ≤ \frac{2ℓ}{n + 1}

proving (2.17). □

Theorem 2.7. For OPUC, dη_{n(j)} converges weakly to dν_{∞} if and only if each of the following converges weakly to dν_{∞}:

(i) dν_{n+1}^{(β_{n})} for any β_{n}. (ii) P(dν_{n+1}), the balayage of dν_{n+1}. (iii) dν_{n+1} if supp(dμ) ≠ ∂𝔻.

Proof. Immediate from Proposition 2.3 and the above remark. □

When I mentioned Theorem 2.4 to V. Totik, he found an alternate proof, using tools more familiar to OP workers, that provides some insight. With his permission, I include this proof. Let me discuss the result for convergence of sequences rather than subsequences and then give some remarks to handle subsequences.

The key to Totik’s proof is Gaussian quadratures which says that if {x_{j}^{(n)}}_{j=1}^{n} are the zeros of p_{n}(x, dμ), and λ_{n−1}(x) = K_{n−1}(x, x)^{−1}, then for any polynomial R_{m} of degree m ≤ 2n − 1,

∫ R_{m}(x) dμ(x) = ∑_{j=1}^{n} λ_{n−1}(x_{j}^{(n)})R_{m}(x_{j}^{(n)})  (2.19)

We will also need the Christoffel variation principle

n ≤ q ⇒ λ_{q}(x) ≤ λ_{n}(x)  (2.20)

which is immediate from (1.12).

Suppose we know dν_{n} → dν_{∞}. Fix a polynomial Q_{m} of degree m with Q_{m} ≥ 0 on cvh(supp(dμ)). Let

R_{m+2n}(x) = \frac{1}{n + 1} Q_{m}(x)K_{n}(x, x)

which has degree m + 2n. Thus, if N = n + m, we have by (2.19) that

∫ Q_{m}(x)[\frac{1}{n + 1} K_{n}(x, x)] dμ

= ∑_{j=1}^{N} λ_{N−1}(x_{j}^{(N−1)})Q_{m}(x_{j}^{(N−1)})(\frac{1}{n + 1})λ_{n}(x_{j}^{(N−1)})^{−1} (2.21)

≤ (\frac{N + 1}{n + 1})\frac{1}{N + 1} ∑_{j=1}^{N} Q_{m}(x_{j}^{(N−1)})  (2.22)

by Q_{m} ≥ 0 and (2.21) (so λ_{N−1}/λ_{n} ≤ 1). As n → ∞, N/n → 1. So, by hypothesis,

RHS of (2.22) → ∫ Q_{m}(y) dν_{∞}(y)  (2.23)

We conclude by (2.21) that

lim sup ∫ Q_{m}(x) dη_{n}(x) ≤ ∫ Q_{m}(y) dν_{∞}(y)  (2.24)

If 0 ≤ Q_{m} ≤ 1 on cvh(supp(dμ)), we can apply this also to 1 − Q_{m} and so conclude for such Q_{m} that

lim ∫ Q_{m} dη_{n}(x) = ∫ Q_{m} dν_{∞}

which implies w-lim dη_{n} = dν_{∞}.

The same inequality (2.22) can be used to show that if dη_{n} ⟶^{w} dη_{∞}, then

lim inf ∫ Q_{m}(y) dν_{m}(y) ≥ ∫ Q_{m}(y) dη_{∞}(y)

and thus, by the same 1 − Q trick, we get dν_{n} → dη_{∞}.

To handle subsequences, we only need to note that, by (2.12), if dη_{n(j)} → dη_{∞}, then dη_{n(j)+ℓ} → dη_{∞} for ℓ = 0, ±1, ±2, . . .. Similarly, by zero interlacing, if dν_{n(j)} → dν_{∞}, then dν_{n(j)+ℓ} → dν_{∞} for fixed ℓ.

By using the operator theoretic proof of Gaussian quadrature (see, e.g., [21, Sect. 1.2]), one sees this proof is closely related to our proof above.

3. Regularity for ∂𝔻: The Erdös–Turán Theorem

Our goal in this section is to prove:

Theorem 3.1. Let dμ on ∂𝔻 have the form (1.7) with w(θ) > 0 for a.e. θ. Then μ is regular, that is, (1.26) holds with E = ∂𝔻 (so C(E) = 1).

Remarks. 1. This is an analog of a theorem for [−1, 1] proven by Erdös and Turän [8]. Our proof here seems to be new.

2. This is, of course, weaker than Rakhmanov’s theorem (see [22, Ch. 9] and references therein), but as we will see, this extends easily to some other situations.

