HIGHER-ORDER SZEGŐ THEOREMS WITH TWO SINGULAR POINTS

BARRY SIMON^{1} AND ANDREJ ZLATOŠ^{2}

Abstract. We consider probability measures, dμ = w(θ)\frac{dθ}{2π} +dμ_{s}, on the unit circle, ∂𝔻, with Verblunsky coefficients, {α_{j}}_{j=0}^{∞}. We prove for θ_{1} ≠ θ_{2} in [0, 2π) and (δβ)_{j} = β_{j+1} that

∫ [1 − cos(θ − θ_{1})][1 − cos(θ − θ_{2})] log w(θ) \frac{dθ}{2π} > −∞

if and only if

∑_{j=0}^{∞} |{(δ − e^{−iθ_{2}})(δ − e^{−iθ_{1}})α}_{j}|^{2} + |α_{j}|^{4} < ∞

We also prove that

∫ (1 − cos θ)^{2} log w(θ) \frac{dθ}{2π} > −∞

if and only if

∑_{j=0}^{∞}|α_{j+2} − 2α_{j+1} + α_{j}|^{2} + |α_{j}|^{6} < ∞

1. Introduction

This paper is a contribution to the theory of orthogonal polynomials on the unit cirle (OPUC); see [6, 15, 16, 18] for background. Throughout, dμ will be a nontrivial probability measure on the unit circle, ∂𝔻, in ℂ, which we suppose has the form

dμ = w(θ) \frac{dθ}{2π} + dμ_{s}  (1.1)

where dμ_{s} is singular with respect to Lebesgue measure dθ on ∂𝔻.

Date: September 16, 2004. 1 Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125.

E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592.

2 Department of Mathematics, University of Wisconsin, Madison, WI 53706. E-mail: andrej@math.wisc.edu.

The Carathéodory and Schur functions, F and f, associated to dμ are given for z ∈ 𝔻 by

F (z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ)  (1.2)
= \frac{1 + zf(z)}{1 − zf(z)}  (1.3)

The Verblunsky coefficients {α_{j}}_{j=0}^{∞} can be defined inductively by the Schur algorithm

f(z) = \frac{α_{0} + zf_{1}(z)}{1 + zᾱ_{0}f_{1}(z)}  (1.4)

which defines α_{0} ∈ 𝔻 and f_{1}. Iterating gives α_{1}, α_{2}, … and f_{2}, f_{3}, . . .. That α_{j} ∈ 𝔻 (rather than just 𝔻̄) follows from the assumption that dμ is nontrivial, that is, has infinite support so f is not a finite Blaschke product. Actually, (1.4) defines what are usually called Schur parameters; the Verblunsky coefficients are defined by a recursion relation on the orthogonal polynomials. The equality of these recursion coefficients and the Schur parameters of (1.4) is a theorem of Geronimus [5]; see [15]. We will use the definition in (1.4).

The most famous result in OPUC is Szegő’s theorem which, in Verblunsky’s format [19], says

log( ∏_{j=0}^{∞}(1 − |α_{j}|^{2})) = ∫ log(w(θ)) \frac{dθ}{2π}  (1.5)

In this expression, both sides are nonpositive (since |α_{j}| < 1, and Jensen’s inequality implies ∫ log(w(θ)) \frac{dθ}{2π} ≤ log(∫ w(θ)\frac{dθ}{2π}) ≤ log(μ(∂𝔻))). Moreover, (1.5) includes the statement that both sides are finite (resp. −∞) simultaneously. Thus (1.5) implies a spectral theory result.

Theorem 1.1.

∫ log(w(θ)) \frac{dθ}{2π} > −∞ ⇔ ∑_{j=0}^{∞} |α_{j}|^{2} < ∞  (1.6)

This form of the theorem has caused considerable recent interest due to work of Deift-Killip [1] and Killip-Simon [7] which motivated a raft of papers [2, 8, 9, 10, 11, 14, 17, 20].

In [15, Section 2.8], Simon found a higher-order analog to (1.6) that allows log(w(θ)) to be singular at a single point:

Theorem 1.2.

∫ (1−cos θ) log(w(θ)) \frac{dθ}{2π} > −∞ ⇔ ∑_{j=0}^{∞} |α_{j+1}−α_{j}|^{2}+|α_{j}|^{4} < ∞ (1.7)

This result allows a single singular point of order 1 in log(w(θ)) at θ = 0. By a simple rotation argument [15], if cos(θ) is replaced by cos(θ − θ_{1}), |α_{j+1} − α_{j}|^{2} is replaced by |α_{j+1} − e^{−iθ_{1}}α_{j}|^{2}.

Our goal in this paper is to analyze two singularities or a single double singularity. We will prove that

Theorem 1.3. For θ_{1} ≠ θ_{2},

∫ (1 − cos(θ − θ_{1}))(1 − cos(θ − θ_{2})) log(w(θ)) \frac{dθ}{2π} > −∞

⇔ ∑_{j=0}^{∞}|{ (δ − e^{−iθ_{2}})(δ − e^{−iθ_{1}})α}_{j}|^{2} + |α_{j}|^{4} < ∞

(1.8)

In this theorem, δ is the operator on sequences

(δα)_{j} = α_{j+1}  (1.9)

We will also prove a result for θ_{1} = θ_{2}.

