TWO EXTENSIONS OF LUBINSKY’S UNIVERSALITY THEOREM

BARRY SIMON^{∗}

Abstract. We extend some remarkable recent results of Lubinsky and Levin–Lubinsky from [−1, 1] to allow discrete eigenvalues outside σ_{ess} and to allow σ_{ess} to first be a finite union of closed intervals and then a fairly general compact set in ℝ (one which is regular for the Dirichlet problem).

1. Introduction

This paper primarily discusses orthogonal polynomials on the real line (OPRL) [36, 9, 25]. To set notation, μ will be a measure of compact support, σ(dμ), on ℝ, positive but not necessarily normalized. Its Lebesgue decomposition is

dμ(x) = w(x) dx + dμ_{s}(x)  (1.1)

where w ∈ L^{1}(ℝ, dx) and μ_{s} is Lebesgue singular. σ_{ess}(dμ) will denote σ(dμ) with isolated points removed and σ_{s}(dμ) = σ(dμ_{s}).

P_{n}(x, dμ) will denote the monic orthogonal polynomials and p_{n}(x, dμ) the orthonormal polynomials. The Jacobi parameters {a_{n}, b_{n}}_{n=1}^{∞} are defined by the recursion relation

xp_{n}(x) = a_{n+1}p_{n+1}(x) + b_{n+1}p_{n}(x) + a_{n}p_{n−1}(x)  (1.2)

We note for later use that

p_{0}(x) = (\frac{1}{μ(ℝ)})^{1/2}  (1.3)

and that (∥ · ∥ means L^{2}(ℝ, dμ) norm)

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

Date: June 11, 2007.

2000 Mathematics Subject Classification. 05E35, 26C10, 26C05.

Key words and phrases. Lubinsky universality, CD kernel.

∗ 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.

The main focus of this paper is the CD (for Christoffel–Darboux) kernel (for x, y ∈ ℝ)

K_{n}(x, y; dμ) = ∑_{j=0}^{n} p_{n}(x, dμ)p_{n}(y, dμ)  (1.5)

We will often drop dμ or consider several measures, say μ, μ^{♯}, and use K_{n}(x, y), K_{n}^{♯}(x, y). K_{n} is the integral kernel of the orthogonal projection in L^{2}(ℝ, dμ) onto the polynomials of degree at most n. So if Q_{n}(x) is such a polynomial, then (the reproducing property)

Q_{n}(x) = ∫ K_{n}(x, y)Q_{n}(y) dμ(y)  (1.6)

and, in particular (an expression that K_{n} is the kernel of a projection),

∫ K_{n}(x, y)K_{n}(y, z) dμ(y) = K_{n}(x, z)  (1.7)

Going back to Faber [6], Fekete [7], and Szegő [35], it has been known that there are deep connections between potential theory and asymptotics of polynomials; see Stahl–Totik [34] and Simon [30]. We will be interested especially in the potential theory associated to E = σ_{ess}(dμ). We will call E ⊂ ℝ a regular set if it is compact, regular for the Dirichlet problem on ℂ and with an equilibrium measure, dρ, of the form ρ(x) dx. Thus, with C(E), the capacity of E,

∫_{E} ρ(y) dy = 1  (1.8)

and

G_{E}(x) = ∫_{E} log|x − y|ρ(y) dy − log C(E)  (1.9)

is continuous on ℂ with

G_{E}(y) = 0 if y ∈ E  G_{E}(z) > 0 if z ∉ E  (1.10)

Stahl–Totik introduce the important notion of regular measure: If μ is a measure of compact support, we call it regular if and only if

lim_{n→∞} (a_{1} … a_{n})^{1/n} = C(E)  (1.11)

where E = σ_{ess}(dμ). (They use E = σ(dμ), but since σ(dμ) \ σ_{ess}(dμ) is countable, it has zero capacity, and so there is no difference.) One reason this is natural is that it is always true that

lim sup_{n→∞} (a_{1} … a_{n})^{1/n} ≤ C(E)  (1.12)

(An elegant way to see this (see Widom [43] or [30]) is to note that when μ(ℝ) = 1, a_{1} … a_{n} = ∥P_{n}∥_{2} ≤ ∥T_{n}∥_{∞} with T_{n} the Chebyshev polynomial for E and use Szegő’s theorem [35] that lim ∥T_{n}∥^{1/n} = C(E).)