The proof will combine the Máté–Nevai theorem (Theorem 1.6) and Proposition 2.5. It is worth noting where the Máté–Nevai theorem comes from. By (1.12), if

Q_{n}(e^{iθ}) = \frac{1}{n + 1} ∑_{j=0}^{n} e^{ij(θ−φ)}  (3.1)

then

λ_{n}(e^{iφ}) ≤ ∫ |Q_{n}(e^{iθ})|^{2} dμ(θ)  (3.2)

Recognizing (n + 1)|Q_{n}|^{2} as the Fejér kernel, (1.22) is a standard maximal function a.e. convergence result.

Proposition 3.2. For any measure on ∂𝔻 for a.e. θ,

lim inf_{n→∞} \frac{1}{n + 1} K_{n}(e^{iθ}, e^{iθ}) ≥ w(θ)^{−1}  (3.3)

On the set where w(θ) > 0,

lim inf_{n→∞} \frac{1}{n + 1} w(θ)K_{n}(e^{iθ}, e^{iθ}) ≥ 1  (3.4)

Remark. If w(θ) = 0, (3.3) is interpreted as saying that the limit is infinite. In that case, of course, (3.4) does not hold.

Proof. (3.3) is immediate from (1.14) and (1.22). If w ≠ 0, ww^{−1} = 1, so (3.3) implies (3.4). □

Proof of Theorem 3.1. The hypothesis w > 0 for a.e. θ implies dμ is not supported on a set of capacity zero. Thus, by Theorem 1.8, regularity holds if we prove that the density of zeros of POPUC converges to \frac{dθ}{2π}. By Proposition 2.5, this follows if we prove that \frac{1}{n+1}K_{n+1} dμ → \frac{dθ}{2π}.

Suppose n(j) → ∞ is a subsequence with \frac{1}{n+1}K_{n+1}(e^{iθ}, e^{iθ})w(θ)\frac{dθ}{2π} → dν_{1} and \frac{1}{n+1}K_{n+1}(e^{iθ}, e^{iθ})dμ_{s} → dν_{2}. Then since \frac{1}{n+1}K_{n} dμ is normalized,

∫ [dν_{1} + dν_{2}] = 1  (3.5)

On the other hand, by Fatou’s lemma and (3.4) for any continuous f ≥ 0 and the hypothesis w(θ) > 0 a.e. θ,

∫ f dν_{1} = lim ∫ f[\frac{1}{n + 1} w(θ)K_{n}(e^{iθ}, e^{iθ})]\frac{dθ}{2π}

≥ ∫ lim inf[f \frac{1}{n + 1} w(θ)K_{n}(e^{iθ}, e^{iθ})]\frac{dθ}{2π}

≥ ∫ f(θ) \frac{dθ}{2π}  (3.6)

Thus,

dν_{1} ≥ \frac{dθ}{2π}  (3.7)

By (3.5), this can only happen if

dν_{1} = \frac{dθ}{2π}  dν_{2} = 0  (3.8)

Compactness of the space of measures proves that \frac{1}{n+1}K_{n+1} dμ → dθ/2π. □

Along the way, we also proved dν_{2} = 0, that is,

Theorem 3.3. Under the hypotheses of Theorem 3.1,

\frac{1}{n + 1} ∫ ∑_{j=0}^{n}|φ_{j}(e^{iθ})|^{2} dμ_{s}(θ) → 0

It would be interesting to see if these methods provide an alternate proof of the theorem of Stahl–Totik [28] that, if for all η > 0,

lim_{n→∞} |\{θ | μ(\{ψ | |e^{iθ} − e^{iψ}| ≤ \frac{1}{n}\}) ≤ e^{−nη}\}| = 0  (3.9)

then μ is regular. The point is that using powers of the Fejér kernel, one can get trial functions localized in an interval of size O(\frac{1}{n}), and off a bigger interval of size O(\frac{1}{n}), it is exponentially small. (3.9) should say that the dominant contribution comes from an O(\frac{1}{n}) interval. On the other hand, the translates of these trial functions are spread over O(\frac{1}{n}) intervals, so one gets lower bounds on \frac{1}{n}K_{n}(e^{iθ}, e^{iθ}) of order μ(e^{iθ}− \frac{c}{n}, e^{iθ} + \frac{c}{n})^{−1} which are then integrated against dμ canceling this inverse and hopefully leading to (3.7), and so regularity.

4. Localization on ∂𝔻

In this section, we will prove Theorem 1.4. Instead of using the Máté–Nevai bound to prove regularity, we combine it with regularity to get information.

