REGULARITY AND THE CESÀRO–NEVAI CLASS

BARRY SIMON

Abstract. We consider OPRL and OPUC with measures regular in the sense of Ullman–Stahl–Totik and prove consequences on the Jacobi parameters or Verblunsky coefficients. For example, regularity on [−2, 2] implies lim_{N→∞} N^{−1}[∑_{n=1}^{N}(a_{n}−1)^{2}+b_{n}^{2}] = 0.

1. Introduction and Background

This paper concerns the general theory of orthogonal polynomials on the real line, OPRL (see [26, 1, 8, 23]), and the unit circle, OPUC (see [26, 9, 18, 19]). Ullman [27] introduced the notion of regular measure on [−2, 2] (he used [−1, 1]; we use the normalization more common in the spectral theory literature): a measure, dμ, on ℝ with

supp(dμ) = [−2, 2]  (1.1)

and ({a_{n}, b_{n}}_{n=1}^{∞} are the Jacobi parameters of dμ)

lim_{n→∞}(a_{1} … a_{n})^{1/n} = 1  (1.2)

Here we will look at the larger class with (1.1) replaced by

σ_{ess}(dμ) = [−2, 2]  (1.3)

(i.e., supp(dμ) is [−2, 2] plus a countable set whose only limit points are a subset of {±2}).

Our goal is to explore what restrictions regularity places on the Jacobi parameters. At first sight, one might think (1.2) is the only restriction but, in fact, the combination of both (1.2) and (1.3) is quite strong. This should not be unexpected. After all, it is well known (going back at least to Nevai [15]; see also [19, Sect. 13.3]) that (1.1) plus lim inf(a_{1} … a_{n}) > 0 implies

Date: November 5, 2007.

2000 Mathematics Subject Classification. 05E35, 47B39.

Key words and phrases. Orthogonal polynomial, regular measure.

Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125.

E-mail: bsimon@caltech.edu. Supported in part by NSF grants DMS–0140592 and DMS-0652919.

To be submitted to J. Approx. Theory.

∑_{n=1}^{∞}(a_{n} − 1)^{2} + b_{n}^{2} < ∞  (1.4)

One can use variational principles to deduce some restrictions on the a’s and b’s. For example, picking φ_{n} to be the vector in ℓ^{2}({1, 2, … })

φ_{n,j} = \begin{cases}\frac{1}{\sqrt{n}} & j ≤ n \\ 0 & j ≥ n + 1 \end{cases}  (1.5)

and using the Jacobi matrix

J = (\begin{matrix}b_{1} & a_{1} & 0 & 0 & ⋯ \\ a_{1} & b_{2} & a_{2} & 0 & ⋯ \\ 0 & a_{2} & b_{3} & a_{3} & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix})  (1.6)

one sees, for example, that (1.3) implies (see also Theorem 1.2 below)

b_{n} ≡ 0 ⇒ lim sup_{n→∞} \frac{1}{n} ∑_{j=1}^{n−1} a_{j} ≤ 1  (1.7)

a_{n} ≡ 1 ⇒ lim_{n→∞} \frac{1}{n} ∑_{j=1}^{n} b_{j} = 0  (1.8)

In fact, we will prove much more:

Theorem 1.1. If μ obeys (1.3) and (1.2), then

lim_{n→∞} \frac{1}{n} ∑_{j=1}^{n}(|a_{j} − 1| + |b_{j}|) = 0  (1.9)

Following the terminology for the OPUC analog of this in Golinskii– Khrushchev [10], we call (1.9) the Cesàro–Nevai condition and {a_{j}, b_{j}}_{j=1}^{∞} obeying (1.9) the Cesàro–Nevai class. It, of course, contains the Nevai class (named after [15]) where |a_{j} − 1| + |b_{j}| → 0.