More generally than (1.12), one has the results of Stahl–Totik [34] (see also [30]) that

Theorem 1.1 ([34]). If E = σ_{ess}(dμ) and μ is regular, then

lim sup_{n→∞} |p_{n}(z, dμ)|^{1/n} ≤ e^{G_{E}(z)}

uniformly on compact subsets of ℂ. In particular, if E is regular, for any ε, there is a δ and C so

sup_{dist(y,E)≤δ} |p_{n}(y, dμ)| ≤ Ce^{εn}  (1.13)

One connection between K and ρ is (an analog of Theorem 8.2.6 of [25]; see Simon [32]):

Theorem 1.2 ([32]). For any regular measure μ,

\frac{1}{n} K_{n}(x, x) dμ → dρ_{E}  (1.14)

the equilibrium measure for E = σ_{ess}(dμ). In (1.14), the convergence is weakly as probability measures on supp(dμ).

If E is regular, if dμ given by (1.1) is regular, and if \frac{1}{n}K_{n}(x, x) has a uniform limit as n → ∞ for x ∈ I some open interval, then by (1.14), that limit must be ρ(x)/w(x) (and w(x) must be continuous and nonvanishing on I). This motivates

Definition. We say that μ has normal limits on a closed interval I = [a, b] if and only if for any x_{n} → x ∈ I,

\frac{1}{n} K_{n}(x_{n}, x_{n}) → \frac{ρ_{E}(x)}{w(x)}  (1.15)

with convergence which is uniform in the sense that for any ε, there is N and δ so that n ≥ N and |x_{n} −x| < δ implies the difference in (1.15) is less than ε.

Normal limits for x_{n} ≡ x has an ancient history for orthogonal polynomials on the unit circle (OPUC) and for E = [−1, 1], going back to Szegő with important contributions by Erdös, Turán, and Freud. This history is discussed in the fundamental paper by Máté–Nevai–Totik [20] who obtained very strong results on pointwise convergence for μ’s supported on [−1, 1] or OPUC supported on ∂𝔻. The refinement of allowing x_{n} → x is one critical idea in a recent wonderful paper of Lubinsky [17] who provides a result on off-diagonal behavior of K_{n} also:

Theorem 1.3 ([17]). Suppose I = [a, b] is a closed interval in (−1, 1) and dμ is a regular measure with support [−1, 1] and that supp(dμ_{s}) ∩ I = ∅. Suppose on I, dμ(x) = w(x) dx with w continuous and nonvanishing. Then μ has normal limits on I, and for x_{0} ∈ I and α, β ∈ ℝ,

lim_{n→∞} \frac{K_{n}(x_{0} + \frac{α}{n}, x_{0} + \frac{β}{n})}{K_{n}(x_{0}, x_{0})} = \frac{sin(πρ(x_{0})(β − α))}{πρ(x_{0})(β − α)}  (1.16)

uniformly if |α|, |β| ≤ A, x_{0} ∈ I for any A > 0.

Remarks. 1. Continuity “on I” here means continuous at each point in I as a function on [−1, 1], that is, continuity at a, b involves values of w outside but near [a, b]. Thus, the continuity hypothesis is nonvacuous if a = b, and the theorem is interesting in that case.

2. The earliest results of the form (1.16) come from the random matrix and Riemann–Hilbert literatures; see [13].

3. Lubinsky [17] does not use ρ(x_{0}) = (π\sqrt{1 − x_{0}^{2}})^{−1} for (1.16), but scales using w(x)K_{n}(x, x) ∼ nρ(x). This gives a form that makes contact with the Riemann–Hilbert literature and is also suitable for end points and Freud weights.

4. As we will explain in Section 4, Levin–Lubinsky [16] use (1.16) to control asymptotics of zeros of p_{n}.

Our goal in this paper is to extend Theorem 1.3 in two ways:

(a) Instead of requiring σ(dμ) = [−1, 1], we want to allow σ_{ess}(dμ) = [−1, 1], as is natural if one makes assumptions on {a_{n}, b_{n}}_{n=1}^{∞} rather than directly on dμ.

(b) We want to replace [−1, 1] by a general finite gap set.

A third important extension involves (1.16) pointwise for a.e. x_{0} ∈ I for situations where dμ obeys a local Szegő condition on I. I had intended to combine Lubinsky’s strategy with ideas of Máté–Nevai– Totik [20] and especially Totik [37], but I was informed by Totik that Findley [8] and he [39] have results along this line. So I decided to focus here only on (a) and (b).

Both of these extensions, while important, are not especially difficult. Because of Lubinsky’s clever inequality (see (4.1)), it is only necessary to find a suitable universal model for E and to control the diagonal kernel.