Proof of Theorem 1.4. As in the proof of Theorem 3.1, we let dν_{1}, dν_{2} be weak limits of \frac{1}{n}K(e^{iθ}, e^{iθ})w(θ)\frac{dθ}{2π} and \frac{1}{n}K_{n}(e^{iθ}, e^{iθ})dμ_{s}(θ). On I, the same arguments as above imply

dν_{1} ↾ I ≥ \frac{dθ}{2π} ↾ I  (4.1)

By regularity, globally

dν_{1} + dν_{2} = \frac{dθ}{2π}  (4.2)

Thus, on I,

ν_{1} ↾ I = \frac{dθ}{2π}  ν_{2} ↾ I = 0  (4.3)

The second implies (1.18). The first implies that

∫_{a}^{b} w(θ) \frac{1}{n + 1} K_{n}(e^{iθ}, e^{iθ}) \frac{dθ}{2π} → (b − a)  (4.4)

Since (3.4) and Fatou imply

∫_{a}^{b}[w(θ) \frac{1}{n + 1} K_{n}(e^{iθ}, e^{iθ}) − 1]_{−} \frac{dθ}{2π} → 0  (4.5)

we obtain (1.19). □

Notice that (4.4) and (3.4) imply a pointwise a.e. result (which we stated as (1.20))

lim inf_{n→∞} \frac{1}{n + 1} w(θ)K_{n}(e^{iθ}, e^{iθ}) = 1  (4.6)

We do not know how to get a pointwise result on lim sup under only the condition w(θ) > 0 (but without a local Szegő condition).

5. Regularity for E ⊂ ℝ: Widom’s Theorem

In this section and the next, our goal is to extend the results of the last two sections to situations where ∂𝔻 is replaced by fairly general closed sets in ℝ. The keys will be Theorem 2.4 and Totik’s Theorem 1.7. We begin with a few remarks on where Theorem 1.7 comes from (see also Section 8).

Since the hypothesis is that supp(dμ) ⊂ E, not = E, it suffices to find E_{n} ⊃ E so that ρ_{E_{n}}(x) → ρ_{E}(x) on I and for which (1.21) can be proven. By using Ẽ_{n} = \{x | dist(x, E) ≤ \frac{1}{n}\}, one first gets approximation by a finite union of intervals and then, by a theorem proven by Bogatyrëv [3], Peherstorfer [17], and Totik [32], one finds Ẽ_{n} ⊂ E_{n} where the E_{n}’s are finite unions of intervals with rational harmonic measure. For rational harmonic measures, one can use as trial polynomials K_{m}(x, x_{0})/K_{m}(x_{0}, x_{0}) where K_{m} is the CD kernel of a measure in the periodic isospectral torus and Floquet theory. (This is the method from Simon [26]; Totik [31] instead uses polynomial mapping.)

Theorem 5.1. Let E ⊂ ℝ be a compact set with ∂E ≡ E \ E^{int} (E^{int} means interior in ℝ) having capacity zero (e.g., a finite version of closed intervals). Let dμ be a measure with σ_{ess}(dμ) = E and

dμ = f(x) dρ_{E} + dμ_{s}  (5.1)

where dμ_{s} is dρ_{E}-singular. Suppose f(x) > 0 for dρ_{E}-a.e. x. Then dμ is regular.

Remarks. 1. In this case, dρ_{E} is equivalent to χ_{E} dx.

2. For any compact E, this is a result of Widom [35]; see also Van Assche [34], Stahl–Totik [28], and Simon [24].

Proof. Essentially identical to Theorem 3.1. By Theorem 1.8 and the fact that dμ is clearly not supported on sets of capacity zero, it suffices to prove that dν_{n} → dρ_{E}. Pick n(j) → ∞ so \frac{1}{n(j)+1}K_{n(j)}(x, x)f(x) dρ_{E} and \frac{1}{n(j)+1}K_{n(j)}(x, x) dμ_{s} separately have limits dν_{1} and dν_{2}. (1.21) says that (given that f(x) > 0 for a.e. x)

lim inf \frac{1}{n(j) + 1} K_{n(j)}(x, x)f(x) ≥ 1  (5.2)

By Fatou’s lemma on E^{int},

dν_{1} ≥ dρ_{E}  (5.3)

Since also ∫ (dν_{1} + dν_{2}) = 1 and ∫_{E^{int}} dρ_{E} = 1 (since C(E \ E^{int}) = 0), we conclude dν_{1} = dρ_{E}, dν_{2} = 0. By compactness of probability measures, \frac{1}{n(j)+1}K_{n(j)}(x, x) dμ ⟶^{w} dρ_{E}, implying regularity. □

6. Localization on ℝ

Here is an analog of Theorem 1.4 for any E ⊂ ℝ.