Theorem 1.4.

∫ (1−cos θ)^{2} log(w(θ)) \frac{dθ}{2π} > −∞ ⇔ ∑_{j=0}^{∞} |α_{j+2}−2α_{j+1}+α_{j}|^{2}+|α_{j}|^{6} < ∞

(1.10)

Again, one can replace cos(θ) by cos(θ − θ_{1}) if {α_{j+2} − 2α_{j+1} + α_{j}}_{j} is replaced by {(δ − e^{−iθ_{1}})^{2}α}_{j}.

Given the form of these theorems, it is natural to conjecture the situation for arbitrarily many singularities:

Conjecture 1.5. For {θ_{k}}_{k=1}^{ℓ} distinct in [0, 2π),

∫ ∏_{k=1}^{ℓ}(1 − cos(θ − θ_{k}))^{m_{k}} log(w(θ)) \frac{dθ}{2π} > −∞

⇔ ∑_{k=0}^{∞}|{ ∏_{k=1}^{ℓ}[δ − e^{−iθ_{k}}]^{m_{k}}α}_{j}|^{2} + |α_{j}|^{2 max(m_{k})+2} < ∞

(1.11)

Independently of our work, Denisov-Kupin [3] have found conditions on the α’s equivalent to the left side of (1.11) being finite. However, their conditions are complicated and even for the case ∑_{k=1}^{ℓ} m_{k} = 2, it is not clear they are equivalent to the ones we have in Theorems 1.3 and 1.4 (although they must be!).

In Section 2, we review the features we need of the relative Szegő function which will play a critical role in our proofs, and we compute its first two Taylor coefficients. In Section 3, we prove Theorem 1.3 in the special case θ_{1} = 0, θ_{2} = π, and in Section 4, we prove Theorem 1.4. With these two warmups done, we turn to the general result, Theorem 1.3, in Section 5. The details of this are sufficiently messy that we do not think this direct approach is likely to yield our conjecture.

We would like to thank S. Denisov for telling us of his work [3].

2. The Relative Szegő Function

In Section 2.9 of [15], Simon introduced the relative Szegő function, defined by

(δ_{0}D)(z) = \frac{1 − ᾱ_{0}f(z)}{ρ_{0}} \frac{1 − zf_{1}(z)}{1 − zf(z)}  (2.1)

where

ρ_{k} = (1 − |α_{k}|^{2})^{1/2}  (2.2)

and f, f_{1} are given by (1.3) and (1.4).

The key property of δ_{0}D we will need and the reason it was introduced is

Theorem 2.1 ([15, Theorem 2.9.3]). Let dμ_{1} be the measure whose Verblunsky coefficients are (α_{1}, α_{2}, …). Let w be given by (1.1) and w_{1} by

dμ_{1} = w_{1}(θ) \frac{dθ}{2π} + dμ_{1,s}  (2.3)

Suppose w(θ) ≠ 0 for a.e. e^{iθ} in ∂𝔻. Then the same is true for w_{1} and

(δ_{0}D)(z) = exp(\frac{1}{4π} ∫ \frac{e^{iθ} + z}{e^{iθ} − z} log(\frac{w(θ)}{w_{1}(θ)})dθ)  (2.4)

As in [7, 14, 17], this is the basis for step-by-step sum rules, as we will see.

To prove Theorems 1.3 and 1.4, we will need to start with computing the first three Taylor coefficients of log((δ_{0}D)(z)).

Theorem 2.2. We have that

log(δ_{0}D(z)) = A_{0} + A_{1}z + A_{2}z^{2} + O(z^{3})  (2.5)

where

A_{0} = log ρ_{0}  (2.6)
A_{1} = α_{0} − α_{1} − ᾱ_{0}α_{1}  (2.7)

A_{2} = \frac{1}{2} α_{0}^{2} − \frac{1}{2} α_{1}^{2} + α_{1} − α_{2} − α_{1}|α_{0}|^{2} + α_{2}|α_{1}|^{2} − ᾱ_{0}α_{2}ρ_{1}^{2} + \frac{1}{2} ᾱ_{0}^{2}α_{1}^{2} (2.8)

Proof. f_{2}(0) = α_{2}, so

f_{1} = \frac{zf_{2} + α_{1}}{1 + zᾱ_{1}f_{2}} = α_{1} + zα_{2}ρ_{1}^{2} + O(z^{2})

Thus

f = \frac{zf_{1} + α_{0}}{1 + zᾱ_{1}f_{1}} = α_{0} + zα_{1}ρ_{0}^{2} + z^{2}ρ_{0}^{2}(α_{2}ρ_{1}^{2} − ᾱ_{0}α_{1}^{2}) + O(z^{3})

Plugging these into (2.1) yields the required Taylor coefficients. □

Remarks. 1. Denisov-Kupin [3] do what is essentially the same calculation using the CMV matrix.

