THE SHARP FORM OF THE STRONG SZEGŐ THEOREM

BARRY SIMON

In memoriam, Robert Brooks (1952–2002)

Abstract. Let f be a function on the unit circle and D_{n}(f) be the determinant of the (n + 1) × (n + 1) matrix with elements \{c_{j−i}\}_{0≤i,j≤n} where c_{m} = f̂_{m} ≡ ∫ e^{−imθ}f(θ)\frac{dθ}{2π}. The sharp form of the strong Szegő theorem says that for any real-valued L on the unit circle with L, e^{L} in L^{1}(\frac{dθ}{2π}), we have

lim_{n→∞} D_{n}(e^{L})e^{−(n+1)L̂_{0}} = exp( ∑_{k=1}^{∞} k|L̂_{k}|^{2})

where the right side may be finite or infinite. We focus on two issues here: a new proof when e^{iθ} → L(θ) is analytic and known simple arguments that go from the analytic case to the general case. We add background material to make this article self-contained.

1. Introduction

Let {c_{m}}_{m=−∞}^{∞} be a two-sided sequence of complex numbers. A Toeplitz matrix is a finite matrix constant along diagonals:

T_{n+1} = (\begin{matrix}c_{0} & c_{1} & c_{2} & … & c_{n} \\ c_{−1} & c_{0} & c_{1} & … & c_{n−1} \\ ⋮ & & & & ⋮ \\ c_{−n} & c_{−n+1} & & … & c_{0} \end{matrix})  (1.1)

It turns out that the natural way to label T is in terms of the Fourier transform of c, that is,

f(θ) = ∑_{m=−∞}^{∞} c_{m}e^{imθ}  (1.2)

Date: January 21, 2004. 1 Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125.

E-mail: bsimon@caltech.edu.

2 Supported in part by NSF grant DMS-0140592. To appear in Proc. Conf. on Geometry and Spectral Theory.

on ∂𝔻 (𝔻 = {z | |z| < 1}, ∂𝔻 = {z | |z| = 1}). f is called the symbol of the Toeplitz matrix.

One can define a symbol as a distribution so long as |c_{m}| is polynomially bounded in m, but we will discuss the case where there is a signed measure, dμ, so that

c_{n} = ∫ e^{−inθ} dμ(θ) ≡ μ̂_{n}  (1.3)

As usual, for f ∈ L^{1}(∂𝔻, \frac{dθ}{2π}), we define f̂_{n} to be the Fourier coefficients of the measure f \frac{dθ}{2π}. We will most often discuss the case where dμ is absolutely continuous, that is, dμ = w(θ)\frac{dθ}{2π} and where w ≥ 0 or even that w = e^{L}.

D_{n}(dμ) is the determinant of T_{n+1}. The strong Szegő theorem says that if L, e^{L} ∈ L^{1} with L real, then

log D_{n}(e^{L} \frac{dθ}{2π} ) ∼ (n + 1)L̂_{0} + ∑_{k=1}^{∞} |k| |L̂_{k}|^{2}  (1.4)

There are a number of remarkable aspects of (1.4). The first term was found in 1915 and the second in 1952. Despite the 37-year break, they were both found by Szegő — the twenty-year old in 1915 [29] and the 57-year old in 1952 [33]! You might wonder about whether (1.4) is the leading term in a systematic 1/n series. In fact, if L is realvalued and if e^{iθ} ↦ L(θ) is analytic in the neighborhood of ∂𝔻, then the error in (1.4) is O(e^{−Bn}) — there are no more terms in the series (this follows from (2.21) and (5.17) below). Lest you be shocked by this, we note that for many models in statistical mechanics, the free energy has a volume term, a surface term, and then, if the interaction is short-range, exponentially small errors.

A second remarkable aspect is the subtlety. Why should log w enter at all, and then in both linear and quadratic terms? There is a fascination with this subject among mathematicians who have extended the result both in the context of function algebras [37, 15, 22] and in the context of pseudodifferential operators on manifolds [38, 14] (see [28] for literally dozens of papers on each aspect).

A third aspect is that there are a remarkable number of applications of this result. Szegő returned to find the second term because of a question raised by Onsager who ran into Toeplitz determinants in his work on the Ising model (see [23, 5] for a discussion of this). They enter in the study of some Coulomb gases [20, 21, 10] and in electrical engineering applications [18, 3]. And they have had a surge of interest recently because of their role in random matrix theory [24].

When Szegő [33] proved (1.4), he assumed L was C^{1+ε}. There were many papers on this subject which improved this incrementally until Ibragimov [16], fifteen years later, proved the following sharp form:

Theorem 1.1 ([16, 12]). Let L be a real-valued function on ∂𝔻 so that L, e^{L} ∈ L^{1}(∂𝔻, \frac{dθ}{2π}). Then

lim_{n→∞} D_{n}(e^{L} \frac{dθ}{2π} )e^{−(n+1)L̂_{0}} = exp( ∑_{k=1}^{∞} |k| |L̂_{k}|^{2})  (1.5)

Thus, (1.4) holds whenever the right side makes sense, that is, L in H^{1/2}, the Sobolev space of order \frac{1}{2}. This should be supplemented with a result of Golinskii-Ibragimov [12]:

Theorem 1.2. If dμ = e^{L} \frac{dθ}{2π} + dμ_{s} with dμ_{s} singular is a positive measure on ∂𝔻 and lim_{n→∞} D_{n}(dμ)e^{−(n+1)L̂_{0}} < ∞, then dμ_{s} = 0.

The combination of these two theorems has a spectral theory consequence that links it up to the theme of the conference and to Bob Brooks’ interests. As we will see in Section 2, probability measures on ∂𝔻 have associated parameters {α_{n}(dμ)}_{n=0}^{∞} called Verblunsky coefficients. It can be shown using Theorems 1.1, 1.2, and 2.4 that

Theorem 1.3. Let dμ be a probability measure on ∂𝔻 and {α_{n}(dμ)}_{n=0}^{∞} its Verblunsky coefficients. Then the following are equivalent:

(i) ∑_{n=0}^{∞} n|α_{n}|^{2} < ∞ (ii) dμ_{s} = 0 and dμ = e^{L} \frac{dθ}{2π} where ∑_{k=1}^{∞} k|L̂_{k}|^{2} < ∞.

This is one of those gems of spectral theory that give necessary and sufficient conditions relating properties of a measure and its inverse spectral parameters.

We have two main themes in this article. First, we wish to note that despite it taking fifteen years to go from L ∈ C^{1+ε} to L ∈ H^{1/2}, there is an elegant and simple argument to do this jump. This combines arguments of Golinskii-Ibragimov [12] and Johansson [17] whose proofs have “easy” halves that handle opposite sides of the extension. It does not appear to be widely appreciated that their arguments can be combined in this way. In fact, these results reduce (1.4) to the case where the L̂_{k} decay exponentially, that is, e^{iθ} ↦ L(θ) is real analytic on ∂𝔻.

Second, we have a new proof of (1.4) in this real analytic case that is perhaps less mysterious than the elaborate calculation in Szegő [33]. From our point of view, the two terms in (1.4) come from two terms in the Christoffel-Darboux formula for z = e^{iθ}.

While the arguments of [12, 17] are simple, they depend on considerable general machinery relating Toeplitz determinants to orthogonal polynomials on the unit circle (OPUC) on the one hand, and to the statistical mechanics of Coulomb gases on the other (both themes go back to Szegő’s early work: [29] has the Coulomb gas representation and the main point of [30, 31] is to discuss the connection to OPUC), so this article, in attempting to be self-contained, provides this background.

Sections 2 and 3 discuss the basics of OPUC. In Section 4, we get the leading term in (1.4), not only for its own sake, but to define in Section 5 the Szegő function which will play a critical role in many aspects of the remainder. These preliminaries allow us to present the Golinskii-Ibragimov half of the extension in Section 6. After proving the Coloumb gas representation in Section 7, we can prove the Johansson half of the extension in Section 8. The final three sections provide the proof of (1.4) in the analytic case: Section 9 has a preliminary proving the Christoffel-Darboux formula, and the last two sections finish the proof.