Theorem 6.1. Let I = [a, b] ⊂ E ⊂ ℝ with a < b and E compact. Let dμ be a measure on ℝ so that σ_{ess}(dμ) = E and μ is regular for E. Suppose

dμ = w(x) dx + dμ_{s}  (6.1)

with dμ_{s} Lebesgue singular. Suppose w(x) > 0 for a.e. x ∈ I. Then

(i) \frac{1}{n+1}K_{n}(x, x) dμ_{s} ⟶^{w} 0

(ii) ∫_{I}|ρ_{E}(x) − \frac{1}{n+1}w(x)K_{n}(x, x)| dx → 0

Proof. By w(x) > 0 a.e. on I and (1.17),

lim inf w(x) \frac{1}{n + 1} K_{n}(x, x) ≥ ρ_{E}(x)

for a.e. x ∈ I. From this, one can follow exactly the proofs in Section 4.

□

7. Comparisons of Density of Zeros

In [33], Totik–Ullman proved the following (we take [−a, a] rather than [a, b] only for notational simplicity):

Theorem 7.1 ([33]). Let dμ be a measure supported on a subset of [−1, 1] where

dμ = w(x) dx + dμ_{s}  (7.1)

with

w(x) > 0 for a.e. x ∈ [−a, a]

for some a ∈ (0, 1). Let dν_{∞} be any limit point of the zero counting measures for dμ. Then on (−a, a), we have that

(2π)^{−1}(1 − x^{2})^{−1/2} dx ≤ dν_{∞}(x) ≤ (2π)^{−1}(a^{2} − x^{2})^{−1/2} dx (7.2)

Our goal here is to prove the following, which we will show implies Theorem 7.1 as a corollary:

Theorem 7.2. Let dμ_{1}, dμ_{2} be two measures on ℝ of compact support. Suppose that for some interval I = (α, β),

(i)  dμ_{1} ≤ dμ_{2}  (7.3)
(ii)  dμ_{1} ↾ (α, β) = dμ_{2} ↾ (α, β)  (7.4)

Let n(j) → ∞ and suppose dν_{n(j)}^{(k)} → dν_{∞}^{(k)} for k = 1, 2, where dν_{n}^{(k)} is the zero counting measure for dμ_{k}. Then on (α, β),

dν_{∞}^{(2)} ↾ (α, β) ≤ dν_{∞}^{(1)} ↾ (α, β)  (7.5)

Proof. By (1.12),

λ_{n}(x, dμ_{1}) ≤ λ_{n}(x, dμ_{2})  (7.6)

so, by (1.14),

\frac{1}{n + 1} K_{n}^{(2)}(x, x) ≤ \frac{1}{n + 1} K_{n}^{(1)}(x, x)  (7.7)

for all x. By (7.4) on (α, β),

\frac{1}{n + 1} K_{n}^{(2)}(x, x) dμ_{2} ≤ \frac{1}{n + 1} K_{n}^{(1)}(x, x) dμ_{1}  (7.8)

By Theorem 2.4, this implies (7.5). □

Proof of Theorem 7.1. Let dμ_{2} = dμ and let dμ_{1} = χ_{(−a,a)}[dμ] so dμ_{1} ≤ dμ_{2} with regularity on [−a, a]. By Theorem 5.1, dμ_{1} is regular for [a, a], so dν_{n}^{(2)} → (2π)^{−1}(a^{2} − x^{2})^{−1/2} dx, the equilibrium measure for [−a, a]. Thus (7.5) implies the second inequality in (7.2).

On the other hand, let dμ_{1} = dμ and let dμ_{2} = [χ_{(−1,1)} −χ_{(−a,a)}] dx+ dμ. Then dμ_{1} ≤ dμ_{2} with equality on (−a, a). Moreover, dμ_{2} is regular for [−1, 1] by Theorem 5.1 and the hypothesis σ(dμ) ⊂ [−1, 1]. Thus, dν_{n}^{(1)} → (2π)^{−1}(1 − x^{2})^{−1/2} dx. Thus (7.5) implies the first inequality in (7.2). □

Remarks. 1. Theorem 7.1 only requires σ_{ess}(dμ) ⊂ [−1, 1].

2. The theorems in [33] are weaker than Theorem 7.1 in one respect and stronger in another. They are weaker in that, because of their dependence on potential theory, they require that one of the comparison measures be regular. On the other hand, they are stronger in that our reliance on weak convergence limits us to open sets like (α, β), while they can handle more general sets.