A key will be to relate K_{n}(x, x) to the Christoffel function,

λ_{n}(x_{0}) = min{∥Q∥_{L^{2}(ℝ,dμ)}^{2} | Q(x_{0}) = 1, deg Q ≤ n}  (1.17)

The minimizer is

Q_{n}(x, x_{0}) = K(x_{0}, x_{0})^{−1} ∑_{j=0}^{n} p_{j}(x)p_{j}(x_{0})  (1.18)

for which

λ_{n}(x_{0}) = K_{(}x_{0}, x_{0})^{−1}  (1.19)

To handle extension (a) will be easy: One can eliminate the point masses distant from E by adding explicit zeros to a trial polynomial and control the point masses near E with some exponential decay.

The key to (b) will be to construct a suitable model that is wellbehaved, and following Lubinsky’s strategy (he uses Legendre polynomials as his model), it will be easy to extend Theorem 1.3. Our model will be the measure associated to a point in the isospectral torus associated to E where the analysis will depend on results of Sodin–Yuditskii [33], Peherstorfer–Yuditskii [23], and Christiansen–Simon–Zinchenko [5].

The most subtle part of the model will be establishing (1.16) which will follow from Jost asymptotics. Jost asymptotics are the key to proving clock behavior for zeros in [27, 28, 15]. In a sense, using the Levin–Lubinsky strategy, we can regard (1.16) as a kind of infinitesimal Jost asymptotics.

To obtain control of the diagonal CD kernel, all we will need is a single model, μ^{♯}, obeying

(i) σ_{ess}(μ^{♯}) = E (ii) w^{♯} is continuous and nonvanishing in E. (iii) For any closed interval I ⊂ E^{int}, and ε > 0,

sup_{x∈I} e^{−εn}K_{n}^{♯}(x, x) → 0  (1.20)

(iv) For any closed interval I ⊂ E^{int},

lim sup_{ε↓0}[ lim sup_{n→∞} \frac{K_{(1+ε)n}^{♯}(x, x)}{K_{n}^{♯}(x, x)}] = 1  (1.21)

It is known [43, 41, 34, 30] that (ii) implies that μ^{♯} is regular. Of course, we will have

\frac{1}{n} K_{n}^{♯}(x, x) → \frac{ρ(x)}{w^{♯}(x)}  (1.22)

from which (1.20) and (1.21) follow. We state the result in this form to allow for future work where the model focuses on a single point in E^{int} where w vanishes or blows up at some rate.

In Section 3, we will prove:

Theorem 1.4. Suppose E is a regular set and there exists a model, μ^{♯}, obeying (i)–(iv). Let μ be a measure with σ_{ess}(μ) = E, μ regular, and w continuous and nonvanishing on I = [a, b] ⊂ E^{int}. Suppose that σ_{s}(μ) ∩ I = ∅. Then for any x_{n} → x ∈ I,

\frac{K_{n}(x_{n}, x_{n})}{K_{n}^{♯}(x_{n}, x_{n})} → \frac{w^{♯}(x)}{w(x)}  (1.23)

uniformly in the sense discussed after (1.15).

Remark. By (1.22), we have normal behavior on I.

In Section 4, we will prove:

Theorem 1.5. Suppose E is a regular set and there exists a model, μ^{♯}, obeying (i)–(iv) so that K^{♯} obeys (1.16) uniformly for x in compacts of E^{int}, and |α|, |β| < A. Let μ be a measure with σ_{ess}(μ) = E, μ regular, and w continuous and nonvanishing on I = [a, b] ⊂ E^{int}. Suppose that σ_{s}(μ) ∩ I = ∅. Then K obeys (1.16) uniformly on I and |α| < A, |β| < A.

Given Theorem 1.4, we will obtain Theorem 1.5 by following Lubinsky’s argument virtually word for word. It will also, following Levin– Lubinsky, imply uniform clock behavior of the zeros in I in the sense of Last–Simon [15] (if a < b).

In Section 2, we will obtain μ^{♯} obeying (i)–(iv) when E is a finite union of intervals and (1.16), therefore accomplishing extensions (a) and (b).

All our arguments extend with little change to finite gap OPUC and to zeros of paraorthogonal polynomials [3, 4, 10, 12, 29, 44].