2. (3.2) and (3.3) below show that (2.4) implies

∫ log(\frac{w(θ)}{w_{1}(θ)})\frac{dθ}{2π} = 2A_{0}  (2.9)

∫ log(\frac{w(θ)}{w_{1}(θ)})e^{−imθ} \frac{dθ}{2π} = \begin{cases}A_{m} & m = 1, 2 \\ Ā_{−m} & m = −1, −2 \end{cases}  (2.10)

3. Singularities at Antipodal Points

As a warmup, in this section we prove the following, which is Theorem 1.3 for θ_{1} = 0, θ_{2} = π. By the remark after Theorem 1.2 this also gives the result for any antipodal θ_{1} and θ_{2}.

Theorem 3.1.

∫ (1−cos^{2}(θ)) log w(θ) \frac{dθ}{2π} > −∞ ⇔ ∑_{j=0}^{∞} |α_{j+2}−α_{j}|^{2}+|α_{j}|^{4} < ∞ (3.1)

Remark. Let α_{j} be given and let β_{j} be the sequence (α_{0}, 0, α_{1}, 0, α_{2}, 0, …). Then (see Example 1.6.14 of [15]), w^{(β)}(θ) = \frac{1}{2}w^{(α)}(\frac{1}{2}θ) and the RHS of (3.1) for β = the RHS of (1.7) for α. Thus (3.1) for β is (1.7) for α. This shows, in particular, that if a result like (3.1) holds, it must involve |α_{j}|^{4}, rather than, say, |α_{j}|^{6}.

We begin by noting that if Q(θ) is real and

Q(θ) = ∑_{n=−∞}^{∞} b_{n}e^{inθ}  (3.2)

then

∫ \frac{e^{iθ} + z}{e^{iθ} − z} Q(θ)\frac{dθ}{2π} = b_{0} + 2 ∑_{n=1}^{∞} b_{n}z^{n}  (3.3)

since (e^{iθ} + z)/(e^{iθ} − z) = 1 + 2 ∑_{n=1}^{∞} z^{n}e^{−inθ}. Thus, by (2.9), (2.10), and

1 − cos^{2}(θ) = \frac{1}{4} (2 − e^{2iθ} − e^{−2iθ})  (3.4)

we have

∫ (1 − cos^{2}(θ)) log(\frac{w(θ)}{w_{1}(θ)})\frac{dθ}{2π} = A_{0} − \frac{1}{2} Re(A_{2})  (3.5)

with A_{0} given by (2.6) and A_{2} by (2.8).

Lemma 3.2. We have that

A_{0} − \frac{1}{2} Re(A_{2}) = B_{0} + C_{0} + D_{0} + F_{0} − F_{1} + G_{0} − G_{2}  (3.6)

where

B_{j} = \frac{1}{2} [log(1 − |α_{j}|^{2}) + |α_{j}|^{2} + \frac{1}{2}|α_{j}|^{4}]  (3.7)
C_{j} = −\frac{1}{4} (1 − |α_{j+1}|^{2})|α_{j} − α_{j+2}|^{2}  (3.8)
D_{j} = −\frac{1}{8} (|α_{j+1}^{2} + α_{j}^{2}|^{2} + 4|α_{j}α_{j+1}|^{2})  (3.9)
F_{j} = −\frac{1}{2} Re(\frac{1}{2} α_{j}^{2} + α_{j+1} − α_{j+1}|α|^{2}) + \frac{1}{4} |α_{j+1}|^{2}|α_{j}|^{2} − \frac{1}{8} |α_{j}|^{4} (3.10)
G_{j} = −\frac{1}{4} |α_{j}|^{2}

Remark. (3.5)/(3.6) is thus the step-by-step sum rule in the spirit of [7, 14, 17].

Proof. This is a straightforward but tedious calculation. The first term in B_{0} is just A_{0} (since log ρ_{j} = \frac{1}{2} log(1 − |α_{j}|^{2})). A_{2} is responsible for the Re(·) terms in F_{0} − F_{1} and the cross-terms in |α_{j} − α_{j+2}|^{2} and |α_{j+1}^{2} + α_{j}^{2}|^{2}. The |α_{j}|^{2} + |α_{j+2}|^{2} term in C_{0} is turned into 2|α_{j}|^{2} by G_{0} − G_{2}, and then cancelled by the |α_{j}|^{2} term in B_{0}. Similarly, the |α_{j}|^{4}+|α_{j+1}|^{4} in D_{0} (after adding the |α_{j}|^{4} terms in F_{0}−F_{1}) cancels the |α_{j}|^{4} term in B_{0}. Finally, the |α_{j+1}|^{2}(|α_{j}|^{2} + |α_{j+2}|^{2}) term in C_{0} (after being turned into 2|α_{j+1}|^{2}|α_{j}|^{2} by the |α_{j+1}|^{2}|α_{j}|^{2} term in F_{0} − F_{1}) cancels the 4|α_{j}α_{j+1}|^{2} term in D_{0}. □