3. [24] has an example of a measure, dμ, on [−1, 1] where w(x) > 0 on [−1, 0] and the zero counting measures include among its limit points the equilibrium measures for [−1, 0] and for [−1, 1]. This shows in the [a, b]-form of Theorem 7.1, both inequalities in (7.2) can be saturated!

8. Totik’s Bound for OPUC

As preparation for applying our strategy to subsets of ∂𝔻, we need to prove an analog of Totik’s bound (1.23) for closed sets on ∂𝔻. Given a, b ∈ ∂𝔻, we let I = (a, b) be the “interval” of all points “between” a and b, that is, going counterclockwise from a to b, so −1 ∈ (e^{iθ}, e^{i(2π−θ)}) for 0 < θ < π but −1 ∉ (e^{i(2π−θ)}, e^{iθ}). Given E ⊂ ∂𝔻 closed, we let dρ_{E} be its equilibrium measure. If I ⊂ E ⊂ ∂𝔻 is a nonempty open interval, then

dρ_{E} ↾ I = ρ_{E}(θ) dm(θ)  (8.1)

where dm = dθ/2π. The main theorem of this section is

Theorem 8.1. Let I ⊂ E ⊂ ∂𝔻 where I = (a, b) is an interval and E is closed. Let dμ be a measure with support in E so that

dμ(θ) = w(θ) dm + dμ_{s}(θ)  (8.2)

Then for dm-a.e. θ ∈ I, we have

lim sup nλ_{n−1}(e^{iθ}) ≤ \frac{w(θ)}{ρ_{E}(θ)}  (8.3)

Following Totik’s strategy [31, 32] for OPRL, we do this in two steps:

Theorem 8.2. (8.3) holds if supp(dμ) ⊂ E^{int} and E is a finite union of intervals whose relative harmonic measures are rational.

Theorem 8.3. For any closed E in ∂𝔻, we can find E_{n} with

(i) Each E_{n} is a finite union of intervals whose relative harmonic measures are rational.

(ii)

E ⊂ E_{n}^{int}  (8.4)

(iii)

C(E_{n} \ E) → 0  (8.5)

Remark. If E = I_{1} ∪ ⋯ ∪ I_{ℓ}, the relative harmonic measures are ρ_{E}(I_{j}) which sum to 1.

Proof of Theorem 8.1 given Theorems 8.2 and 8.3. By general principles, since I is an interval, ρ_{E_{n}}(θ) and ρ_{E}(θ) are real analytic with bounded derivatives. (8.5) implies dρ_{E_{n}} ⟶^{w} dρ_{E} and then the bounded derivative implies ρ_{E_{n}}(θ) → ρ_{E}(θ) uniformly on compact subsets of I. By Theorem 8.2, LHS of (8.3) ≤ w(θ)/ρ_{E_{n}}(θ) for each n. Since ρ_{E_{n}}(θ) → ρ_{E}(θ), we obtain (8.3). □

Our proof of Theorems 8.2 and 8.3 diverges from the Totik strategy in two ways. He obtains Theorem 8.2 by using polynomial maps. Instead, following Simon [26], we use Floquet solutions.

Second, Totik shows if E has ℓ gaps, one can find E_{n} with rational relative harmonic measures also with ℓ gaps obeying (8.4) and (8.5). This is a result with rather different proofs by Bogatyrëv [3], Peherstorfer [17], and Totik [32]. I believe any of these proofs will extend to OPUC, but we will settle for a weaker result—our E_{n}’s will contain up to 2ℓ intervals, the first ℓ each containing one of the ℓ intervals of E and an additional ℓ or fewer exponentially small intervals. This will allow us to “get away” with following only the easier part of Peherstorfer’s strategy.

Proof of Theorem 8.2. By Theorem 11.4.5 of [22], E is the essential spectrum of an isospectral torus of Verblunsky coefficients periodic up to a phase, that is, for suitable p (chosen so that pρ_{E}(I_{j}) is an integer for each j),

α_{n+1} = λα_{n}

Let μ_{E} be the measure associated to a point on the isospectral torus.