Noting that supp(dμ) bounded implies

A = sup_{n}(|a_{n} − 1| + |b_{n}|) < ∞  (1.10)

and that, by the Schwarz inequality,

(\frac{1}{n} ∑_{j=1}^{n} |a_{j} − 1| + |b_{j}|)^{2} ≤ \frac{2}{n} ∑_{j=1}^{n}(a_{j} − 1)^{2} + (b_{j})^{2}

≤ 2A \frac{1}{n} ∑_{j=1}^{n}(|a_{j} − 1| + |b_{j}|)  (1.11)

we see

(1.9) ⇔ \frac{1}{n} ∑_{j=1}^{n}(a_{j} − 1)^{2} + (b_{j})^{2} → 0  (1.12)

While Theorem 1.1 has a lot of information, it is not the whole story. For example, if a_{n} ≡ 1, then by the same variational principle, for any j_{k} → ∞,

\frac{1}{n} ∑_{j_{k}}^{j_{k}+n} b_{j} → 0

It would be interesting to see what else can be said.

A major theme we explore is what can be said if [−2, 2] is replaced by a more general set, 𝔢. In Section 5, we define Nevai and CN classes for finite gap sets 𝔢 and state a general conjecture which we prove in the special case where d has p components, each of harmonic measure 1/p, that is, the periodic case with all gaps open.

In Section 3, we extend Theorem 1.1 to the matrix OPRL case on [−2, 2], and in Section 6, we use this and ideas of Damanik–Killip– Simon [6] to obtain the result in the last paragraph. Section 4 has a brief discussion of OPUC.

We should close by noting an earlier result of Máté–Nevai–Totik [14] related to—but neither stronger nor weaker than—Theorem 1.2:

Theorem 1.2 ([14]). Suppose μ obeys (1.1) and a_{n} → 1 as n → ∞. Then b_{n} → 0 as n → ∞.

Remarks. 1. μ need only obey (1.3) as seen by Remark 3 below. 2. This strengthens (1.8). There is no similar strengthening of (1.7).

3. One way of seeing this is as follows: By Last–Simon [13], any right limit of a J obeying (1.3) has σ(J_{r}) ⊂ [−2, 2] and has a_{n} ≡ 1. By a result of Killip–Simon [11] (see also [3, 4, 5]), any such J_{r} has b_{n} ≡ 0. By compactness, b_{n} → 0 for the original J.

It is a pleasure to thank Paul Nevai and Christian Remling for useful correspondence.

2. OPRL on [−2, 2]

Our goal here is to prove Theorem 1.1.

Lemma 2.1. Suppose a_{n} ∈ (0, ∞) is a sequence so that

(i)  lim inf_{N→∞} (a_{1} … a_{N})^{1/N} ≥ 1  (2.1)
(ii)  lim sup_{N→∞} \frac{1}{N} ∑_{n=1}^{N} a_{n}^{2} ≤ 1  (2.2)

Then, as N → ∞,

\frac{1}{N} ∑_{n=1}^{N} a_{n} → 1  \frac{1}{N} ∑_{n=1}^{N} a_{n}^{2} → 1  (2.3)

\frac{1}{N} ∑_{n=1}^{N}(a_{n} − 1)^{2} → 0  (2.4)

Proof. By concavity of log x for all x ∈ (0, ∞),

log x ≤ x − 1

so (2.1) implies

lim inf_{N→∞} \frac{1}{N} ∑_{n=1}^{N} a_{n} ≥ 1 + lim inf_{N→∞} log(a_{1} … a_{N})^{1/N} ≥ 1

Thus,

lim sup \frac{1}{N} ∑_{n=1}^{N}(a_{n} − 1)^{2} ≤ 1 − 2 + 1 = 0

so (2.4) holds.

By the Schwarz inequality,

\frac{1}{N} ∑_{n=1}^{N} |a_{n} − 1| ≤ [ \frac{1}{N} ∑_{n=1}^{N}(a_{n} − 1)^{2}]^{1/2} → 0

which implies the first limit in (2.3). (2.4) and that limit imply (2.3). □

Proposition 2.2. Let {a_{n}, b_{n}}_{n=1}^{∞} be the Jacobi parameters for a regular measure with σ_{ess}(J) = [−2, 2]. Then

\frac{1}{N} [2 ∑_{n=1}^{N−1} a_{n}^{2} + ∑_{n=1}^{N} b_{n}^{2}] → 2  (2.5)

as N → ∞.