By iterating (3.5)/(3.6) and noting the cancellations from the telescoping F_{j} − F_{j+1} and G_{j} − G_{j+2} yields

\begin{aligned}∫ (1− & cos^{2}(θ)) log(\frac{w(θ)}{w_{2m}(θ)})\frac{dθ}{2π} \\ & = F_{0} − F_{2m} + G_{0} + G_{1} − G_{2m} − G_{2m+1} + ∑_{j=0}^{2m−1}(B_{j} + C_{j} + D_{j}) \end{aligned} ; (3.11)

As a final preliminary, we need,

Lemma 3.3. (i) |F_{j}| ≤ \frac{13}{8}; |G_{j}| ≤ \frac{1}{4} (ii) |α_{j}| < \frac{1}{2} ⇒ c_{1}|α_{j}|^{6} ≤ −B_{j} ≤ c_{2}|α_{j}|^{6} for some c_{2} > c_{1} > 0. (iii) |α_{j+1}|^{4} + |α_{j}|^{4} ≤ −8D_{j} ≤ 4(|α_{j+1}|^{4} + |α_{j}|^{4})

Proof. (i) follows from |α_{j}| ≤ 1, (ii) from − log(1 − x) = ∑_{j=1}^{∞} x^{j}/j, and (iii) by noting that 2 Re(α_{j}^{2}α_{j+1}^{2}) + 2|α_{j}^{2}α_{j+1}^{2}| ≥ 0 and repeated use of |xy| ≤ \frac{1}{2}(|x|^{2} + |y|^{2}). □

Proof of Theorem 3.1. We follow the strategy of [7] as modified by [17]. Suppose first that the RHS of (3.1) holds. Let w^{(n)} be the weight for the n^{th} Bernstein-Szegő approximation with Verblunsky coefficients (α_{0}, α_{1}, …, α_{n−1}, 0, …, 0, …). By (3.11) and (w^{(n)})_{2m} ≡ 1 for large m,

∫ (1−cos^{2}(θ)) log(w^{(n)}(θ)) \frac{dθ}{2π} = F_{0}^{(n)}+G_{0}^{(n)}+G_{1}^{(n)}+∑_{j=0}^{n−1}(B_{j}^{(n)}+C_{j}^{(n)}+D_{j}^{(n)})

so, by Lemma 3.3, |α_{j}|^{6} ≤ |α_{j}|^{4} → 0, and RHS of (3.1),

inf_{n}[∫ (1 − cos^{2}(θ)) log(w^{(n)}(θ))\frac{dθ}{2π} ] > −∞  (3.12)

Up to a constant, ∫ (1 − cos^{2}(θ)) log w(θ)\frac{dθ}{2π} is an entropy and so upper semicontinuous [7]. Thus (3.12) implies

∫ (1 − cos^{2}(θ)) log w(θ) \frac{dθ}{2π} > −∞  (3.13)

Conversely, suppose (3.13) holds. Since ∫ (1−cos^{2}(θ)) log(w_{2m}(θ))\frac{dθ}{2π} is an entropy up to a constant, it is bounded above [7], and so the left side of (3.11) is bounded below as m varies.

Since F and G are bounded and B, C, D are negative, we conclude

∑_{j=0}^{∞} −(B_{j} + C_{j} + D_{j}) < ∞

Since ∑(−D_{j}) < ∞, Lemma 3.3 implies ∑|α_{j}|^{4} < ∞. This implies α_{j} → 0, so ∑(−C_{j}) < ∞ implies ∑|α_{j} − α_{j+2}|^{2} < ∞. □

4. Singularity of Order 2

Our goal here is to prove Theorem 1.4. Since

(1 − cos θ)^{2} = \frac{1}{4} (2 − e^{iθ} − e^{−iθ})^{2}
= \frac{3}{2} − e^{iθ} − e^{−iθ} + \frac{1}{4} e^{2iθ} + \frac{1}{4} e^{−2iθ}

we see, by (2.9)/(2.10) that

∫ log(\frac{w(θ)}{w_{1}(θ)})(1 − cos θ)^{2} \frac{dθ}{2π} = 3A_{0} − 2 Re(A_{1}) + \frac{1}{2} Re(A_{2}) (4.1)

with A_{0}, A_{1}, A_{2} given by (2.6)–(2.8).

Lemma 4.1. The RHS of (4.1) = H_{0} + I_{0} + J_{0} + K_{0} − K_{1} + L_{0} − L_{2} where

H_{j} = \frac{3}{2} [log(1 − |α_{j}|^{2}) + |α_{j}|^{2}]
I_{j} = −\frac{1}{4} |α_{j+2} − 2α_{j+1} + α_{j}|^{2}
J_{j} = \frac{1}{4} (α_{j}ᾱ_{j+2} + ᾱ_{j}α_{j+2})|α_{j+1}|^{2} + \frac{1}{8} (α_{j}^{2}ᾱ_{j+1}^{2} + ᾱ_{j}^{2}α_{j+1})
K_{j} = −2 Re(α_{j}) + \frac{1}{4} Re(α_{j}^{2})
+ \frac{1}{2} Re(α_{j+1}) − \frac{1}{2} Re(α_{j+1}|α_{j}|^{2}) + Re[ᾱ_{j+1}α_{j}] − |α_{j}|^{2}
L_{j} = −\frac{1}{4} |α_{j}|^{2}