During the preparation of this manuscript, I learned that Totik [39] was also working on extending Lubinsky universality to general sets. After I finished the above and Sections 2–4 below, Totik and I exchanged manuscripts. His technical methods are different from what I do in Section 2. After I got his manuscript, I realized that Lubinsky’s inequality ((4.1) below) is so strong that it is easy to go from finite gap to general compact sets and prove

Theorem 1.6. Let E ⊂ ℝ be an arbitrary regular compact set so that I = [a, b] ⊂ E^{int}. Let μ be a measure with σ_{ess}(μ) = E, μ regular in the sense of Stahl–Totik [34], and μ ↾ [a − ε, b + ε] is purely absolutely continuous with w = \frac{dμ}{dx} continuous and nonvanishing on [a, b]. Let ρ_{E}(x) dx be the density for the equilibrium measure for E restricted to I (it is not hard to see ρ_{E} is purely a.c. on I; see [30]). Then uniformly for x ∈ [a, b],

K_{n}(x, x) → \frac{ρ_{E}(x)}{w(x)}  (1.24)

and uniformly for x_{0} ∈ [a, b], |α|, |β| ≤ A, one has (1.16), and so (as in Section 4 following [16]), clock behavior for the zeros.

Remark. (1.24) is not new. It is essentially in Totik [37].

The proof of this theorem is sketched in Section 5. We will also note there that it suffices to prove the results in Section 2 when each interval has rational harmonic measure so that one can use Floquet theory in place of the more subtle analysis of [33, 23, 5].

It is a pleasure to thank D. Lubinsky, P. Nevai, and V. Totik for useful correspondence.

2. Models

Let E be a finite gap set, that is,

E = ⋃_{j=1}^{k+1}[α_{j}, β_{j}]  (2.1)

where α_{1} < β_{1} < α_{2} < ⋯ < β_{k+1} are reals. Associated with any such E is an isospectral torus of Jacobi matrices defined by the fact that their m-functions are Herglotz functions extendable to minimal degree meromorphic functions on the two-sheeted Riemann surface associated to [∏_{j=1}^{k+1}(z − α_{j})(z − β_{j})]^{1/2}. For OPUC, this is discussed, for example, in [26], and for OPRL in [31].

The spectral measure μ^{♯} for any such Jacobi matrix has the form (see [33, 23, 5])

dμ^{♯} = w^{♯}(x) dx + dμ_{s}  (2.2)

where dμ_{s} is a pure point measure with at most one pure point in each gap of E and none in E and

w^{♯} > 0  (2.3)

and real analytic on E^{int}.

Our goal in this section is the prove the following

Theorem 2.1. Let μ^{♯} be the probability measure associated to a Jacobi matrix in the isospectral torus for E. Then, uniformly for x in any interval [a, b] in E^{int}, we have (i)

K_{n}^{♯}(x, x) = \frac{ρ_{E}(x)}{w(x)} n + O(1)  (2.4)

(ii) Uniformly in |α|, |β| < L,

\frac{w(x)}{n} K_{n}^{♯}(x + \frac{α}{n} , x + \frac{β}{n}) = \frac{sin(πρ_{E}(x)(β − α))}{π(β − α)} + O(\frac{1}{n}) (2.5)

We obtain this from results [33, 23, 5] on Jost solutions, that is, solutions of

a_{n+1}u_{n+2} + (b_{n+1} − x)u_{n+1} + a_{n}u_{n} = 0  (2.6)

an equation solved by

u_{n} = p_{n−1}(x)

Jost solutions, u_{n}(x), solve (2.6) for x ∈ E and obey (i)

u_{n}(x) = e^{inθ(x)}f_{n}(x)  (2.7)

(ii) On any I ⊂ E^{int}, f_{n} is analytic in x and its derivatives are uni-

formly bounded in n and x ∈ I. (iii)

θ′(x) = πρ_{E}(x)  (2.8)

(iv)

u_{0}(x) = 1  (2.9)

(v)

\overline{u_{1}(x)} ≠ u_{1}(x)  (2.10)

(vi) The Wronskian

a_{n}(u_{n+1}ū_{n} − ū_{n+1}u_{n})  (2.11)

is n-independent (but x-dependent). (vii) While we do not need it, we note that f_{n}(x) is almost periodic in

n. Because of (2.9)/(2.10), u_{n} and ū_{n} are linearly independent, so p_{n} is

a linear combination of u_{n+1} and ū_{n+1}, and thus,

p_{n}(x) = \frac{u_{n+1} − \overline{u_{n+1}(x)}}{u_{1}(x) − ū_{1}(x)}  (2.12)

Define

g_{n}(x) = \frac{e^{iθ(x)}f_{n+1}(x)}{[u_{1}(x) − ū_{1}(x)]}  (2.13)

so (2.12) becomes

p_{n}(x) = g_{n}(x)e^{inθ(x)} + \overline{g_{n}(x)} e^{−inθ(x)}  (2.14)

and constancy of the Wronskian becomes

a_{n+1}[g_{n+1}(x)ḡ_{n}e^{iθ(x)} − g_{n}(x) \overline{g_{n+1}(x)} e^{−iθ(x)}] = c(x)  (2.15)

x-dependent but not n-dependent.