Proof. Let {x_{j}^{(N)}}_{j=1}^{N} be the zeros of the OPRL p_{N}(x) associated to the Jacobi parameters. Let dρ_{[−2,2]} be the equilibrium measures for [−2, 2] (see [12, 16, 22] for potential theory notions). Since regularity implies that the density of zeros converges to dρ_{[−2,2]} (see [25, 22]), we have

\frac{1}{N} ∑_{n=1}^{N}(x_{j}^{(N)})^{2} → ∫ x^{2}dρ_{[−2,2]}(x)  (2.6)

Since {x_{j}^{(N)}}_{j=1}^{N} are the eigenvalues of the finite Jacobi matrix

J_{N;F} = (\begin{matrix}b_{1} & a_{1} \\ a_{1} & b_{2} & ... \\ & ... & ... & ... \\ & & ... & b_{N−1} & a_{N−1} \\ & & & a_{N−1} & b_{N} \end{matrix})  (2.7)

we have that

LHS of (2.6) = \frac{1}{N} Tr(J_{N;F}^{2} )
= \frac{1}{N} [ ∑_{n=1}^{N} b_{n}^{2} + 2 ∑_{n=1}^{N−1} a_{n}^{2}]  (2.8)

Thus (2.5) is equivalent to

∫ x^{2}dρ_{[−2,2]}(x) = 2  (2.9)

This can be seen either by using the explicit formula for dρ_{[−2,2]} (and ∫_{0}^{π}(2 cos θ)^{2} \frac{dθ}{π} = 2) or by considering the special case a_{n} ≡ 1, b_{n} ≡ 0 since the limit in (2.5) is the same for all regular J’s. □

Proof of Theorem 1.1. By regularity,

lim inf_{N→∞} (a_{1} … a_{N})^{1/N} = 1  (2.10)

and by Proposition 2.2 and

lim sup \frac{1}{N} ∑_{n=1}^{N−1} a_{n}^{2} ≤ 1  (2.11)

By Lemma 2.1, we have (2.4), and this and (2.5) imply

\frac{1}{N} ∑_{n=1}^{N} b_{n}^{2} → 0  (2.12)

By(1.12), we get (1.9). □

3. MOPRL on [−2, 2]

In this section, both for its own sake and because of the application in Section 6, we want to consider matrix-valued measures for [−2, 2]. Our reference for the associated OPRL will be [7] which discusses regular measures. ℓ is fixed and finite, and we have a block Jacobi matrix of the form

J = (\begin{matrix}B_{1} & A_{1} & 0 & ⋯ \\ A_{1}^{†} & B_{2} & A_{2} & ⋯ \\ 0 & A_{2}^{†} & B_{3} & ⋯ \\ ⋮ & ⋮ & ⋮ & ... \end{matrix})  (3.1)

where A_{j} and B_{j} are ℓ × ℓ matrices and ^{†} is Hermitian conjugate. One requires each A_{j} is nonsingular.

Two sets of Jacobi parameters, {A_{j}, B_{j}}_{j=1}^{∞} and {Ã_{j}, B̃_{j}}_{j=1}^{∞}, are called equivalent if there exist ℓ × ℓ unitaries, u_{1} ≡ 1, u_{2}, u_{3}, … so that

B̃_{j} = u_{j}^{†}B_{j}u_{j}  Ã_{j} = u_{j}^{†}A_{j}u_{j+1}  (3.2)

It is known (see [7, Thm. 2.11]) that there is a one-one correspondence between nontrivial ℓ×ℓ matrix-valued measures, dμ, (with nontriviality suitably defined) and equivalence classes of Jacobi parameters.

{A_{j}, B_{j}}_{j=1}^{∞} is called type 1 (resp. type 3) if each A_{j} is positive (resp. A_{j} is lower triangular and positive on diagonal). Moreover ([7, Thm. 2.8]), each equivalence class has exactly one representative of type 1 and one of type 3. An ℓ × ℓ matrix-valued measure is called regular ([7, Ch. 5]) for [−2, 2] if and only if

σ_{ess}(dμ) = [−2, 2]  (3.3)

and

[ ∏_{n=1}^{N} |det(A_{n})|]^{1/N} → 1  (3.4)

Our basic result for such MOPRL is:

Theorem 3.1. If {A_{n}, B_{n}}_{n=1}^{∞} are the Jacobi parameters for an ℓ × ℓ matrix-valued measure which is regular for [−2, 2] and are either of type 1 or type 3, then

\frac{1}{N} ∑_{n=1}^{N} ∥A_{n} − 1∥ + ∥B_{n}∥ → 1  (3.5)

Remark. (3.5) does not hold for all equivalent {Ã_{n}, B̃_{n}}_{n=1}^{∞}, but it is easy to see that

\frac{1}{N} ∑_{n=1}^{N} ∥A_{n}^{∗}A_{n} − 1∥ + ∥B_{n}∥ → 0  (3.6)

is equivalence class independent and implied by (3.5) for the type 1 or type 3 representative.