I have written a comprehensive book on OPUC [28] and everything in this paper appears there, but it seemed sensible, given the fact that the material is spread through a long book, to pull out exactly what is needed to prove Ibragimov’s theorem.

While (1.4) is sharp in one sense, it is not the end of the story by any means. First, there is a simple argument of Johansson [17] that drops the requirement of reality from L: if e^{L}, L ∈ L^{1} and L ∈ H^{1/2}, then an extension (1.4) holds in the sense that

e^{−(n+1)L̂_{0}}D_{n}(e^{L} \frac{dθ}{2π} ) → exp( ∑_{k=1}^{∞} |k|L̂_{k}L̂_{−k})

There are also subtleties in extending (1.4) to allow matrix-valued symbols, to allow complex w’s with nonzero winding number, and to determine the leading behavior when L ∉ L^{1}. The reader can consult [28] for references on all these issues.

Over the course of studying asymptotics of Toeplitz determinants, I have learned a lot in discussions with Percy Deift, Rowan Killip, and Irina Nenciu, and I would like to thank them for their insights.

Bob Brooks was a substantial mathematician and wonderful person. We lost him too soon. I’m glad to dedicate this article to his memory.

2. Verblunsky Coefficients and Toeplitz Matrices

If c_{n} are the moments of a measure, μ, it is natural to form the monic orthogonal polynomials, Φ_{n}(z; dμ), defined by

Φ_{n}(z) = z^{n} + lower order  (2.1)
⟨z^{j}, Φ_{n}⟩_{μ} = 0 if j = 0, 1, …, n − 1  (2.2)

where

⟨f, g⟩ = ∫ \overline{f(e^{iθ})} g(e^{iθ}) dμ(θ)  (2.3)

is the L^{2}(∂𝔻, dμ) inner product. In order to form Φ_{n} for all n, we need the measures dμ to be nontrivial, that is, not supported on a finite set of points.

The matrix elements c_{k−ℓ} = ∫ e^{i(ℓ−k)θ} dμ = ⟨z^{k}, z^{ℓ}⟩_{μ}, so D_{n}(dμ) is a Gram determinant. Such determinants allow change of basis, that is, if P_{k}(z) = z^{k} + lower order, det(⟨P_{k}, P_{ℓ}⟩)_{0≤k,ℓ≤n} = D_{n}(dμ). We can take P_{k} = Φ_{k}, in which case, ⟨P_{k}, P_{ℓ}⟩ is a diagonal matrix (!), and so,

Theorem 2.1.

D_{n}(dμ) = ∏_{j=0}^{n} ∥Φ_{j}∥_{dμ}^{2}  (2.4)

Φ_{j} is the orthogonal projection in L^{2} of z^{j} onto the orthogonal complement of [1, …, z^{j−1}], the span of {1, …, z^{j−1}}. Since multiplication by z is unitary, zΦ_{j} is the projection of z^{j+1} onto [z, …, z^{j}]^{⊥} while Φ_{j+1} is the projection of z^{j+1} onto [1, …, z^{j}]^{⊥}, so

∥Φ_{j+1}∥ ≤ ∥zΦ_{j}∥ = ∥Φ_{j}∥  (2.5)

Thus, ∥Φ_{j}∥ is decreasing in j. It follows that

Theorem 2.2.

(a)  lim D_{n}(dμ)^{1/n+1} = lim \frac{D_{n+1}(dμ)}{D_{n}(dμ)} = lim_{n→∞} ∥Φ_{n}∥^{2}  (2.6)
(b)  \frac{D_{n+1}}{D_{n}} ≤ \frac{D_{n}}{D_{n−1}}  (2.7)

These ideas all go back to Szegő [30, 31]. The next idea, which is a set of recursion relations of the Φ_{n}, was first written down by Szegő [32], but the basic parameters occurred in a related context in Verblunsky [35, 36]. To state them, we need to define the reversed polynomials

Φ_{n}^{∗}(z) = z^{n} \overline{Φ_{n}(1/z̄)}  (2.8)

Theorem 2.3. For any nontrivial measure, dμ, there exists a sequence of numbers {α_{n}}_{n=0}^{∞} so that

Φ_{n+1}(z) = zΦ_{n}(z) − ᾱ_{n}Φ_{n}^{∗}(z)  (2.9)

Moreover, α_{n} ∈ 𝔻 and

∥Φ_{n+1}∥^{2} = (1 − |α_{n}|^{2})∥Φ_{n}∥^{2}  (2.10)

and

∥Φ_{n}∥^{2} = c_{0}(dμ) ∏_{j=0}^{n−1} (1 − |α_{j}|^{2})  (2.11)

Remarks. 1. In the next section, we will see that μ ↦ {α_{n}}_{n=0}^{∞} is one-one for μ’s which are normalized.

2. There is a converse going back to Verblunsky [35] (see [28] for many other proofs) that the map from probability measures to ×_{n=0}^{∞} 𝔻 by μ ↦ {α_{n}}_{n=0}^{∞} is onto.

3. The α_{n} are called the Verblunsky coefficients for dμ.

4. Applying Q^{∗}(z) = z^{n+1} \overline{Q(1/z̄)} to (2.9) yields

Φ_{n+1}^{∗}(z) = Φ_{n}^{∗}(z) − α_{n}zΦ_{n}(z)  (2.12)

5. The proof below is a variant of one of Atkinson [1]. Szegő’s proof first proves the Christoffel-Darboux formula (see Section 9) and uses that to prove (2.9).

Proof. Let V_{n}g = z^{n}ḡ on L^{2}(dμ). V_{n} is anti-unitary, maps P_{n}, the polynomials of degree n, to themselves, and maps Φ_{n} to Φ_{n}^{∗}. Since Φ_{n} is the unique element (up to constants) of P_{n} orthogonal to {1, z, …, z^{n−1}} and V_{n} is anti-unitary, Φ_{n}^{∗} is the unique element of P_{n} orthogonal to {V_{n}1, …, V_{n}z^{n−1}} = {z^{n}, z^{n−1}, …, z}.

Now, for j = 1, …, n,

⟨z^{j}, zΦ_{n}⟩ = ⟨z^{j−1}, Φ_{n}⟩ = 0

and clearly, ⟨z^{j}, Φ_{n+1}⟩ = 0. Thus,

⟨z^{j}, Φ_{n+1} − zΦ_{n}⟩ = 0

for j = 1, …, n. Since Φ_{n} and Φ_{n+1} are monic, Φ_{n+1} − zΦ_{n} ∈ P_{n}. So, by the first part of the proof, (2.9) holds for a suitable constant α_{n}. Thus

α_{n} = −\overline{Φ_{n+1}(0)}  (2.13)

Since Φ_{n}^{∗} ⊥ Φ_{n+1}, we have

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

which implies (2.10). (2.10) in turn implies |α_{n}| < 1. (2.11) follows by induction and ∥Φ_{0}∥^{2} = ∥1∥^{2} = c_{0}(dμ). □

This leads to

Theorem 2.4. Suppose ∫ dμ = 1. We have

F (dμ) = lim_{n→∞} \frac{D_{n+1}(dμ)}{D_{n}(dμ)} = ∏_{j=0}^{∞} (1 − |α_{j}|^{2})  (2.14)

where the product always converges although the limit may be zero. If F (dμ) > 0, then

G_{n} = \frac{D_{n}}{F^{n+1}}  (2.15)

obeys

G_{n+1} ≥ G_{n}  (2.16)

The limit always exists (but may be infinite) and is given by

G(dμ) = lim_{n→∞} G_{n}(dμ) = ∏_{j=0}^{∞} (1 − |α_{j}|^{2})^{−j−1}  (2.17)

Remarks. 1. In particular,

F > 0 ⇔ ∑_{n=0}^{∞} |α_{n}|^{2} < ∞  (2.18)

G < ∞ ⇔ ∑_{n=0}^{∞} (n + 1)|α_{n}|^{2} < ∞  (2.19)

2. (2.14) in a sense goes back to Verblunsky [36]. (2.17) seems to have only been noted by Baxter [4] many years later.