Proof. The non-cross-terms in I_{0} are

−\frac{1}{4} (|α_{2}|^{2} + 4|α_{1}|^{2} + |α_{0}|^{2}) = −\frac{3}{2} |α_{0}|^{2} + (|α_{0}|^{2} − |α_{1}|^{2}) + \frac{1}{4} (|α_{0}|^{2} − |α_{2}|^{2})

which cancel the |α_{0}|^{2} term in H_{0}, the final |α_{j}|^{2} term in K_{0} − K_{1}, and the L_{0} − L_{2} term.

The cross-terms in I_{0} are

−\frac{1}{2} Re(ᾱ_{2}α_{0}) − Re(ᾱ_{2}α_{1} + ᾱ_{1}α_{0})

= −\frac{1}{2} Re(ᾱ_{2}α_{0}) + 2 Re(ᾱ_{0}α_{1}) − Re(ᾱ_{0}α_{1}) + Re(ᾱ_{1}α_{2})

The first term comes from the piece of \frac{1}{2} Re(A_{2}) (since ᾱ_{0}α_{2}ρ_{1}^{2} = ᾱ_{0}α_{2}(1 − |α_{1}|^{2}), the second from the last term in −2 Re(A_{1}), and the last two are cancelled by the Re(ᾱ_{j+1}α_{j}) term in K_{0} − K_{1}.

The α_{0} − α_{1} term in A_{0} leads to the first term in K_{1} − K_{0}. The first term in J_{0} comes from the second half of ᾱ_{0}α_{2}ρ_{1}^{2} = ᾱ_{0}α_{2} − ᾱ_{0}α_{2}|α_{1}|^{2} (the first half in this expression gave a cross-term in I_{j}). The second term in J_{0} is the \frac{1}{2}ᾱ_{0}^{2}α_{1}^{2} term in A_{2}.

The remaining terms in A_{2} give precisely the remaining terms in K_{0} − K_{1}. □

Lemma 4.2. The RHS of (4.1) = H̃_{0} + Ĩ_{0} + J̃_{0} + K̃_{0} − K̃_{1} + L̃_{0} − L̃_{2}, where

H̃_{j} = \frac{3}{2} [log(1 − |α_{j}|^{2}) + |α_{j}|^{2} + \frac{1}{2}|α_{j}|^{4}]
Ĩ_{j} = I_{j}
J̃_{j} = −\frac{1}{4} |α_{j+1}|^{2}|α_{j} − α_{j+2}|^{2} − \frac{1}{8} |α_{j+1}^{2} − α_{j}^{2}|^{2} − \frac{1}{4} (|α_{j+1}|^{2} − |α_{j}|^{2})^{2}
K̃_{j} = K_{j} − \frac{3}{8} |α_{j}|^{2} − \frac{1}{4} |α_{j+1}|^{2}|α_{j}|^{2}
L̃_{j} = L_{j}

Proof. The non-cross-terms in the last two terms in J̃_{0} give

−\frac{3}{8} (|α_{0}|^{4} + |α_{1}|^{4}) = −\frac{3}{4} |α_{0}|^{4} + \frac{3}{8}(|α_{0}|^{4} − |α_{1}|^{4})

The first term cancels the H̃_{0} − H_{0} term, and the second, the first term in (K_{0} − K̃_{0}) − (K_{1} − K̃_{1}).

The cross-term in −\frac{1}{4}(|α_{j+1}|^{2} − |α_{j}|^{2})^{2} and the non-cross-terms in −\frac{1}{4}|α_{j+1}|^{2}|α_{j} − α_{j+2}|^{2} combine to −\frac{1}{4}|α_{j+2}|^{2}|α_{j+1}|^{2} + \frac{1}{4}|α_{j+1}|^{2}|α_{j}|^{2} and are cancelled by the second term in (K_{0} − K̃_{0}) − (K_{1} − K̃_{1}). The cross-term in −\frac{1}{8}|α_{j+1}^{2} − α_{j}^{2}|^{2} is the second term in J_{0} and finally, the cross-term in −\frac{1}{4}|α_{j+1}|^{2}|α_{j} − α_{j+2}|^{2} is the first term in J_{0}. □

Lemma 4.3. (i) |K̃_{j}| ≤ \frac{47}{8}; |L̃_{j}| ≤ \frac{1}{4} (ii) |α_{j}| < \frac{1}{2} ⇒ d_{1}|α_{j}|^{6} ≤ −H̃_{j} ≤ d_{2}|α_{j}|^{6} for some d_{2} > d_{1} > 0. (iii) J̃_{j} ≤ 0 (iv) ∑_{j=0}^{∞}(−Ĩ_{j}) + |α_{j}|^{6} < ∞ ⇒ ∑_{j=0}^{∞}|α_{j+1} − α_{j}|^{3} < ∞

(v) ∑_{j=0}^{∞}(−Ĩ_{j}) + |α_{j}|^{6} < ∞ ⇒ ∑_{j=0}^{∞}(−J̃_{j}) < ∞

Remark. (iv) is essentially a discrete version of the inequality of Gagliardo [4] and Nirenberg [12].