We will also need the CD formula

K_{n}^{♯}(x, y) = a_{n+1}[\frac{p_{n+1}(x)p_{n}(y) − p_{n}(x)p_{n+1}(y)}{x − y}]  (2.16)

for x ≠ y and its limit at x = y,

K_{n}^{♯}(x, x) = a_{n+1}[p′_{n+1}(x)p_{n}(x) − p_{n+1}(x)p′_{n}(x)]  (2.17)

Proof of Theorem 2.1. By (ii), (iii), and (v), and

p′_{n}(x) = inθ′(x)[g_{n}(x)e^{inθ(x)} − \overline{g_{n}(x)} e^{−inθ(x)}] + O(1)  (2.18)

where O(1) is bounded uniformly in x ∈ I and in n. Thus, by (2.14), (2.15), (2.17), and (2.18),

K_{n}^{♯}(x, x) = 2inθ′(x)c(x) + O(1)  (2.19)

Therefore, \frac{1}{n}K_{n}(x, x) converges uniformly on I to 2iθ′(x)c(x), so by (1.14),

2iθ′(x)c(x) = \frac{ρ_{E}(x)}{w(x)}  (2.20)

and (2.19) is (2.4).

Similarly, by (2.14), (2.8), and (ii),

p_{n}(x + \frac{α}{n}) = g_{n}(x)e^{inθ(x)}e^{iαπρ(x)} + \overline{g_{n}(x)} e^{−inθ(x)}e^{−iαπρ(x)} + O(\frac{1}{n})

(2.21)

uniformly in |α| < A.

Plugging this into (2.16) shows

K_{n}(x + \frac{α}{n} , x + \frac{β}{n}) = 2in \frac{sin((β − α)πρ(x))}{β − α} c(x) + O(1) (2.22)

By (2.20) and (2.8), c(x) = 1/2πiw(x), so (2.22) is (2.5). □

3. Asymptotics of the Diagonal CD Kernel

Our goal here is to prove Theorem 1.4. The key is an idea of Nevai [21] that lets one exponentially localize CD minimizers, augmented by the regularity ideas of Máté–Nevai–Totik [20], and a simple extension that accommodates discrete spectrum.

Lemma 3.1. Let E be a regular subset of ℝ. Let μ, μ^{♯} be two measures both regular for E. Let I be a closed interval in E^{int} with I ∩ [σ_{s}(μ) ∪ σ_{s}(μ^{♯})] = ∅. Fix C_{1} larger than the diameter of E. Then for all sufficiently small δ and each ε > 0, there is a constant C_{2} and positive integer J depending on μ, μ^{♯}, and ε so that for all m and ℓ,

λ_{n}(x_{0}, μ^{♯}) ≤ sup_{|y−x_{0}|≤δ}(\frac{w^{♯}(y)}{w(y)})λ_{m}(x_{0}, μ) + C_{2}e^{εm}(1 − \frac{δ^{2}}{C_{1}^{2}} )^{2ℓ} (3.1)

for all x_{0} ∈ I where n = m + 2ℓ + 2J.

Remarks. 1. While we apply this to w positive and continuous near x_{0}, it is stated in a way that should be applicable to situations where w and w^{♯} vanish or blow up in the same way (the sup in (3.1) is then interpreted as an essential sup).

2. One should be able, as in Lubinsky [18], to accommodate end points with these methods.

Proof. Let Q_{m}(x, x_{0}; μ) be the optimal trial function for the CD problem at x_{0}, that is,

Q_{m}(x, x_{0}; μ) = K_{m}(x_{0}, x_{0})^{−1} ∑_{n=0}^{m} p_{m}(x, μ_{0})p_{m}(x_{0}, μ_{0})  (3.2)

By (1.13), there is a δ_{1} and C_{3} so that

sup_{\begin{matrix}dist(y,E)≤δ_{1} \\ x_{0}∈I \end{matrix}} |Q_{m}(x, x_{0}; μ)| ≤ C_{3} e^{εm/2}  (3.3)

We use here the fact that by (1.13) for dist(y, E) ≤ δ_{1} with δ_{1} suitable,

|p_{m}(y)| ≤ ce^{m/6}

and m ≤ 6e^{m/6}. By shrinking δ_{1}, we can also be sure that

x_{0} ∈ I, dist(y, E) ≤ δ_{1} ⇒ |x_{0} − y| ≤ C_{1}  (3.4)

We now define J to be the number of points, x, in supp(μ^{♯}) with dist(x, E) ≥ δ_{1}, which is finite since σ_{ess}(dμ^{♯}) = E, and let {x_{j}}_{j=1}^{J} be those points. We take for our trial polynomials for λ_{n}(x_{0}, μ^{♯}) with n = m + 2ℓ + 2J,