3. If c_{0} ≠ 1, F (dμ) = c_{0} ∏_{j=0}^{∞}(1 − |α_{j}|^{2}) while G(dμ) is still given by (2.17). Indeed, G_{n}(dμ) = G_{n}(dμ/ ∫ dμ).

4. (2.7) says log D_{n} is concave in n. The monotonicity of G_{n} is a standard fact about concave functions with finite asymptotics.

Proof. By (2.4) and then (2.11),

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

(if ∥Φ_{1}∥^{2} = c_{0} = 1). From this and |α_{j}| < 1, (2.14) is immediate. If F is nonzero,

\frac{D_{n+1}}{D_{n}} \frac{1}{F} = ∏_{j=n+1}^{∞} (1 − |α_{j}|^{2})^{−1}

so that

G_{n} = \frac{c_{0}}{F} ∏_{k=1}^{n} [\frac{D_{k}}{D_{k−1}} \frac{1}{F} ]

= ∏_{j=0}^{∞} (1 − |α_{j}|^{2})^{− min(n,j)−1}  (2.21)

from which G_{n+1} ≥ G_{n} and (2.17) are immediate. □

3. Bernstein-Szegő Approximations

Given a nontrivial probability measure, dμ, with Verblunsky coefficients {α_{n}}_{n=0}^{∞}, we will identify the measures, dμ^{(N)}, with

α_{j}(dμ^{(N)}) = \begin{cases}α_{j}(dμ) & j = 0, 1, …, N − 1 \\ 0 & j ≥ N \end{cases}  (3.1)

and see dμ^{(N)} → dμ weakly. In many ways, the general proof of the strong Szegő theorem will play off this approximation and the distinct approximation obtained by truncating the Fourier series for L in e^{L} \frac{dθ}{2π}. As a preliminary, we need

Theorem 3.1. Φ_{n}(z) has all its zeros in 𝔻. Φ_{n}^{∗}(z) has all its zeros in ℂ\𝔻̄.

Remark. This proof is due to Landau [19]. See [28] for many other proofs of this fact.

Proof. Let Φ_{n}(z_{0}) = 0. Then π_{n−1} = Φ_{n}(z)/(z − z_{0}) is a polynomial of degree n − 1, and so ⟨π_{n−1}, Φ_{n}⟩ = 0. Since (z − z_{0})π_{n−1} = Φ_{n},

∥π_{n−1}∥^{2} = ∥zπ_{n−1}∥^{2} = ∥z_{0}π_{n−1} + Φ_{n}∥^{2}
= |z_{0}|^{2}∥π_{n−1}∥^{2} + ∥Φ_{n}∥^{2}

or

(1 − |z_{0}|^{2})∥π_{n−1}∥^{2} = ∥Φ_{n}∥^{2}  (3.2)

from which we conclude |z_{0}| < 1, that is, the zeros of Φ_{n} lie in 𝔻. Since Φ_{n}^{∗}(z_{0}) = 0 if and only if Φ_{n}(1/z̄_{0}) = 0, the zeros of Φ_{n}^{∗} lie in ℂ\𝔻̄. □

Given μ, define a measure, μ̃^{(N)}, by

dμ̃^{(N)} = \frac{dθ}{2π|Φ_{N}(e^{iθ}; dμ)|^{2}}  (3.3)

which can be defined since Φ_{N}(e^{iθ}) ≠ 0 for all θ by Theorem 3.1. We have Lemma 3.2. For j ≥ 0,

Φ_{N+j}(z; dμ̃^{(N)}) = z^{j}Φ_{N}(z; dμ)  (3.4)

Moreover,

α_{ℓ}(dμ̃^{(N)}) = 0  (3.5)

for ℓ ≥ N.

Proof. Since

|Φ_{N}(e^{iθ})|^{2} = e^{−iNθ}Φ_{N}(e^{iθ})Φ_{N}^{∗}(e^{iθ})  (3.6)

we have for k ∈ ℤ, k ≤ N − 1 (includes k < 0),

∫_{0}^{2π} e^{−ikθ} \frac{Φ_{N}(e^{iθ})}{|Φ_{N}(e^{iθ})|^{2}} \frac{dθ}{2π} = \frac{1}{2πi} ∮_{|z|=1} z^{N−k} \frac{dz}{zΦ_{N}^{∗}(z)}

= 0

since N − k − 1 ≥ 0 and Φ_{N}^{∗}(z)^{−1} is analytic in a neighborhood of 𝔻̄ by Theorem 3.1. This says

⟨z^{ℓ}, z^{j}Φ_{N}⟩_{}_{μ̃^{(N)}} = 0  (3.7)

for ℓ = 0, 1, …, N + j − 1. Since z^{j}Φ_{N} is monic, (3.4) holds.

By (2.13), if Φ_{k+1}(0) = 0, then α_{k} = 0, so (3.4) implies Φ_{N+j}(0; dμ̃^{(N)}) = 0, which in turn implies (3.5). □

Theorem 3.3 (Geronimus [11]). Let dμ, dν be two nontrivial measures on ∂𝔻. Suppose that for some fixed N,

Φ_{N}(z; dμ) = Φ_{N}(z; dν)  (3.8)

Then

Φ_{j}(z; dμ) = Φ_{j}(z; dν)  j = 0, 1, …, N − 1  (3.9)
α_{j}(dμ) = α_{j}(dν)  j = 0, 1, …, N − 1  (3.10)
\frac{c_{j}(dμ)}{c_{0}(dμ)} = \frac{c_{j}(dν)}{c_{0}(dν)}  j = 0, 1, …, N  (3.11)
∥Φ_{j}∥_{dμ}^{2} = c_{0}(dμ)c_{0}(dν)^{−1}∥Φ_{j}∥_{dν}^{2}  j = 0, 1, …, N  (3.12)

Proof. (2.9) and (2.12) can be written in matrix form

(\begin{matrix}Φ_{j+1}(z) \\ Φ_{j+1}^{∗}(z) \end{matrix}) = (\begin{matrix}z & −ᾱ_{j} \\ −zα_{j} & 1 \end{matrix}) (\begin{matrix}Φ_{j}(z) \\ Φ_{j}^{∗}(z) \end{matrix})  (3.13)

The 2 × 2 matrix in (3.13) has an inverse z^{−1}ρ_{j}^{−2} ( \begin{matrix}1 & ᾱ_{j} \\ α_{j}z & z \end{matrix} ) where

ρ_{j} = (1 − |α_{j}|^{2})^{1/2}  (3.14)

Thus (3.14) implies the inverse Szegő recursions:

Φ_{j}(z) = ρ_{j}^{−2} \frac{[Φ_{j+1}(z) + ᾱ_{j}Φ_{j+1}^{∗}]}{z}  (3.15)
Φ_{j}^{∗}(z) = ρ_{j}^{−2}[Φ_{j+1}^{∗}(z) + α_{j}Φ_{j+1}(z)]  (3.16)

(3.8) implies

α_{N}(dμ) = \overline{−Φ_{N}(0; dμ)} = \overline{−Φ_{N}(0; dν)} = α_{N}(dν)

and thus, by (3.15) with j = N − 1, we have (3.9) for j = N − 1. By iterating this argument, we conclude that (3.9) and (3.10) hold.

We need only prove (3.11) and (3.12), assuming c_{0}(dμ) = c_{0}(dν) = 1 since with dμ̃ = dμ/c_{0}(dμ), we have c_{n}(dμ) = c_{0}(dμ)c_{n}(dμ̃) and ∥Φ_{j}∥_{dμ}^{2} = c_{0}(dμ)∥Φ_{j}∥_{dμ̃}^{2}. (3.12) is immediate from (2.11).

We prove (3.11) when c_{0}(dμ) = c_{0}(dν) by induction and noting ⟨1, Φ_{k}⟩ = 0 yields a formula for c_{k} in terms of the coefficients of Φ_{k} and c_{0}, c_{1}, …, c_{k−1}. □

Remark. The last paragraph of the proof shows that the α’s determine the c’s and proves that the map of μ to α is one-one.