Proof. (i) follows from |α_{j}| < 1, (ii) is just (ii) of Lemma 3.3 (since H̃_{j} = 3B_{j}), and (iii) is trivial.

To prove (iv), we let δ be given by (1.9) and let

∂ = δ − 1  (4.2)

so since δ^{∗} = δ^{−1} (δ is unitary on ℓ^{2}), we have

∂^{∗} = δ^{∗} − 1 = −δ^{−1}∂ = −δ^{∗}∂  (4.3)

As a result, if α is a finite sequence, then

∑_{n} |(∂α)_{n}|^{3} = ∑_{n}(∂α)_{n}(∂ᾱ)_{n}|∂α|_{n}

= − ∑_{n}(δα)_{n}[∂{(∂ᾱ)|∂α|}]_{n}  (4.4)

Moreover, we have a discrete Leibnitz rule,

∂(fg) = (δf)(δg) − fg
= (δf)∂g + (∂f)g  (4.5)

and since |a − b| ≥ |a| − |b| by the triangle inequality,

|∂|f|| ≤ |∂f|  (4.6)

which is a discrete Kato inequality.

By (4.5),

∂{(∂ᾱ)|∂α|} = [δ(∂ᾱ)]∂|∂α| + (∂^{2}ᾱ)|∂α|

so, by (4.6),

|∂{(∂ᾱ)|∂α|}| ≤ |∂^{2}α| |δ(∂ᾱ)| + |∂^{2}α| |∂α|

Using Hölder’s inequality with \frac{1}{6} + \frac{1}{2} + \frac{1}{3} = 1 and (4.4), we get

∥∂α∥_{3}^{3} ≤ 2∥α∥_{6}∥∂^{2}α∥_{2}∥∂α∥_{3}

(because ∥δα∥_{p} = ∥α∥_{p}), so

∑_{n} |(∂α)_{n}|^{3} ≤ 2^{3/2}( ∑_{n} |α_{n}|^{6})^{1/4}( ∑_{n} |(∂^{2}α)_{n}|^{2})^{3/4}  (4.7)

Having proven (4.7) for α’s of finite support, we get it for any α with the right side finite since ∑_{n}|α_{n}|^{6} < ∞ implies α_{n} → 0, which allows one to cut off α at N and take N → ∞ in (4.7). But (4.7) implies (iv).

To prove (v), we control the individual terms in ∑(−J̃_{j}). First,

∥|α|^{2}|δ^{2}α − α|^{2}∥_{1} ≤ ∥ α^{2}∥_{3} ∥|δ^{2}α − α|^{2}∥_{3/2}

(by Hölder’s inequality with \frac{1}{3} + \frac{2}{3} = 1)

≤ 4∥α∥_{6}^{2} ∥∂α∥_{3}^{2} < ∞

(by first using ∥δ^{2}α − α∥_{3} ≤ 2∥∂α∥_{3} and then (iv)). Next,

|α_{j+1}^{2} − α_{j}^{2}|^{2} ≤ (|α_{j+1}| + |α_{j+1}|)^{2}|α_{j+1} − α_{j}|^{2}

can be controlled as the first term was and the final term is controlled in the same way since |α_{j+1}|^{2} − |α_{j}|^{2} ≤ |α_{j+1}^{2} − α_{j}^{2}|. □

Proof of Theorem 1.4. Suppose first that the right-hand side of (1.10) holds, that is, α ∈ ℓ^{6} and ∂^{2}α ∈ ℓ^{2}. Iterate n times (4.1)/Lemma 4.2 for the n^{th} Bernstein-Szegő approximation (with weight w^{(n)}) to obtain

inf_{n} [∫ (1 − cos θ)^{2} log(w^{(n)}(θ)) \frac{dθ}{2π} ] > −∞

since the left side is just

inf_{n} [K̃_{0}^{(n)} + L̃_{0}^{(n)} + L̃_{1}^{(n)} + ∑_{j=0}^{n−1}(H̃_{j}^{(n)} + Ĩ_{j}^{(n)} + J̃_{j}^{(n)})]

which is finite by Lemma 4.3 and the hypothesis. Again we have that ∫ (1 − cos θ)^{2} log w(θ)\frac{dθ}{2π} is an entropy up to a constant and so upper semicontinuous. Thus RHS of (1.10) ⇒ LHS of (1.10).