Q(x) = Q_{m}(x, x_{0}; μ)(1 − \frac{(x − x_{0})^{2}}{C_{1}^{2}} )^{ℓ} ∏_{j=1}^{J}(1 − [\frac{(x − x_{0})}{(x_{j} − x_{0})}]^{2}) (3.5)

Pick δ so small that {x | dist(x, I) ≤ δ} is disjoint from σ_{s}(μ)∪σ_{s}(μ_{0}) and δ < δ_{1}. If y ∈ supp(dμ^{♯}) and |y − x_{0}| ≥ δ, we use (3.3) to see

|Q(y)| ≤ C_{4} e^{εm/2}(1 − \frac{δ^{2}}{C_{1}^{2}} )^{ℓ}  (3.6)

where

C_{4} = C_{3} max_{\begin{matrix}such y \\ x_{0}∈I \end{matrix}} |∏_{j=1}^{J}(1 − [\frac{(y − x_{0})}{(x_{j}x_{0})}]^{2})|

(3.3) does not hold at x_{j} but the product ∏_{j=1}^{J} vanishes at such points. Thus, ∫_{|y−x_{0}|≥δ}|Q(y)|^{2} dμ^{♯}(y) is bounded by the second term in (3.1).

On the other hand, since the second two factors in (3.4) are bounded by 1 on [x_{0} − δ, x_{0} + δ], |Q(y)| ≤ |Q_{m}(y)|, so, since there is no singular spectrum there,

∫_{|y−x_{0}|≤δ}|Q(y)|^{2} dμ^{♯} ≤ sup_{|x−y|≤δ} [\frac{w^{♯}(y)}{w(y)}] ∫_{|y−x_{0}|≤δ}|Q_{m}(y)|^{2} dμ

≤ sup_{|x−y|≤δ} [\frac{w^{♯}(y)}{w(y)}]λ_{m}(x_{0}, μ)

since adding the part of the integral with |y − x_{0}| > δ only makes the integral larger. □

Proof of Theorem 1.4. Let μ^{∗} = μ^{♯} be the model obeying (i)–(iv). Once δ is fixed, we can pick η so (1 − \frac{δ^{2}}{C_{1}^{2}} )^{2η} ≤ e^{−2}. Then in (3.1), we pick ℓ = ηεm. The second term in (3.1) is thus O(e^{−εm}).

Divide by λ_{m}(x, μ^{♯}). By (1.20), the second term in (3.1) goes to zero, and so (3.1) implies

lim inf \frac{λ_{m}(x_{m}, μ)}{λ_{m}(x_{m}, μ^{∗})} ≥ sup_{|x_{0}−y|≤2δ}(\frac{w^{∗}(y)}{w(y)}) lim inf (\frac{λ_{n}^{∗}}{λ_{m}^{∗}} )  (3.7)

Here we used x_{m} − x_{0} → 0 so |x_{0} − y| ≤ 2δ for m large.

Taking ε ↓ 0, by (1.21), the lim inf on the right goes to 1. Thus, by continuity, taking δ ↓ 0, we have

lim inf \frac{λ_{m}(x_{m}, μ)}{λ_{m}(x_{m}, μ^{∗})} ≥ \frac{w^{∗}(x_{0})}{w(x_{0})}  (3.8)

Now we interchange μ and μ^{∗} in (3.1) but now divide by λ_{n}(x_{n}, μ^{∗}) and use the same arguments to get

lim sup \frac{λ_{n}(x_{n}, μ)}{λ_{n}(x_{n}, μ^{∗})} ≤ \frac{w^{∗}(x_{0})}{w(x_{0})}  (3.9)

(3.8) and (3.9) complete the proof. All the arguments are uniform in x_{0} ∈ I. □

4. Off-Diagonal CD Asymptotics and Clock Behavior

In this section, we will prove Theorem 1.5 and note its consequences for zeros of the OPRL. Given Theorem 1.4, this is essentially a straightforward translation of [17] and [16]. We note that earlier Freud [9] had noted that the universality result, (1.16), (which he only had under very restrictive assumptions) implies clock behavior of zeros.

Proof of Theorem 1.5. Let μ ≤ μ^{∗}. Then, as noted by Lubinsky [17], for any x, y,

|K_{n}(x, y) − K_{n}^{∗}(x, y)|^{2} ≤ K_{n}^{∗}(y, y)[K_{n}(x, x) − K_{n}^{∗}(x, x)]  (4.1)

This critical result—which we dub Lubinsky’s inequality—is proven in a few lines in [17].