To succinctly state this section’s final result, we introduce

φ_{N}(z; dμ) = \frac{Φ_{N}(z; dμ)}{∥Φ_{N}∥}  (3.17)

the orthonormal polynomials. By (2.11),

φ_{N}(z; dμ) = ∏_{j=0}^{N−1} ρ_{j}^{−1}c_{0}(dμ)^{−1/2}Φ_{N}(z; dμ)  (3.18)

We define

κ_{n} = ( ∏_{j=0}^{n−1} ρ_{j}^{−1})(c_{0}(dμ))^{−1/2} = ∥Φ_{n}∥^{−1}  (3.19)

and

κ_{∞} = lim_{n→∞} ∥Φ_{n}∥^{−1}  (3.20)

which exists (but it may be +∞) by ρ_{j} < 1. The infinite product in (3.19), and so κ_{∞}, is finite if and only if

κ_{∞} < ∞ ⇔ ∑_{j=0}^{∞} |α_{j}|^{2} < ∞  (3.21)

We also note the translation of (2.9)/(2.12) from Φ to φ:

φ_{n+1}(z) = ρ_{n}^{−1}(zφ_{n}(z) − ᾱ_{n}φ_{n}^{∗}(z))  (3.22)

φ_{n+1}^{∗}(z) = ρ_{n}^{−1}(φ_{n}^{∗}(z) − α_{n}zφ_{n}^{∗}(z))  (3.23)

Theorem 3.4. Let dμ be a nontrivial probability measure on ∂𝔻. Define

dμ^{(N)} = \frac{dθ}{2π|φ_{N}(e^{iθ})|^{2}}  (3.24)

Then dμ^{(N)} is a probability measure on ∂𝔻 for which (3.1) holds. As N → ∞, dμ^{(N)} → dμ weakly.

Remark. We call measures of the form (3.24) BS measures and dμ^{(N)} the BS approximation.

Proof. Since dμ^{(N)} is a multiple of dμ̃^{(N)}, we have the bottom half of (3.1) by (3.5). Since (3.4) holds for j = 0, Theorem 3.3 and (3.10) imply the top half of (3.1).

Since Φ_{N} = ∥Φ_{N}∥_{μ}φ_{N}, we clearly have

∥Φ_{N}∥_{μ^{(N)}}^{2} = ∥Φ_{N}∥_{μ}^{2}

so, by (3.13), c_{0}(dμ^{(N)}) = c_{0}(dμ) = 1, that is, dμ_{N} is a probability measure.

By the above and (3.11), we have

c_{j}(dμ^{(N)}) = c_{j}(dμ) j = 0, 1, …, N  (3.25)

This and its complex conjugate implies that for any Laurent polynomial, f (polynomial in z and z^{−1}),

lim_{N→∞} ∫ f(e^{iθ}) dμ^{(N)} = ∫ f(e^{iθ}) dμ  (3.26)

since the left side is equal to the right for N large. Since Laurent polynomials are dense in C(∂𝔻), (3.26) holds for all f, that is, we have weak convergence. □

We note we have proven that

α_{j}(dμ^{(N)}) = \begin{cases}α_{j} & j ≤ N − 1 \\ 0 & j ≥ N \end{cases}  (3.27)

The ideas of this section go back to Geronimus [11] and were rediscovered in [9] and [7]. In particular, the use of inverse recursion to prove Geronimus’ theorem is taken from [7].

4. Szegő’s Theorem

In this section, our main goal is to prove

Theorem 4.1. For any w ∈ L^{1}(\frac{dθ}{2π}),

lim_{N→∞} \frac{1}{N} log D_{N}(w) = ∫ log w(θ) \frac{dθ}{2π}  (4.1)

Remarks. 1. Since log(x) ≤ x − 1 and w(θ) ∈ L^{1}, ∫ max(0, log w(x))\frac{dθ}{2π} < ∞, so the integral on the right side of (4.1) is either convergent or diverges to −∞, in which case (4.1) says D_{n}(w)^{1/n} → 0.

2. This was conjectured by Pólya [26] and proven by the twentyyear old Szegő in 1915 [29]. Our proof here is essentially the one Szegő presented in [30, 31].

3. The same result is true for the symbol dμ = w(θ)\frac{dθ}{2π} + dμ_{s}, that is, the limit is independent of dμ_{s}. This extension was first proven by Verblunsky [36]. We will not prove this more general result here ([28] has four different proofs in Chapter 2) since it is peripheral to our main result.

The first half of the theorem is a simple use of Jensen’s inequality:

Proposition 4.2. Let w = e^{L} with w, L ∈ L^{1}. Then

∥Φ_{n}∥_{w \frac{dθ}{2π}}^{2} ≥ exp(∫ L(θ) \frac{dθ}{2π} )  (4.2)

In particular,

D_{n}(w) ≥ exp(∫ (n + 1)L(θ) \frac{dθ}{2π} )  (4.3)

Proof. (4.3) follows from (4.2) and (2.4). To prove (4.2), we write

∥Φ_{n}∥^{2} = ∥Φ_{n}^{∗}∥^{2} = ∫ exp(2 log|Φ_{n}^{∗}(e^{iθ})| + L(e^{iθ})) \frac{dθ}{2π}
≥ exp(∫ [2 log|Φ_{n}^{∗}(e^{iθ})| + L(e^{iθ})]) \frac{dθ}{2π}  (4.4)

by Jensen’s inequality.

By Theorem 3.1, log(Φ_{n}^{∗}(z)) is analytic in 𝔻, so

∫ log|Φ_{n}^{∗}(e^{iθ})| \frac{dθ}{2π} = Re ∫ log(Φ_{n}^{∗}(e^{iθ})) \frac{dθ}{2π}
= log|Φ_{n}^{∗}(0)| = 0

since Φ_{n} monic implies Φ_{n}^{∗}(0) = 1. □

The other half of the theorem depends on a variational principle noted by Szegő:

Proposition 4.3. We have for any w ∈ L^{1}(∂𝔻, \frac{dθ}{2π}),

lim_{n→∞} [D_{n}(w)]^{1/n} = inf\{∫ |f(e^{iθ})|^{2}w(θ) \frac{dθ}{2π} | f ∈ H^{∞}(𝔻); f(0) = 1\}

(4.5)

Remark. H^{∞}(𝔻) is the Hardy space of bounded analytic functions on 𝔻. By general principles [8, 27], for \frac{dθ}{2π} a.e. e^{iθ} ∈ ∂𝔻, lim_{r↑1} f(re^{iθ}) exists, and that is what we mean by f(e^{iθ}) in (4.5).

Proof. Since ∥Φ_{n}^{∗}∥ = ∥Φ_{n}∥, by (2.6),

LHS of (4.5) = lim_{n→∞} ∥Φ_{n}^{∗}∥^{2}
= inf_{n} ∥Φ_{n}^{∗}∥^{2}  (4.6)

by (2.5). By the argument at the start of the proof of Theorem 2.3,

Φ_{n}^{∗} = π_{n}1

where π_{n} is the projection in the space of polynomials, P_{n}, of degree n onto the orthogonal component of the span of z, z^{2}, z^{3}, …, z^{n}. This span is {P ∈ P_{n} | P (0) = 0}, so by standard geometry,

∥Φ_{n}^{∗}∥^{2} = inf{∥P ∥^{2} | P ∈ P_{n}; P (0) = 1}  (4.7)

proving again that ∥Φ_{n}^{∗}∥^{2} is decreasing in n and proving (4.5) if H^{∞} is replaced by the set of all polynomials.

To complete the proof, we need only show that for any f ∈ H^{∞} with f(0) = 1, there are polynomials P_{ℓ}(z) so that P_{ℓ}(0) = 1 and

∫ |P_{ℓ}(e^{iθ})|^{2}w(θ) \frac{dθ}{2π} → ∫ |f(e^{iθ})|^{2}w(θ) \frac{dθ}{2π}  (4.8)

If f is analytic in a neighborhood of 𝔻, the Taylor approximations converge uniformly on 𝔻̄, so (4.8) holds. For general f, by the dominated convergence theorem,

lim_{r↑1} ∫ |f(re^{iθ})|^{2}w(θ) \frac{dθ}{2π} = ∫ |f(e^{iθ})|^{2}w(θ) \frac{dθ}{2π}

so, by a two-step approximation, we find P_{ℓ}’s so (4.8) holds. □

Proof of Theorem 4.1. We will prove that for any ε > 0, there is an f in H^{∞} with f(0) = 1 and

∫ |f(e^{iθ})|^{2}w(θ) \frac{dθ}{2π} ≤ exp(∫ log(w(θ) + ε) \frac{dθ}{2π} )  (4.9)

so taking ε ↓ 0 yields the opposite inequality to (4.3).