By Floquet theory (see [22, Sect. 11.2]), for z ∈ E^{int}, φ_{n} is a sum of two functions each periodic up to a phase and, by [22, Sect. 11.12], on compact subsets, K, of E^{int},

sup_{z∈K,n} |φ_{n}(z)| < ∞  (8.6)

It follows from the Christoffel–Darboux formula (see [21, Sect. 2.2]) that

sup_{z,w∈K} |K_{n}(z, w)| ≤ C|z − w|^{−1}  (8.7)

The almost periodicity of φ implies uniformly on K, \frac{1}{n+1}K_{n}(z, z) has a finite nonzero limit, and then, by Theorem 2.7, the limit must be ρ_{E}(θ)/w_{E}(θ), that is, uniformly on K,

lim_{n→∞} \frac{1}{n} K_{n}(e^{iθ}, e^{iθ}) = \frac{ρ_{E}(θ)}{w_{E}(θ)}  (8.8)

where w_{E}(θ) is the weight for dρ_{E}(θ). Moreover, as proven in Simon [26], for any A > 0, uniformly on e^{iθ} ∈ K and n|φ − θ| < A,

Q_{n}(e^{iφ}) ≡ \frac{K_{n}(e^{iθ}, e^{iφ})}{K_{n}(e^{iθ}, e^{iθ})} = \frac{sin(nρ_{E}(θ)(θ − φ))}{n(θ − φ)ρ_{E}(θ)} (1 + O(1))  (8.9)

Now use Q_{n}(e^{iφ}) as a trial function in (1.12). By (8.7) and (8.8), by taking A large, the contribution of n|φ − θ| > A can be made arbitrarily small. Maximal function arguments and (8.9) show that the contribution of the region n|φ−θ| < A to nλ_{n−1} is close to w(θ)/ρ_{E}(θ). This proves (8.3) for dμ. □

Given E ⊂ ∂𝔻 compact, define

Ẽ_{n} = \{e^{iθ} ∈ 𝔻 | dist(e^{iθ}, E) ≤ \frac{1}{n}\}  (8.10)

It is easy to see that C(Ẽ_{n} \ E) → 0 and Ẽ_{n} is a union of ℓ(n) < ∞ closed intervals. It thus suffices to prove Theorem 8.3 when E is already a union of finitely many ℓ disjoint closed intervals, and it is that we are heading towards. (Parenthetically, we note that we could dispense with this and instead prove the analog of Theorem 8.2 for a finite union of intervals using Jost solutions for the isospectral torus associated to such finite gap sets, as in Simon [26].)

Define P_{n} to be the set of monic polynomials all of whose zeros lie in ∂𝔻. Since

e^{iθ/2} + e^{iφ}e^{−iθ/2} = e^{iφ/2}[e^{i(θ−φ)/2} + e^{−i(θ−φ)/2}]

if P ∈ P_{n}, there is a phase factor e^{iη} so

z^{−n/2}e^{iη}P (z) is real on ∂𝔻  (8.11)

Define the restricted Chebyshev polynomials, T̃_{n}, associated to E ⊂ ∂𝔻 by requiring that T̃_{n} miminize

∥P_{n}∥_{E} = sup_{z∈E} |P_{n}(z)|  (8.12)

over all P_{n} ∈ P_{n}. We will show that for n large, 2T̃_{2n}/∥T̃_{2n}∥_{E} are the rotated discriminants associated to sets E_{n} that approximate an E which is a finite union of intervals. An important input is

Lemma 8.4. Let I = (z_{0}, z_{1}) be an interval in ∂𝔻. For any small φ, let I_{φ} = (z_{0}e^{iφ}, z_{1}e^{−iφ}), so for φ > 0, I_{φ} is smaller than I. For φ < 0,

|(z − z_{0}e^{iφ})(z − z_{1}e^{−iφ})|

decreases on ∂𝔻 \ I and increases on I as φ decreases in (−ε, 0).

Proof. Take z_{1} = z̄_{0} and then use some elementary calculus. □

Theorem 8.5. Let E be a finite union of disjoint closed intervals on ∂𝔻, E = I_{1} ∪ ⋯ ∪ I_{ℓ} and let ∂𝔻 \ E = G_{1} ∪ ⋯ ∪ G_{ℓ} has ℓ gaps. Then (i) Each T̃_{n} has at most one zero in each G_{ℓ}. (ii) If z_{j}^{(n)} is the zero of T̃_{n} in G_{j}, then on any compact K ⊂ G_{j}, we

have

lim_{n→∞} inf_{z∈K}(|\frac{T̃_{n}(z)}{(z − z_{j}^{(n)})∥F_{n}∥_{E}} | )^{1/n} > 1  (8.13)

(iii) At any local maximum, z̃, of |T̃_{n}(z)| in some I_{j}, we have

|T̃_{n}(z̃)| = ∥T̃_{n}∥_{E}  (8.14)

Proof. (i) If there are two zeros in some G_{j}, we can symmetrically move the zeros apart. Doing that increases T̃_{n} on G_{j} which is disjoint from E, but it decreases ∥T̃_{n}∥_{E}, contradicting the minimizing definition.