Given μ and x_{0}, let μ̃ be that multiple of the model μ^{♯} with w̃(x_{0}) = w(x_{0}). Let μ^{♯} = sup(μ, μ̃). By the lemma below, μ^{♯} is regular. Thus, by Theorem 1.4, we see that

\frac{K_{n}(x_{0} + \frac{α}{n}, x_{0} + \frac{α}{n})}{K_{n}^{∗}(x_{0} + \frac{β}{n}, x_{0} + \frac{β}{n})} → 1  (4.2)

and

\frac{K̃_{n}(x_{0} + \frac{α}{n}, x_{0} + \frac{α}{n})}{K_{n}^{∗}(x_{0} + \frac{β}{n}, x_{0} + \frac{β}{n})} → 1  (4.3)

uniformly if |α|, |β| < A and x_{0} ∈ I.

From this and (4.1), we find (dividing by K_{n}^{∗}(y, y)) that

\frac{K_{n}(x_{0} + \frac{α}{n}, x_{0} + \frac{β}{n})}{K̃_{n}(x_{0} + \frac{α}{n}, x_{0} + \frac{β}{n})} → 1  (4.4)

also uniformly in |α|, |β| < A, x_{0} ∈ I. (1.16) for K̃_{n} and (4.2)–(4.4) imply (1.16) for K_{n}. □

Lemma 4.1. Suppose μ, μ^{∗} are two measures with σ_{ess}(μ) = σ_{ess}(μ^{∗}) ≡ E and μ ≤ μ^{∗}. Then μ regular implies μ^{∗} is regular.

Proof. Regularity means

lim_{n→∞} ∥P_{n}( · , dμ^{∗})∥^{1/n} = C(E)  (4.5)

By (1.12),

lim sup ∥P_{n}( · , dμ^{∗})∥^{1/n} ≤ C(E)  (4.6)

By μ ≤ μ^{∗} and ∥P_{n}( · , dν)∥ = min{∥Q_{n}∥_{L^{2}(dν)} | deg Q_{n} = n, Q_{n}(x) = x^{n} + lower order},

∥P_{n}( · , dμ)∥ ≤ ∥P_{n}( · , dμ^{∗})∥  (4.7)

(4.5), (4.7), and

lim_{n→∞} ∥P_{n}( · , dμ)∥^{1/n} = C(E)  (4.8)

imply (4.5). □

Last–Simon [15] define clock behavior and uniform clock behavior. Theorem 1.5 implies

Theorem 4.2. Let μ be a measure obeying the hypothesis of Theorem 1.5 (so, in particular, E must have a suitable model). If a = b, there is clock behavior for the zeros of p_{n}(x, dμ) at a. If a < b, there is uniform clock behavior on I. The density of zeros in the clock behavior is ρ_{E}(x).

Remark. In particular, E can be a finite gap set by Theorem 2.1. Thus, we recover and vastly generalize the results of [28].

Proof. We need only follow the ideas of [16]: Step 1. By the CD formula, \frac{p_{n+1}(y)}{p_{n}(y)} = \frac{p_{n+1}(x)}{p_{n}(x)} for y ≠ x if and only

if K_{n}(x, y) = 0. Thus, with γ ≡ \frac{p_{n+1}(x_{0})}{p_{n}(x_{0})}, we see by (1.16) that there is a zero of p_{n+1}(y) − γp_{n}(y) within \frac{1}{nρ(x_{0})} + o(\frac{1}{n}) of x_{0}. Since zeros of p_{n+1} − γp_{n} and of p_{n} interlace, we conclude there are zeros of p_{n}(x) within \frac{1}{nρ(x_{0})} + o(\frac{1}{n}) of x_{0} ∈ I. Step 2. By the CD formula, for x ≠ y, if p_{n}(x) = 0, then p_{n}(y) = 0 if and only if K_{n}(x, y) = 0. Thus, by (1.16), there are no two zeros of p_{n}(x) within (1 − ε)\frac{1}{nρ(x_{0})} for any ε > 0, that is, we have an O(\frac{1}{n}) lower bound. Step 3. By the CD formula and (1.16), if p_{n}(x_{0} + \frac{α}{n}) = 0, there exist zeros which are at x_{0} + \frac{α}{n} + \frac{k}{nρ(x_{0})} + o(\frac{1}{n}) for |k| ≤ K, and, by Step 2, they are unique.

All these arguments are uniform in x_{0}, so we have uniform clock behavior. □

5. General Sets

As explained in the introduction, this section was written after I saw [39] and realized my results plus Lubinsky’s inequality easily allowed one to obtain universality for intervals with continuous a.c. weight in arbitrary compact sets, and also allowed an alternative to Section 2 that only requires Floquet theory.