Define

g(z) = exp(− ∫ log(w(θ) + ε)(\frac{e^{iθ} + z}{e^{iθ} − z} ) \frac{dθ}{4π} )  (4.10)

and f(z) = g(z)/g(0), so f(0) = 1. Moreover, |g(z)| ≤ ε^{−1/2} by the fact that the Poisson kernel

P_{r}(θ, φ) = Re(\frac{e^{iθ} + re^{iφ}}{e^{iθ} − re^{iφ}} )

is nonnegative and ∫ \frac{dθ}{2π}P_{r}(θ, φ) = 1. By standard maximal function arguments (see [27]), |g(e^{iθ})| = |w(θ) + ε|^{−1/2}, so

∫ |f(e^{iθ})|^{2}w(θ) \frac{dθ}{2π} ≤ g(0)^{−2} = RHS of (4.9)

□

5. The Szegő Function

When w(θ) = e^{L(θ)} with L ∈ L^{1}, Szegő [30, 31] introduced a natural function, D(z) on 𝔻, which will play a critical role in several places below:

D(z) = exp(∫ (\frac{e^{iθ} + z}{e^{iθ} − z} )L(θ) \frac{dθ}{4π} )  (5.1)

Do not confuse D_{n} and D(z). Both symbols are standard, but the objects are very different.

Theorem 5.1. (a) If (2.18) holds, then for |z| < 1,

D(z) = D(0) exp( ∑_{k=1}^{∞} L̂_{k}z^{k})  (5.2)

D(0) = [c_{0}(w \frac{dθ}{2π} )]^{1/2} ∏_{n=0}^{∞} (1 − |α_{n}|^{2})^{1/2}  (5.3)

(b) D(z) lies in H^{2}(𝔻).

(c) lim_{r↑1} D(re^{iθ}) ≡ D(e^{iθ}) exist for a.e. θ and

|D(e^{iθ})|^{2} = w(θ)  (5.4)

(d) D is nonvanishing on 𝔻.

Proof. (a) We get (5.2) and

D(0) = exp(\frac{1}{2} L_{0})  (5.5)

from (5.1) and

\frac{e^{iθ} + z}{e^{iθ} − z} = 1 + 2 ∑_{j=0}^{∞} (e^{−iθ}z)^{n}

uniformly in e^{iθ} ∈ ∂𝔻 and z ∈ {|z| < r}. (5.3) then follows from (2.14), (4.1), and (5.5).

(b) Let D^{(M)}(z) be given by (5.1) with L(θ) replaced by min(L(θ), M). Then D^{(M)} ∈ H^{∞} and |D^{(M)}(e^{iθ})|^{2} = min(w(θ), e^{2M}),

sup_{0<r<1} ∫_{0}^{2π} |D^{(M)}(re^{iθ})|^{2} \frac{dθ}{2π} = ∫_{0}^{2π} min(w(θ), e^{2M}) \frac{dθ}{2π}

≤ ∫_{0}^{2π} w(θ) \frac{dθ}{2π}

Thus, taking M → ∞, we see D ∈ H^{2}.

(c) is immediate from properties of boundary values of the Poisson integral in (5.1).

(d) is trivial from (5.1). □

The following simple but powerful L^{2} calculation goes back to Szegő [30, 31] (it has a version when dμ_{s} ≠ 0; see [28]):

Theorem 5.2. Let dμ = w\frac{dθ}{2π} where w = e^{L}, L ∈ L^{1}. Then, as n → ∞,

(i)  ∫ |Dφ_{n}^{∗}(e^{iθ}) − 1|^{2} \frac{dθ}{2π} → 0  (5.6)
(ii)  ∫ |φ_{n}^{∗}(e^{iθ}) − D^{−1}(e^{iθ})|^{2} dμ → 0  (5.7)
(iii)  φ_{n}^{∗}(z) → D(z)^{−1}  (5.8)

uniformly on compact subsets of 𝔻.

Proof. (i) By (5.4), ∫ |Dφ_{n}^{∗}|^{2} \frac{dθ}{2π} = ∫ |φ_{n}^{∗}|^{2} dμ = 1, so (5.6) is equivalent to

Re ∫ φ_{n}^{∗}(e^{iθ})D(e^{iθ}) \frac{dθ}{2π} → 1  (5.9)

Dφ_{n}^{∗} is in H^{2}, so the Cauchy integral formula applies, that is,

LHS of (5.9) = φ_{n}^{∗}(0)D(0)
= κ_{n}κ_{∞}^{−1}  (5.10)

since Φ_{n}^{∗}(0) = 1, φ_{n} = κ_{n}Φ_{n} and (5.3) and (3.19) imply D(0) = κ_{∞}^{−1}. (5.10) implies (5.9).

(ii) is immediate from (5.6) and dμ = w\frac{dθ}{2π} = |D|^{2} \frac{dθ}{2π} by (5.4).

(iii) (5.6) says Dφ_{n}^{∗} → 1 in ℍ^{2} and so, a fortiori, uniformly on compact subsets of 𝔻. □

We need to extend this result to a neighborhood of 𝔻̄ when L is real analytic. As a preliminary, we note:

Lemma 5.3. Let dμ = w\frac{dθ}{2π} be a probability measure with log w ∈ L^{1}(\frac{dθ}{2π}). Then

α_{n} = −κ_{∞} ∫ \overline{Φ_{n+1}(e^{iθ})} D(e^{iθ})^{−1} dμ(θ)  (5.11)

Proof. We will prove for m ≥ n + 1 that

α_{n} = −κ_{m} ∫ \overline{Φ_{n+1}(e^{iθ})} φ_{m}^{∗}(e^{iθ}) dμ(θ)  (5.12)

from which (5.11) follows from (5.7).

φ_{m}^{∗} is orthogonal to {z^{ℓ}}_{ℓ=1}^{m}, so if P is any polynomial of degree at most m with P (0) = 0,

∫ \overline{P (e^{iθ})} φ_{m}^{∗}(e^{iθ}) dμ(θ) = 0  (5.13)

(5.12) follows from (5.13) by taking

P (z) = ᾱ_{n}φ_{m}^{∗}(z) + κ_{m}Φ_{n+1}(z)

which has P (0) = ᾱ_{n}κ_{m} + κ_{m}(−ᾱ_{n}) = 0. □

The following is due to Nevai-Totik [25]; it is needed in Section 11:

Theorem 5.4. Suppose that dμ = e^{L} \frac{dθ}{2π} and e^{iθ} ↦ L(θ) is analytic in a neighborhood of ∂𝔻. Then φ_{n}^{∗}(z) → D(z)^{−1} uniformly in a neighborhood of 𝔻̄. Moreover, the Verblunsky coefficients obey |α_{n}| ≤ C_{2}e^{−An/2} for some A > 0.