Proof. We consider type 3 first. By Thm. 5.2 of [7], the density of zeros converges to the equilibrium measure, so analogously to (2.5),

\frac{1}{Nℓ}[2 ∑_{n=1}^{N−1} Tr(A_{n}^{∗}A_{n}) + ∑_{n=1}^{N} Tr(B_{n}^{∗}B_{n})] → 2  (3.7)

In the type 3 case, (3.4) says

[ ∏_{n=1}^{N} ∏_{j=1}^{ℓ}(A_{n})_{jj}]^{1/Nℓ} → 1  (3.8)

so as in the proof of Theorem 1.1, we find

\frac{1}{Nℓ} ∑_{n=1}^{N} ∑_{j=1}^{ℓ} |(A_{n})_{jj} − 1|^{2} → 0  (3.9)

and then that

\frac{1}{N} ∑_{n=1}^{N} Tr(B_{n}^{∗}B_{n}) → 0  (3.10)

and

\frac{1}{N} ∑_{n=1}^{N}|Tr(A_{n}^{∗}A − 1)| → 0  (3.11)

In the type 1 case, one uses the inequality

A ≥ 0 ⇒ det(A) ≤ ∏_{j=1}^{ℓ} A_{jj}  (3.12)

(see Simon [20, Cor. 8.10]) and the fact that Lemma 2.1 only requires an inequality in (2.1). □

4. OPUC

Here we will prove two results about OPUC. Recall dμ on ∂𝔻 with σ_{ess}(dμ) = 𝔢 is called regular if and only if

lim_{N→∞}( ∏_{j=0}^{N−1} ρ_{j})^{1/N} = C(𝔢)  (4.1)

the capacity of 𝔢 where ρ_{j} = (1−|α_{j}|^{2})^{1/2} and {α_{j}}_{j=0}^{∞} are the Verblunsky coefficients.

Theorem 4.1. Let dμ be a measure of ∂𝔻 regular for 𝔢 = ∂𝔻. Then, as N → ∞,

\frac{1}{N} ∑_{j=0}^{N−1} |α_{j}| → 0  (4.2)

Remark. This is the original CN class of [10].

Proof. C(∂𝔻) = 1, so by Lemma 2.1 and

\frac{1}{N} ∑_{j=0}^{N−1} ρ_{j}^{2} ≤ 1  (4.3)

we obtain

\frac{1}{N} ∑_{j=0}^{N−1}(1 − ρ_{j}^{2}) → 0  (4.4)

which implies (4.2) by the Schwarz inequality. □

For a ∈ (0, 1), let Γ_{a} be the arc

{z ∈ ∂𝔻 | z = e^{iθ}, π ≥ |θ| > 2 arcsin(a)}  (4.5)

which has capacity a. Then

Theorem 4.2. Let dμ be a measure on ∂𝔻, regular for 𝔢 = Γ_{a}. Then as N → ∞,

(a)  \frac{1}{N} ∑_{j=0}^{N−1} (|α_{j}| − a)^{2} → 0  (4.6)

(b)  \frac{1}{N} ∑_{j=0}^{N−1} |α_{j+1} − α_{j}|^{2} → 0  (4.7)

For any k,

(c)  \frac{1}{N} ∑_{j=0}^{N−1} min_{θ}( ∑_{ℓ=1}^{k} |α_{j+ℓ} − ae^{iθ}|^{2}) → 0  (4.8)

Remark. The isospectral torus for Γ_{a} is exactly {{α_{j} ≡ ae^{iθ}}}_{θ∈[0,2π)}, that is, the constant sequence of Verblunsky coefficients, so (c) involves an approach to an isospectral torus.