For the opposite direction, as in the last section, we use iterated (4.1)/Lemma 4.2 plus the fact that ∫ (1 − cos θ)^{2} log(w_{2m}(θ))\frac{dθ}{2π} is bounded from above to conclude

∑_{j=0}^{∞} −(H̃_{j} + Ĩ_{j} + J̃_{j}) < ∞

Since each is positive, ∑(−H̃_{j}) < ∞, which implies ∑|α_{j}|^{6} < ∞ by (ii) of Lemma 4.3, and ∑_{j=0}^{∞}(−Ĩ_{j}) < ∞, which implies ∂^{2}α ∈ ℓ^{2}. □

5. The General Case

Finally, we turn to the general case of Theorem 1.3, and we define

I_{m} ≡ ∫ [1 − cos(θ − θ_{1})][1 − cos(θ − θ_{2})] log(\frac{w(θ)}{w_{m}(θ)})\frac{dθ}{2π} (5.1)

Using (2.9) and (2.10), we obtain

I_{1} = \frac{4 + e^{i(θ_{1}−θ_{2})} + e^{−i(θ_{1}−θ_{2})}}{4} A_{0}−Re [(e^{iθ_{1}}+e^{iθ_{2}})A_{1}]+ \frac{1}{2} Re [e^{i(θ_{1}+θ_{2})}A_{2}]

(5.2)

The situation is now somewhat more complicated than in the previous sections and it will be more convenient to work with I_{m} from the start, only keeping track of the essential components of the sums (analogs of ∑(B_{j} + C_{j} + D_{j}) and ∑(H̃_{j} + Ĩ_{j} + J̃_{j}) above) and ignore the ones that are always bounded and hence irrelevant for us (analogs of F_{0} −F_{1} +G_{0} +G_{1} −G_{m} −G_{m+1} and K̃_{0} −K̃_{m} +L̃_{0} +L̃_{1} −L̃_{m} +L̃_{m+1}).

Hence substituting (2.6)–(2.8) in (5.2) and iterating, we obtain

I_{m} = C_{α,m} + \frac{4 + e^{i(θ_{1}−θ_{2})} + e^{−i(θ_{1}−θ_{2})}}{4} ∑_{j=0}^{m−1} log(1 − |α_{j}|^{2})

+ ∑_{j=0}^{m−1} Re \{(e^{iθ_{1}} + e^{iθ_{2}})α_{j+1}ᾱ_{j} − \frac{1}{2}e^{i(θ_{1}+θ_{2})}[α_{j+2}ᾱ_{j}(1 − |α_{j+1}|^{2}) − \frac{1}{2}α_{j+1}^{2}ᾱ_{j}^{2}]\}

where

C_{α,m} ≡ − Re [(e^{iθ_{1}} + e^{iθ_{2}})(α_{0} − α_{m})]

+ \frac{1}{2} Re [e^{i(θ_{1}+θ_{2})}(\frac{1}{2}α_{0}^{2} − \frac{1}{2}α_{m}^{2} + α_{1} − α_{m+1} − α_{1}|α_{0}^{2}| + α_{m+1}|α_{m}|^{2})]

We let

β_{j} ≡ α_{j}e^{i(θ_{1}+θ_{2})j/2}

and

a ≡ \frac{1}{2}(e^{i(θ_{1}−θ_{2})/2} + e^{−i(θ_{1}−θ_{2})/2}) ∈ (−1, 1)

We will assume a ≠ 0 since the case when θ_{1} and θ_{2} are antipodal follows from Theorem 3.1. With C_{β,m} ≡ C_{α,m} and all the sums taken from 0 to m − 1, the above becomes

I_{m} =C_{β,m} + (\frac{1}{2} + a^{2}) ∑ log(1 − |β_{j}|^{2}) + a ∑ [β_{j+1}β̄_{j} + β̄_{j+1}β_{j}]
− \frac{1}{4} ∑ [β_{j+2}β̄_{j}(1 − |β_{j+1}|^{2}) + β̄_{j+2}β_{j}(1 − |β_{j+1}|^{2})]
+ \frac{1}{8} ∑ [β_{j+1}^{2}β̄_{j}^{2} + β̄_{j+1}^{2}β_{j}^{2}]  (5.3)

In the following manipulations with the sums, we will use C_{β,m} as a general pool/depository of terms that will be added/left over in order to keep all the sums from 0 to m − 1. Its value will therefore change along the argument, but it will always depend on a few β_{j}’s with j close to 0 or m only (i.e., it will gather all the “irrelevant” terms) and will always be bounded by a universal constant.

Lemma 5.1. With C_{β,m} universally bounded, we have

I_{m} =C_{β,m} + (\frac{1}{2} + a^{2}) ∑ [ log(1 − |β_{j}|^{2}) + |β_{j}|^{2} + \frac{1}{2}|β_{j}|^{4}]
− \frac{1}{4} ∑(1 − |β_{j+1}|^{2})|β_{j+2} − 2aβ_{j+1} + β_{j}|^{2}

− \frac{1}{4} ∑ |β_{j+1}|^{2}|β_{j+2} − 2aβ_{j+1}|^{2} − \frac{1}{4} ∑ |β_{j+1}|^{2}|β_{j} − 2aβ_{j+1}|^{2}

− \frac{1}{8} ∑ |β_{j+1}^{2} − β_{j}^{2}|^{2} + \frac{1}{2}a^{2} ∑ |β_{j}|^{4}  (5.4)

with all the sums taken from 0 to m − 1.