Proof. Analyticity says |L̂_{k}| ≤ Ce^{−A|k|} for some A > 0. So, by (5.2), D(z) is analytic and nonvanishing in a disk of radius e^{A}. In particular, if

D(z)^{−1} = ∑_{j=0}^{∞} d_{j,−1}z^{j}  (5.14)

then

|d_{j,−1}| ≤ C_{1}e^{−A|j|/2}  (5.15)

Plug (5.14) into (5.11) and note that

∫ \overline{Φ_{n+1}(e^{iθ})} e^{ikθ} dμ(θ) = 0

for k = 0, 1, …, n. Thus

|α_{n}| ≤ κ_{∞} ∑_{k=n+1}^{∞} |d_{k,−1}| |∫ \overline{Φ_{n+1}(e^{iθ})} e^{ikθ} dμ|

≤ κ_{∞}∥Φ_{n+1}∥ ∑_{k=n+1}^{∞} |d_{k,−1}|

≤ κ_{∞} ∑_{k=n+1}^{∞} |d_{k,−1}|  (5.16)

since ∥Φ_{n+1}∥ ≤ 1. So, by (5.15),

|α_{n}| ≤ C_{2}e^{−A|n|/2}  (5.17)

By (2.12) and |Φ_{n}^{∗}(e^{iθ})| = |Φ_{n}(e^{iθ})|, we have

sup_{θ} |Φ_{n+1}^{∗}(e^{iθ})| ≤ (1 + |α_{n}|) sup_{θ} |Φ_{n}^{∗}(e^{iθ})|

≤ ∏_{j=0}^{n} (1 + |α_{j}|)

≤ exp( ∑_{j=0}^{∞} |α_{j}|) = C_{4} < ∞  (5.18)

by iteration. C_{4} < ∞ follows from (5.17). Since Φ_{n}^{∗} is analytic, we get

sup_{z∈𝔻̄}|Φ_{n}^{∗}(z)| ≤ C_{4}  (5.19)

and thus, since Φ_{n}^{∗}(z) = z^{n} \overline{Φ_{n}(1/z̄)}, we get

z ∈ ℂ\𝔻 ⇒ |Φ_{n}(z)| ≤ C_{4}|z|^{n}  (5.20)

Returning to (2.12),

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

≤ C_{4}C_{2} ∑_{n=0}^{∞} |ze^{−A/2}|^{n}

showing Φ_{n}^{∗}, and so φ_{n}^{∗}, converges uniformly in {z | |z| ≤ e^{A/4}}. Since the limit is D^{−1} in 𝔻, it is D^{−1} in this larger disk. □

Finally, in terms of D, we want to rewrite the second term in the Szegő asymptotic formula (1.4):

Theorem 5.5. Let dμ = e^{L(θ)} \frac{dθ}{2π} with L ∈ L^{1}. Let L̂_{k} be given by (1.3). Then

∑_{k=1}^{∞} k|L̂_{k}|^{2} = \frac{1}{π} ∫_{|z|≤1} |D(z)|^{−2}|\frac{∂D}{∂z}|^{2} d^{2}z  (5.21)

where both sides can be infinite.

Proof. (5.21) follows by taking r ↑ 1 in

∑_{k=1}^{∞} k |L̂_{k}|^{2}r^{2k} = \frac{1}{π} ∫_{|z|≤r} |D(z)|^{−2}|\frac{∂D}{∂z}|^{2} d^{2}z  (5.22)

(by using monotone convergence). To prove (5.22), note that by (5.2),

log[\frac{D(z)}{D(0)}] = ∑_{k=1}^{∞} L̂_{k}z^{k}

converges uniformly in |z| < R and that |D|^{−2}|\frac{∂D}{∂z}|^{2} = |\frac{∂}{∂z} log D(z)|^{2}. Thus (5.22) follows from

\frac{1}{π} ∫_{|z|≤r} z̄^{k−1}z^{ℓ−1} d^{2}z = k^{−1}δ_{kℓ}r^{2k}

and

\frac{∂}{∂z} log D(z) = ∑_{k=1}^{∞} kL̂_{k}z^{k−1}

□

6. Extending the Strong Szegő Theorem, Part I

With the Szegő function and Fatou’s lemma, we have the tools for the Golinskii-Ibragimov [12] half of the extension theorem:

Theorem 6.1 ([12]). Suppose for any BS measure, dμ, we know that

G(dμ) = exp( ∑_{k=1}^{∞} k|L̂_{k}|^{2})  (6.1)

Then for any measure dμ = e^{L} \frac{dθ}{2π} with L ∈ L^{1}, we have

G(dμ) ≥ exp( ∑_{k=1}^{∞} k|L̂_{k}|^{2})  (6.2)

Remark. In particular, G < ∞ ⇒ ∑ k|L̂_{k}|^{2} < ∞.

Proof. Let dμ^{(N)} be the BS approximations to dμ. Let D^{(N)}(z) be the D function for dμ^{(N)} and let

M^{(N)} = \frac{1}{π} ∫_{|z|≤1} |D^{(N)}(z)|^{−2} |\frac{∂D^{(N)}}{∂z} |^{2} d^{2}z  (6.3)

and similarly for M and D.

By (2.17) and (3.27),

G(dμ^{(N)}) = ∏_{j=0}^{N−1} (1 − |α_{j}|^{2})^{−j−1}

so G(dμ^{(N)}) is monotone increasing to G(dμ), that is,

G(dμ) = lim_{N→∞} G(dμ^{(N)})  (6.4)

On the other hand, since φ_{N+j}(z, dμ^{(N)}) = z^{j}φ_{N}(z, dμ^{(N)}), φ_{N+j}^{∗}(z, dμ^{(N}) = φ_{N}^{∗}(z, dμ^{(N)}) = φ_{N}^{∗}(z, dμ), so by (5.8),

D^{(N)}(z) = φ_{N}^{∗}(z, dμ)^{−1}  (6.5)

and thus, by (5.8) again,

D^{(N)}(z) → D(z)  (6.6)

uniformly on compacts. By analyticity, the same thing is true for derivatives, so

|D(z)|^{−2} |\frac{∂D}{∂z}|^{2} = lim_{N→∞} |D^{(N)}(z)|^{−2} |\frac{∂D^{(N)}}{∂z} |^{2}

Thus, by Fatou’s lemma,

M ≤ lim inf M^{(N)}  (6.7)

By hypothesis,

G(dμ^{(N)}) = exp(M^{(N)})  (6.8)

(6.8), (6.7), and (6.4) imply that

G(dμ) ≥ exp(M)

which is (6.2). □

7. The Coulomb Gas Representation

In this section, we will provide an integral formula for D_{n} that will be critical in the next section. This formula appeared in Szegő’s first paper [29] on asymptotics of Toeplitz determinants. He used it there for a minor technical purpose — and for us, too, it plays a relatively minor role. That said, it plays a central role in two proofs of the strong Szegő theorem and several applications.

While we will not pursue the Coulomb gas picture, if one uses |z_{j} − z_{k}| = exp(log|z_{j}−z_{k}|), the formula we give for D_{n} says it is the partition function of a two-dimensional gas, and this point of view is the basis of Johansson’s proof [17]. It also explains some interest in Toeplitz matrices in the physics literature [20, 21, 10].

Using Weyl’s relation that Haar measure restricted to the classes of 𝕌(n+1), the group of n×n unitary matrices, is essentially (2π)^{−n−1}[(n+ 1)!]^{−1} ∏_{k<j}|e^{iθ_{k}} − e^{iθ_{j}}|^{2} dθ_{0} … dθ_{n} with {e^{iθ_{j}}}_{j=0}^{n} the eigenvalues of U ∈ 𝕌(n + 1), one can use the formula below to rewrite D_{n} as an integral over 𝕌(n+1). This is both the starting point of the Bump-Diaconis [6] proof and of the many applications of Toeplitz matrices in the theory of random matrices [24].

It is, of course, well-known that ∏_{k<j}(z_{k} − z_{j}) is a Vandermonde determinant. Expanding two such products, we get

| ∏_{0≤k<j≤n} (z_{k} − z_{j})|^{2} = ∑_{π,σ∈Σ_{n+1}} (−1)^{π}(−1)^{σ} ∏_{j=0}^{n} z̄_{j}^{π(j)}z_{j}^{σ(j)}  (7.1)

where Σ_{n+1} is the permutations of {0, …, n} to itself.