Proof. By regularity and the connection between zeros of POPUC and eigenvalues of finite CMV matrices as defined in [21], one has that

\frac{1}{N} ∑_{n=0}^{N−1} −ᾱ_{n+1}α_{n} → c  (4.9)

where c is the first moment of the equilibrium measure, that is, ∫ z dρ_{Γ_{a}}(z). Specializing to the case α_{n} ≡ a to evaluate c, we see that

\frac{1}{N} ∑_{n=0}^{N−1} ᾱ_{n+1}α_{n} → a^{2}  (4.10)

On the other hand, by regularity,

\frac{1}{N} ∑_{n=0}^{N−1} log(1 − |α_{n}|^{2}) → log(1 − |a|^{2})  (4.11)

and by concavity of log,

log(1 − x) − log(1 − |a|^{2}) ≤ \frac{1}{1 − |a|^{2}} (|a|^{2} − x)

so

lim inf \frac{1}{N} ∑_{n=0}^{N−1} (|a|^{2} − |α_{n}|^{2}) ≥ 0  (4.12)

and thus,

lim sup \frac{1}{N} ∑_{n=0}^{N−1} |α_{n}|^{2} ≤ a^{2}  (4.13)

By (4.10) and the Schwarz inequality,

lim inf \frac{1}{N} ∑_{n=0}^{N−1} |α_{n}|^{2} ≥ a^{2}  (4.14)

so

\frac{1}{N} ∑_{n=0}^{N−1} |α_{n}|^{2} → a^{2}  (4.15)

For y ∈ (0, 1] (by Taylor’s theorem with remainder and max_{(0,1]} \frac{d^{2}}{dy^{2}} log(y) = −1),

log(y) − log(1 − |a|^{2}) − [\frac{y − (1 − |a|^{2})}{1 − |a|^{2}}] ≤ −\frac{1}{2} (y − (1 − |a|^{2}))^{2}

so (4.11) and (4.15) imply

\frac{1}{N} ∑_{n=0}^{N−1} ||α|^{2} − a^{2}| → 0

which implies (4.7).

(4.15) and (4.10) imply (4.7). Finally, (4.6) and (4.7) imply (4.8). □

5. The Nevai and CN Classes

In [19], I proposed using approach to an isospectral torus as a replacement for the Nevai class when [−2, 2] is replaced by the spectrum of a periodic Jacobi matrix. This idea was then implemented in Last–Simon [13] and Damanik–Killip–Simon [6]. The latter discussed extending this notion to a general finite gap set, and this idea was further developed in Remling [17].

𝔢 will denote a finite gap set, that is,

𝔢 = [α_{1}, β_{1}] ∪ [α_{2}, β_{2}] ∪ ⋯ ∪ [α_{ℓ+1}, β_{ℓ+1}] ⊂ ℝ  (5.1)

where

α_{1} < β_{1} < α_{2} < β_{2} < ⋯ < β_{ℓ+1}  (5.2)

Given such a set, there is a natural torus, T_{𝔢}, of almost periodic Jacobi matrices, discussed, for example, in [24, 2]; it can be described [17] as the restriction to {1, … } of the two-sided reflectionless Jacobi matrices, J^{♯}, with

σ(J^{♯}) = 𝔢  (5.3)

All J ∈ T_{𝔢} have

σ_{ess}(J) = 𝔢  (5.4)

T_{𝔢} is a torus in the uniform topology as well as the product topology.

Given a pair of bounded Jacobi parameters, J = {a_{n}, b_{n}}_{n=1}^{∞}, J̃ = {ã_{n}, b̃_{n}}_{n=1}^{∞}, define d_{m}(J, J̃) by

d_{m}(J, J̃) = ∑_{k=0}^{∞} e^{−|k|}(|a_{m+k} − ã_{m+k}| + |b_{m+k} − b̃_{m+k}|)  (5.5)

If T_{𝔢} is an isospectral torus, let

d_{m}(J, T_{𝔢}) = inf_{J̃∈T_{𝔢}} d_{m}(J, J̃)  (5.6)

Definition. If 𝔢 ⊂ ℝ is a finite gap set, we define the Nevai class N(𝔢) to be those J’s with

lim_{m→∞} d_{m}(J, T_{𝔢}) = 0  (5.7)

This is equivalent (by compactness) to saying all the right limits of J lie in T_{𝔢}.