Proof of Theorem 1.6. ℝ \ E as an open set is a union of countably many maximal open intervals whose total size, after the two semiinfinite ones are removed, is finite. Thus, for any q > 0, only finitely many are of size larger than \frac{2}{q}, so for any positive integer q, E_{q} = \{x | dist(x, E) ≤ \frac{1}{q}\} is a finite gap compact set.

Let ρ(x) be the equilibrium density for E (restricted to I) and ρ_{q}(x) for E_{q}. By balayage ideas, dρ_{q} ↾ I is nonincreasing, and by some potential theory using the real analyticity of ρ_{q} and ρ on I,

ρ_{q}(x) ↑ ρ(x)  (5.1)

uniformly on I (see [2, 11, 14, 24, 30, 34, 40] for background on the needed potential theory).

For each q, pick a multiple, c_{q}p_{q}(x), of the equilibrium measure so sup_{I} c_{q}p_{q} < inf_{I} w(x), and let μ_{q} = u∨c_{q}p_{q}, the measure theoretic max. This is regular for E_{q} by Lemma 4.1. Thus, if x_{m} → x_{0} ∈ I,

K_{m}^{(q)}(x_{m}, x_{m}) → \frac{ρ_{q}(x)}{w(x)}

where w = \frac{dμ}{dx}. The trial functions for μ_{q} used in proving (3.8) can be used for μ, that is, we can get upper bounds directly and see that

lim sup nλ_{n}(x_{n}, μ) ≤ \frac{w(x)}{ρ_{q}(x)}  (5.2)

for each q. Since (5.1) is uniform on I, we see

lim sup nλ_{n}(x_{n}, μ) ≤ \frac{w(x)}{ρ(x)}  (5.3)

uniformly for x ∈ I and x_{n} → x (as in Section 3, this means that for any ε, there are N and δ so if n ≥ N and |x_{n} − x| ≤ δ, then nλ_{n}(x_{n}, μ) ≤ \frac{w(x)}{ρ(x)+ε} + ε).

We can use the polynomials p_{n}(x, μ) as trial functions for μ_{q} and so still get (3.1), but unlike in Section 3, we cannot take ε to zero for q fixed and so not take \frac{ℓ}{n} → 0. Instead for a fixed q, there is η(q) and we have to take ℓ ≥ nη(q). But as q → ∞, η(q) → 0. We obtain

(1 + η(q)) lim inf nλ_{n}(x_{n}, μ) ≥ \frac{w(x)}{ρ_{q}(x)}  (5.4)

Since η(q) → 0 and ρ_{q}(x) → ρ(x), we therefore prove

lim nλ_{n}(x_{n}, μ) = \frac{w(x)}{ρ(x)}  (5.5)

The limit argument is essentially one used several years ago by Totik [37], the only difference being that we make uniform assumptions (i.e., continuity) on w and conclude uniformity in (5.5) with variable points.

Now use (4.1) with K^{∗} = K^{q}, the CD kernel, for μ_{q} ≥ μ by construction. Replace (x, y) by (x_{0} + \frac{α}{n}, x_{0} + \frac{β}{n}). Divide by \frac{nρ(x_{0})}{w(x_{0})} and take n → ∞ using (1.15) and (1.16) for μ_{q}, and (5.4) to get

lim sup_{n→∞} |\frac{K_{n}(x_{0} + \frac{α}{n}, x_{0} + \frac{β}{n})}{K_{n}(x_{0}, x_{0})}−\frac{sin ∏ ρ_{q}(x_{0})(β − α)}{∏ ρ(x_{0})(β − α)}|^{2} ≤ \frac{ρ_{q}(x_{0})}{ρ(x_{0})} [1−\frac{ρ_{q}(x_{0})}{ρ(x_{0})}]

(5.6)

Note the mixed \frac{ρ_{q}}{ρ} in the ratio on the left side. Taking q → ∞ yields the desired limit result. □

As a final remark, we note that we can use the same approximation idea to go from finite gap E’s with all rational harmonic measures to general finite gap E’s. For it is a result of Bogatyrëv [1], Peherstorfer [22], and Totik [38] that any finite gap E with ℓ gaps can be approximated by rational harmonic measure sets E_{q} ⊃ E so |E_{q} \ E| → 0. The arguments above can get results for general E from the E_{q}. The point of this remark is that the construction in Section 2 relies on Jost solutions. For E’s with rational harmonic measures, the Jacobi parameters are periodic, and Jost solutions can be constructed with Floquet theory rather than the more elaborate methods of [33, 23, 5].

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