Setting z_{j} = e^{iθ_{j}} and integrating dμ(θ_{0}) … dμ(θ_{n}), we get that

∫ |π(z_{k} − z_{j})|^{2} dμ(θ_{0}) … dμ(θ_{n}) = ∑_{π,σ∈Σ_{n+1}} (−1)^{π}(−1)^{σ} ∏_{j=0}^{n} T_{π(j)σ(j)}

(7.2)

where T_{kℓ} are the matrix elements of the (n + 1) × (n + 1) Toeplitz matrix. Now

∏_{j=0}^{n} T_{π(j)σ(j)} = ∏_{j=0}^{n} T_{j(σπ^{−1})(j)}

and (−1)^{π}(−1)^{σ} = (−1)^{σπ^{−1}}. Thus, summing over σ for π fixed and then over π, we see the right side of (7.2) is (n+1)!D_{n}(dμ). Specializing to dμ = e^{L} \frac{dθ}{2π}, we have proven Theorem 7.1 (Coulomb Gas Representation for D_{n}). Let e^{L}, L ∈ L^{1}(∂𝔻, \frac{dθ}{2π}). Then with z_{k} = e^{iθ_{k}},

D_{n}(e^{L} \frac{dθ}{2π} ) = [(n + 1)!]^{−1} ∫_{∂𝔻^{n+1}} | ∏_{0≤k<j≤n} (z_{k} − z_{j})|^{2}e^{∑_{j=0}^{n} L(θ_{j})} ∏_{j=0}^{n} \frac{dθ_{j}}{2π}

8. Extending the Strong Szegő Theorem, Part II

The Coulomb representation and Fatou’s lemma are the tools for Johansson’s half of the extension theorem. A measure of the form e^{L} \frac{dθ}{2π} where L is a real Laurent polynomial (i.e., ∑_{k=−n}^{n} L̂_{k}e^{ikθ} with L̂_{−k} = ¯L̂_{k}) is called a GI measure (after its early use in [12]). If dμ = e^{L} \frac{dθ}{2π} with L ∈ L^{1}, we define the GI approximations, dμ_{(N)}, by

dμ_{(N)} = exp( ∑_{|k|≤N} L̂_{k}e^{ikθ}) \frac{dθ}{2π} ≡ exp(L_{(N)}(θ)) \frac{dθ}{2π}  (8.1)

Theorem 8.1 ([17]). Suppose for any GI measure, dμ, we know that

G(dμ) = exp( ∑_{k=1}^{∞} k|L̂_{k}|^{2})  (8.2)

Then for any measure dμ = e^{L} \frac{dθ}{2π} with L ∈ L^{1}, we have

G(dμ) ≤ exp( ∑_{k=1}^{∞} k|L̂_{k}|^{2})  (8.3)

Remark. In particular, if ∑_{k=1}^{∞} k|L̂_{k}|^{2} < ∞, then G(dμ) < ∞.

Proof. Without loss (since we can multiply dμ by a constant), suppose L̂_{0} = 0. Given dμ = e^{L} \frac{dθ}{2π}, let dμ_{(N)} be its GI approximations and let

C_{(N),m} = [(n+1)!]^{−1} ∫_{𝔻^{m+1}} ∏_{0≤j<k≤m} |z_{j}−z_{k}|^{2}e^{∑_{j=0}^{m} L_{(N)}(θ_{j})} ∏_{j=0}^{m} \frac{dθ_{j}}{2π} (8.4)

so, of course, C_{(N),m} = D_{m}(dμ_{(N)}) by Theorem 7.1. Let C_{m} be the integral with L_{(N)} replaced by L.

There is nothing to prove in (8.3) if ∑_{k=1}^{∞} k|L̂_{k}|^{2} = ∞, so suppose the sum is finite. Thus, L_{(N)} → L in L^{2}, and so, we can find a subsequence L_{(N_{j})} → L pointwise. By Fatou’s lemma, for each m,

C_{m} ≤ lim inf_{j→∞} C_{(N_{j}),m}  (8.5)

By Theorem 7.1, (2.16), L̂_{0} = 0, and the assumption for GI measures,

C_{(N_{j}),m} = G_{m}(dμ_{(N_{j})})

≤ G(dμ_{(N_{j})})

= exp( ∑_{k=1}^{N_{j}} k|L̂_{k}|^{2})

≤ exp( ∑_{k=1}^{∞} k|L̂_{k}|^{2})

By (8.5) and L̂_{0} = 0,

G_{m}(dμ) = C_{m} ≤ exp( ∑_{k=1}^{∞} k|L̂_{k}|^{2})

Taking m → ∞ yields (8.3). □

Theorems 6.1 and 8.1 reduce the proof of the sharp form of the strong Szegő theorem to proving the result for BS and GI measures. In both cases, dμ = e^{L} \frac{dθ}{2π} with e^{iθ} ↦ L(θ) analytic in a neighborhood of ∂𝔻. In the last three sections, we will prove the strong Szegő theorem in that case.

9. The CD Formula

If u_{1}, u_{2} solve −u′′_{j}+V u_{j} = λ_{j}u_{j} with u_{j}(0) = 0, then (λ̄_{1}−λ_{2})ū_{1}u_{2} = (ū_{1}u′_{2} − u_{2}ū′_{1})′, so

(λ̄_{1} − λ_{2}) ∫_{0}^{a} \overline{u_{1}(x)} u_{2}(x) dx = ū_{1}(a)u′_{2}(a) − ū′_{1}(a)u_{2}(a)  (9.1)

The Christoffel-Darboux formula is just the analog of this Wronskian relation that many undergraduates learn!

Theorem 9.1 (CD Formula). Let

K_{n}(z, ζ) = ∑_{j=0}^{n} \overline{φ_{j}(ζ)} φ_{j}(z)  (9.2)

Then

K_{n}(z, ζ) = \frac{\overline{φ_{n+1}^{∗}(ζ)} φ_{n+1}^{∗}(z) − \overline{φ_{n+1}(ζ)} φ_{n+1}(z)}{1 − ζ̄z}  (9.3)
= \frac{\overline{φ_{n}^{∗}(ζ)} φ_{n}^{∗}(ζ) − zζ̄\overline{φ_{n}(ζ)} φ_{n}(z)}{1 − ζ̄z}  (9.4)

Proof. By using (3.22)/(3.23) and their conjugates to write φ_{n+1} and φ_{n+1}^{∗} in terms of φ_{n} and φ_{n}^{∗}, we get

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

(9.5)

where we used ρ_{n}^{−2}(1 − |α_{n}|^{2}) = 1, and that the φ^{∗}φ cross terms cancel.

First of all, (9.5) says that (9.3) is equivalent to (9.4). In addition, if we let Q_{n}(z, ζ) be what we know is the common value of the right sides of (9.3) and (9.4), then subtracting (9.3) for n − 1 from (9.4) for n, we see that

Q_{n}(z, ζ) − Q_{n−1}(z, ζ) = (1 − zζ̄)[ \frac{\overline{φ_{n}(ζ)} φ_{n}(z)}{1 − zζ̄}]
= K_{n}(z, ζ) − K_{n−1}(z, ζ)

so we need only prove Q_{0}(z, ζ) = K_{0}(z, ζ). Since (9.4) for n = 0 is 1 and that is K_{0}(z, ζ), (9.3) is proven. □

Corollary 9.2. For e^{iθ} ∈ ∂𝔻,

K_{n}(e^{iθ}, e^{iθ}) ≡ ∑_{j=0}^{n} |φ_{j}(e^{iθ})|^{2} = −\frac{∂}{∂r} |φ_{n+1}^{∗}(re^{iθ})|^{2}|_{r=1}+(n+1)|φ_{n+1}^{∗}(e^{iθ})|^{2}

(9.6)

Proof. In (9.3), take z = ζ = re^{iθ} and take r ↑ 1. If we note |φ_{n+1}(re^{iθ})|^{2} = r^{2n+2}|φ_{n+1}^{∗}(r^{−1}e^{iθ})|^{2}, we get two terms: the one from the −(r^{2n+2} − 1)/(1 − r^{2}) gives the (n + 1)|φ_{n+1}^{∗}(e^{iθ})|^{2} term in (9.6), and the other terms give the derivative in (9.6). □

The CD formula for OPUC is due to Szegő [32]. Our proof is taken from Golinskii [13].

Remark. Integrating both sides of (9.6) yields

∫ \frac{∂}{∂r} |φ_{n+1}^{∗}(re^{iθ})|^{2}|_{r=1} dμ(θ) = 1  (9.7)

10. The Feynman-Hellman Theorem for Toeplitz Determinants

Here we want to begin with a simple but useful formula for \frac{∂}{∂λ}∥Φ_{n}∥_{dμ_{λ}}^{2}, where the measure dμ_{λ} depends smoothly on λ. This appeared in [28], strongly motivated by closely related ideas of Baik et al. [2].