(ii) The zero counting measure for T̃_{n} converges to the equilibrium measure on E. For standard T_{n}’s and E ⊂ ℝ, this result is proven in [1, 20, 24]. A small change implies this result for T̃_{n}. This, in turn, says that uniformly on G_{j},

|\frac{T̃_{n}(z)}{(z − z_{j}^{(n)})∥T̃_{n}∥_{E}} |^{1/n} → exp(−Φ_{ρ_{E}}(z))  (8.15)

which implies (8.13).

(iii) Since z^{−n/2}T̃_{n}(z) is real up to a phase, the local maxima of |T̃_{n}(z)| on ∂𝔻 alternate with the zeros of T̃_{n}(z). If a local maximum is smaller than ∥T̃_{n}∥_{E}, we move the nearest zeros, say z_{0}, z_{1}, apart. That decreases ∥T̃_{n}∥_{E\(z_{0},z_{1})} and increases ∥T̃_{n}∥_{(z_{0},z_{1})}. Since the latter is assumed smaller than ∥T̃_{n}∥_{E}, it decreases ∥T̃_{n}∥_{E} overall, violating the minimizing definition. Thus, |T̃_{n}(z)| ≥ ∥T̃_{n}∥_{E}. But since z̃ ∈ E, |T̃_{n}(z)| ≤ ∥T̃_{n}∥_{E}. □

Now define

Δ_{n}(z) = \frac{2e^{iφ_{n}} T̃_{2n}(z)}{∥T̃_{2n}∥_{E}}  (8.16)

where φ_{n} is chosen to make Δ_{n} real on ∂𝔻. By (8.14), maxima in E occur with Δ_{n}(z) = ±2, and by (8.13), maxima in ∂𝔻 \ E occur at points where |Δ_{n}(z)| > 2. Thus, up to a phase, Δ_{n}(z) looks like a discriminant. So, by Theorem 11.4.5 of [22], E_{n} ≡ Δ_{n}^{−1}([−2, 2]) is the essential spectrum of a CMV matrix whose Verblunsky coefficients obey α_{m+p} = λα_{m} for |λ| = 1.

Proof of Theorem 8.3. E_{n} has at most 2ℓ components, ℓ containing I_{1}, …, I_{ℓ} (call them I_{1}^{(n)}, …, I_{ℓ}^{(n)}), and ℓ possible components, J_{1}^{(n)}, …, J_{ℓ}^{(n)}, one in each gap. Since capacities are bounded by \frac{1}{4} times Lebesgue measure, it suffices to show that

∑_{j=1}^{ℓ} |I_{j}^{(n)} \ I_{j}| + |J_{j}^{(n)}| → 0

to prove (8.5) and complete the proof. Since, on ∂𝔻,

−Φ_{ρ_{E}}(x) ≥ c dist(x, E)^{1/2}  (8.17)

by (8.15), we have |I_{j}^{(n)} \ I_{j}| → 0 and |J_{j}^{(n)}| → 0. □

9. Theorems for Subsets of ∂𝔻

Given Theorem 8.1 and our strategies in Sections 3–6, we immediately have

Theorem 9.1. Let E ⊂ ∂𝔻 with ∂E = E \E^{int} (E^{int} means interior in ∂𝔻) having capacity zero. Let dμ be a measure on ∂𝔻 with σ_{ess}(dμ) = E and

dμ = f(x) dρ_{E} + dμ_{s}  (9.1)

where dμ_{s} is dρ_{E}-singular. Suppose f(x) > 0 for dρ_{E}-a.e. x. Then dμ is regular.

Theorem 9.2. Let I ⊂ E ⊂ ∂𝔻 with I a nonempty closed interval and E closed. Let dμ be a measure on ∂𝔻 so σ_{ess}(dμ) = E and μ is regular for E. Suppose

dμ = w(θ) \frac{dθ}{2π} + dμ_{s}  (9.2)

and w(θ) > 0 for a.e. e^{iθ} ∈ I. Then

(i)  \frac{1}{n + 1} K_{n}(e^{iθ}, e^{iθ}) dμ_{s}(θ) ⟶^{w} 0
(ii)  ∫_{I} |ρ_{E}(θ) − \frac{1}{n + 1} w(θ)K_{n}(e^{iθ}, e^{iθ})| \frac{dθ}{2π} → 0

where dρ(θ) = ρ_{E}(θ)\frac{dθ}{2π} on I.

References

[1] V. V. Andrievskii and H.-P. Blatt, Discrepancy of Signed Measures and Polynomial Approximation, Springer Monographs in Mathematics, Springer-Verlag, New York, 2002.

[2] J. Avron and B. Simon, Almost periodic Schrödinger operators. II. The integrated density of states, Duke Math. J. 50 (1983), 369–391.