It is a theorem of Last–Simon [13] that

J ∈ N(𝔢) ⇒ σ_{ess}(J) = 𝔢  (5.8)

and of Remling [17] that

σ_{ess}(J) = σ_{ac}(J) = 𝔢 ⇒ J ∈ N(𝔢)  (5.9)

It is not hard to see that

J ∈ N(𝔢) ⇒ J is regular for 𝔢

Analogously, we define the Cesàro–Nevai class, CN(𝔢), as those J with

\frac{1}{N} ∑_{m=1}^{N} d_{m}(J, T_{𝔢}) → 0  (5.10)

A main conjecture we make in this note is:

Conjecture 5.1. If J is regular for 𝔢, that is, σ_{ess}(J) = 𝔢, and (a_{1} … a_{N})^{1/N} → C(𝔢), then J ∈ CN(𝔢).

In the next section, we will prove this for a special class of 𝔢’s. Of course, we make a similar conjecture for finite gap OPUC. Indeed, Theorem 4.2 is the case of OPUC with one gap!

6. Generic Periodic Spectrum

Our goal is to prove:

Theorem 6.1. Let 𝔢 be a finite gap set so that each [α_{j}, β_{j}] has harmonic measure (ℓ + 1)^{−1} (equivalently, there is a J_{0} with period ℓ + 1 so 𝔢 = σ_{ess}(J_{0})). Let J be a Jacobi matrix with regular spectral measure so that σ_{ess}(J) = 𝔢. Then J ∈ CN(𝔢).

We use p for ℓ + 1, the period of J_{0}.

Following [6], we exploit Δ_{J_{0}}(J) where Δ_{J_{0}} is the discriminant [6, 23] of J_{0}, a polynomial of degree p. If J is any Jacobi matrix, Δ_{J_{0}}(J) is a p × p block Jacobi matrix of type 3. We use A_{J_{0},k}(J) and B_{J_{0},k}(J) to denote the p × p matrix blocks in Δ(J).

[6] proved the following theorem (their Thm. 11.12); here ∥·∥ is the Hilbert–Schmidt norm.

Theorem 6.2 ([6]). Fix J_{0} periodic with σ_{ess}(J_{0}) = 𝔢 and J an arbitrary bounded Jacobi matrix. Then

∑_{k=1}^{∞} ∥A_{J_{0},k}(J) − 1∥_{2}^{2} + ∥B_{J_{0},k}(J)∥_{2}^{2} < ∞  (6.1)

if and only if

∑_{k=1}^{∞} d_{k}(J, T_{𝔢})^{2} < ∞  (6.2)

Because this comparison is local, the exact same proof shows

Theorem 6.3. Let J_{0} be periodic with σ_{ess}(J_{0}) = 𝔢 and J an arbitrary Jacobi matrix. Then

lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} [∥A_{J_{0},k}(J) − 1∥_{2}^{2} + ∥B_{J_{0},k}(J)∥^{2}] = 0  (6.3)

if and only if

lim_{N→∞} \frac{1}{N} ∑_{k=1}^{N} d_{k}(J, T_{𝔢})^{2} → 0  (6.4)

With this and Theorem 3.1, we can prove Theorem 6.1.

Proof of Theorem 6.1. Δ(x) has the form

Δ(x) = (a_{0,1} a_{0,2} … a_{0,p})^{−1}x^{p} + lower order

so the diagonal matrix elements of Δ(J) are

\frac{a_{j}a_{j+1} … a_{j+p}}{a_{0,j} … a_{0,j+p}} ≡ α_{jj}

If J is regular, for 𝔢,

[\frac{a_{1} … a_{n}}{C(𝔢)^{n}} ]^{1/n} → 1  (6.5)

But a_{0,j} … a_{0,j+p} = C(𝔢)^{p} for periodic Jacobi matrices, so (6.5) implies

(α_{11}α_{22} … α_{nn})^{1/n} → 1

which implies that Δ(J) is a regular block Jacobi matrix.

By Theorem 3.1, (6.3) holds and so, by Theorem 6.3, we have the CN(𝔢) condition (6.4). □

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