Theorem 10.1. Let dμ_{λ} be a family of positive measures on ∂𝔻 which are C^{1} in λ in the sense that λ ↦ c_{n}(dμ_{λ}) is C^{1} for each n. Then ∥Φ_{n}( · , dμ_{λ})∥_{μ_{λ}} is C^{1} and

\frac{d}{dλ} log ∥Φ_{n}∥_{μ_{λ}}^{2} = ∫ |φ_{n}(e^{iθ}, dμ_{λ})|^{2} \frac{dμ_{λ}(θ)}{dλ}  (10.1)

Proof. Since \frac{d}{dλ} log ∥Φ_{n}∥^{2} = (\frac{d}{dλ}∥Φ_{n}∥^{2})/∥Φ_{n}∥^{2}, it suffices to prove that

\frac{d}{dλ} ∥Φ_{n}∥_{μ_{λ}}^{2} = ∫ |Φ_{n}(e^{iθ}, dμ_{λ})|^{2} \frac{dμ_{λ}(θ)}{dλ}  (10.2)

Since

∥Φ_{λ}∥^{2} = ∫ \overline{Φ_{n}(e^{iθ}, dμ_{λ})} Φ_{n}(e^{iθ}, dμ_{λ}) dμ_{λ}  (10.3)

its derivative is a sum of three terms, of which one term is the right side of (10.2) and the other two are

∫ \overline{\frac{∂}{∂λ} Φ_{n}(e^{iθ}, dμ_{λ})} Φ_{n}(e^{iθ}, dμ_{λ}) dμ_{λ}  (10.4)

and its conjugate.

But, for all λ, Φ_{n} is monic, so \frac{∂}{∂λ}Φ_{n} is a polynomial of degree at most n − 1 and so orthogonal to Φ_{n}. It follows that the term in (10.4) is zero. □

Remark. Those familiar with the Feynman-Hellman theorem [34] will see a metaphysical link to this proof.

We want to do two things with (10.1). First, we sum several logs in log D_{n}(dμ) = ∑_{j=0}^{n} log ∥Φ_{j}∥^{2} (by (2.4)), and second, we make an explicit choice of dμ_{λ}:

Theorem 10.2. Let w_{t} be a family of L^{1} functions, C^{1} in t so log w_{t} is also C^{1} in t, and w_{0} = 1. Then

log D_{n}(w_{1} \frac{dθ}{2π} ) = (n + 1) log(∥Φ_{n+1}∥_{t=1}^{2})

− ∫_{0}^{1} dt ∫ [\frac{d}{dt} (log w_{t})]( \frac{∂}{∂r} |φ_{n+1}^{∗}(re^{iθ}; dμ_{t})|^{2}|_{r=1}) dμ_{t}(θ)

(10.5)

Proof. We have that

\frac{d}{dt} w_{t} \frac{dθ}{2π} = \frac{w′_{t}}{w_{t}} dμ_{t} = \frac{d}{dt} (log w_{t}) dμ_{t}

Using this, (2.4), (10.1), and the definition (9.2) of K, we get

\frac{d}{dt} log D_{n}(w_{t} \frac{dθ}{2π} ) = ∫ [\frac{d}{dt} (log w_{t})]K(e^{iθ}, e^{iθ}; dμ_{t}) dμ_{t}  (10.6)

Now use (9.6) to get two terms. If we integrate dt, the first term gives the integral in (10.5). The second term can be integrated using (10.1) to give the first term in (10.5). □

11. Completion of the Proof

We complete the proof, following [28], by showing:

Theorem 11.1. Let dμ(θ) = e^{L(θ)} \frac{dθ}{2π} where e^{iθ} ↦ L(θ) is analytic in a neighborhood of ∂𝔻. Then

lim_{n→∞} [log D_{n}(dμ) − (n + 1)L̂_{0}] = ∑_{k=1}^{∞} k|L̂_{k}|^{2}  (11.1)

Proof. Without loss, we suppose ∫ dμ(θ) = 1. We claim first that (under the analyticity assumption)

lim_{n→∞} (n + 1)[log ∥Φ_{n+1}∥^{2} − L̂_{0}] = 0  (11.2)

For in terms of the Verblunsky coefficients,

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

(by (2.11)), while (4.1) and (2.14) say that

eL̂_{0} = ∏_{j=0}^{∞} (1 − |α_{j}|^{2})

Thus

∥Φ_{n+1}∥^{2}e^{−L̂_{0}} = ∏_{j=n+1}^{∞} (1 − |α_{j}|^{2})^{−1}
= 1 + O(e^{−An/2})

by (5.17). This proves (11.2).

Thus, by (10.5), (11.1) is equivalent to there being a choice w_{t} with

− ∫_{0}^{1} dt ∫ (\frac{d}{dt} log w_{t}) \frac{∂}{∂r} |φ_{n+1}^{∗}(re^{iθ}; dμ_{t})|^{2}|_{r=1} dμ_{t}(θ) → ∑_{k=1}^{∞} k|L̂_{k}|^{2}

(11.3)

Define for 0 ≤ t ≤ 1,

c_{t} = log[∫ e^{tL(θ)} \frac{dθ}{2π} ]  (11.4)

and

w_{t}(θ) = exp(tL(θ) − c_{t})  (11.5)

so ∫ dμ_{t} = 1 also. Thus, the Szegő function for μ_{t} is given by

log D_{t}(z) = t log D(z) − \frac{1}{2} c_{t}  (11.6)
= t ∑_{k=1}^{∞} z^{k}L̂_{k} + \frac{1}{2} tL̂_{0} − \frac{1}{2} c_{t}  (11.7)

It follows that D_{t}(z) is analytic in a t-independent disk, and then the Verblunsky coefficients obey

|α_{n}(dμ_{t})| ≤ Ce^{−An/2}

uniformly in t ∈ [0, 1], and then, from the proof of Theorem 5.4, that φ_{n}^{∗}(z, dμ_{t}) → D_{t}(z)^{−1} uniformly in t ∈ [0, 1] and z in {z | |z| ≤ e^{A/4}}. Thus, the \frac{∂}{∂r} derivative in the integral on the left side of (11.3) converges to a \frac{∂}{∂r} derivative of D_{t}.

We also write

\frac{∂}{∂r} |φ_{n+1}^{∗}|^{2} = 2|φ_{n+1}^{∗}|^{2} Re \frac{∂}{∂r} log(φ_{n+1}^{∗}) → −2|D_{t}|^{−2} Re(t \frac{∂}{∂r} log D)

(11.8)

and we can use |D_{t}|^{−2} dμ_{t} = \frac{dθ}{2π}.

By (11.5),

\frac{d}{dt} log w_{t} = L(θ) − \frac{d}{dt} c_{t}  (11.9)

Since the \frac{d}{dt}c_{t} is θ-independent and by using (9.7), we see the \frac{d}{dt}c_{t} can be integrated to c_{1} − c_{0} = 0. Thus, in (11.3) we can replace \frac{d}{dt} log w_{t} by L(θ). The result is that

LHS of (11.3) → ∫_{0}^{1} dt ∫ 2t L(θ) Re[\frac{∂}{∂r} log D] \frac{dθ}{2π}  (11.10)

The only t-dependence is the 2t and ∫_{0}^{1} 2t dt = 1, so

RHS of (11.10) = ∫ [ ∑_{k=−∞}^{∞} L̂_{k}e^{ikθ}] Re( ∑_{k=1}^{∞} kL̂_{k}e^{ikθ}) \frac{dθ}{2π}

= ∫ ( ∑_{k=−∞}^{∞} L̂_{k}e^{ikθ})[\frac{1}{2} ∑_{k=−∞}^{∞} |k|L̂_{k}e^{ikθ}] \frac{dθ}{2π}

= \frac{1}{2} ∑_{k=−∞}^{∞} |k|¯L̂_{k}L̂_{k} = ∑_{1}^{∞} k|L̂_{k}|^{2}

as claimed. □

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