[3] A. B. Bogatyrëv, On the efficient computation of Chebyshev polynomials for several intervals, Sb. Math. 190 (1999), 1571–1605; Russian original in Mat. Sb. 190 (1999), no. 11, 15–50.

[4] M. J. Cantero, L. Moral, and L. Velázquez, Measures and para-orthogonal polynomials on the unit circle, East J. Approx. 8 (2002), 447–464.

[5] M. J. Cantero, L. Moral, and L. Velázquez, Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl. 362 (2003), 29–56.

[6] M. J. Cantero, L. Moral, and L. Velázquez, Measures on the unit circle and unitary truncations of unitary operators, J. Approx. Theory 139 (2006), 430– 468.

[7] E. B. Davies and B. Simon, unpublished.

[8] P. Erdös and P. Turán, On interpolation. III. Interpolatory theory of polynomials, Ann. of Math. (2) 41 (1940), 510–553.

[9] G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford-New York, 1971.

[10] Ya. L. Geronimus, Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval, Consultants Bureau, New York, 1961.

[11] L. Golinskii, Quadrature formula and zeros of para-orthogonal polynomials on the unit circle, Acta Math. Hungar. 96 (2002), 169–186.

[12] L. Golinskii and S. Khrushchev, Cesàro asymptotics for orthogonal polynomials on the unit circle and classes of measures, J. Approx. Theory 115 (2002), 187–237.

[13] W. B. Jones, O. Njåstad, and W. J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc. 21 (1989), 113–152.

[14] A. Máté and P. Nevai, Bernstein’s inequality in L^{p} for 0 < p < 1 and (C, 1) bounds for orthogonal polynomials, Ann. of Math. (2) 111 (1980), 145–154.

[15] A. Máté, P. Nevai, and V. Totik, Strong and weak convergence of orthogonal polynomials, Am. J. Math. 109 (1987), 239–281.

[16] A. Máté, P. Nevai, and V. Totik, Szegő’s extremum problem on the unit circle, Ann. of Math. 134 (1991), 433–453.

[17] F. Peherstorfer, Deformation of minimal polynomials and approximation of several intervals by an inverse polynomial mapping, J. Approx. Theory 111 (2001), 180–195.

[18] E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, Math. USSR Sb. 32 (1977), 199–213.

[19] E.A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, II, Math. USSR Sb. 46 (1983), 105–117.

[20] E. B. Saff and V. Totik, Logarithmic Potentials With External Eields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 316, Springer-Verlag, Berlin, 1997.

[21] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series, 54.1, American Mathematical Society, Providence, RI, 2005.

[22] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, AMS Colloquium Series, 54.2, American Mathematical Society, Providence, RI, 2005.

[23] B. Simon, CMV matrices: Five years after, to appear in the Proceedings of the W. D. Evans 65th Birthday Conference.

[24] B. Simon, Equilibrium measures and capacities in spectral theory, to appear in Inverse Problems and Imaging.

[25] B. Simon, Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle, to appear in J. Math. Anal. Appl.

[26] B. Simon, Two extensions of Lubinsky’s universality theorem, preprint

[27] B. Simon, Szegő’s Theorem and Its Descendants: Spectral Theory for L^{2} Perturbations of Orthogonal Polynomials, in preparation; to be published by Princeton University Press.

[28] H. Stahl and V. Totik, General Orthogonal Polynomials, in “Encyclopedia of Mathematics and its Applications,” 43, Cambridge University Press, Cambridge, 1992.

[29] G. Szegő, Beiträge zur Theorie der Toeplitzschen Formen, I, II, Math. Z. 6 (1920), 167–202; 9 (1921), 167–190.

[30] G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., 23, American Mathematical Society, Providence, RI, 1939; 3rd edition, 1967.

[31] V. Totik, Asymptotics for Christoffel functions for general measures on the real line, J. Anal. Math. 81 (2000), 283–303.

[32] V. Totik, Polynomial inverse images and polynomial inequalities, Acta Math. 187 (2001), 139–160.

[33] V. Totik and J. L. Ullman, Local asymptotic distribution of zeros of orthogonal polynomials, Trans. Amer. Math. Soc. 341 (1994), 881–894.

[34] W. Van Assche, Invariant zero behaviour for orthogonal polynomials on compact sets of the real line, Bull. Soc. Math. Belg. Ser. B 38 (1986), 1–13.

[35] H. Widom, Polynomials associated with measures in the complex plane, J. Math. Mech. 16 (1967), 997–1013.

[36] M.-W. L. Wong, First and second kind paraorthogonal polynomials and their zeros, to appear in J. Approx. Theory.