Remarks. 1. This enables us to prove the “⇐” part of (1.8) (even if a = 0) since

|{(δ − e^{−iθ_{2}})(δ − e^{−iθ_{1}})α}_{j}| = |β_{j+2} − 2aβ_{j+1} + β_{j}|  (5.5)

But to prove the other implication, we first need to deal with the last sum in (5.4), which has the “wrong” sign.

2. Note that we actually did not need to exclude the case a = 0 since then the last sum in (5.4) vanishes and an examination of (5.4) shows that lim_{m→∞} I_{m} > −∞ if and only if the RHS of (1.8) holds. An argument from the proofs of Theorems 1.3 and 1.4 then gives the “⇒” part of (1.8).

Proof. Multiplying out the terms in the second, third, and fourth sums of (5.4) and after obvious cancellations, we are left with

−\frac{1}{4} ∑ [|β_{j+1}|^{2}(4a^{2}|β_{j+1}|^{2} − β_{j+2}β̄_{j} − β̄_{j+2}β_{j}) + |β_{j+2} − 2aβ_{j+1} + β_{j}|^{2}]

But this is just

−\frac{1}{4} ∑ [|β_{j+2}|^{2} + 4a^{2}|β_{j+1}|^{2} + |β_{j}|^{2} + 4a^{2}|β_{j+1}|^{4}]  (5.6)

plus the second and third sums in (5.3), the latter written as

\frac{1}{2}a ∑[β_{j+2}β̄_{j+1} + β̄_{j+2}β_{j+1} + β_{j+1}β̄_{j} + β̄_{j+1}β_{j}]

(with C_{β,m} keeping the change). Adding the fifth and sixth sums in (5.4) to (5.6) and subtracting the last sum in (5.3), we obtain

−\frac{1}{4} ∑(2 + 4a^{2})|β_{j}|^{2} − \frac{1}{8} ∑(2 + 4a^{2})|β_{j}|^{4}

(again replacing all |β_{j+1}| and |β_{j+2}| by |β_{j}| and adding the difference to C_{β,m}). But this together with the first sum in (5.4) gives exactly the first sum in (5.3). □

We define

γ_{j} ≡ β_{j+2} − 2aβ_{j+1} + β_{j}

then the second, third, and fourth sums in (5.4) involve |γ_{j}|, |γ_{j} − β_{j}| and |γ_{j} − β_{j+2}|. Using |x − y|^{2} ≥ |x|^{2} + |y|^{2} − 2|x||y| for the last two, we obtain (with a new C_{β,m})

(−8)I_{m} ≥C_{β,m} + ∑ O(|β_{j}|^{6}) + ∑(2 + 2|β_{j+1}|^{2})|γ_{j}|^{2}
+ 4 ∑ |β_{j+1}|^{2}|β_{j}|^{2} − 4 ∑ |β_{j+1}|^{2}(|β_{j+2}| + |β_{j}|)|γ_{j}|
+ ∑ |β_{j+1}^{2} − β_{j}^{2}|^{2} − 4a^{2} ∑ |β_{j+1}|^{4}  (5.7)

since

log(1 − |β_{j}|^{2}) + |β_{j}|^{2} + \frac{1}{2}|β_{j}|^{4} = O(|β_{j}|^{6})

Next, we use −4xy ≥ −8x^{2} − \frac{1}{2}y^{2} with x = |β_{j+1}|^{2}(|β_{j+2}| + |β_{j}|) and y = |γ_{j}| to estimate the fourth sum by ∑ O(|β_{j}|^{6}) − \frac{1}{2} ∑ |γ_{j}|^{2}. Also,

−4a^{2} ∑ |β_{j+1}|^{4} = − ∑ |β_{j+1}|^{2}|β_{j+2} + β_{j} − γ_{j}|^{2}
≥ − ∑ |β_{j+1}|^{2}|β_{j+2} + β_{j}|^{2} − ∑ |β_{j+1}|^{2}|γ_{j}|^{2}
− 2 ∑ |β_{j+1}|^{2}|β_{j+2} + β_{j}||γ_{j}|
≥ C_{β,m} − 4 ∑ |β_{j+1}|^{2}|β_{j}|^{2} − ∑ |β_{j+1}|^{2}|γ_{j}|^{2}
− ∑ O(|β_{j}|^{6}) − \frac{1}{4} ∑ |γ_{j}|^{2}

again using −2xy ≥ −4x^{2} − \frac{1}{4}y^{2}. Plugging these into (5.7), we have

(−8)I_{m} ≥ C_{β,m} + ∑ O(|β_{j}|^{6}) + ∑ (\frac{5}{4} + |β_{j+1}|^{2})|γ_{j}|^{2} + ∑ |β_{j+1}^{2} − β_{j}^{2}|^{2}

The last sum is just ∑ \frac{1}{2}(|β_{j+2}^{2} − β_{j+1}^{2}|^{2} + |β_{j+1}^{2} − β_{j}^{2}|^{2}) plus a piece that goes into C_{β,m}. Letting ε ≡ \frac{1}{3} min\{2|a|, 2 − 2|a|\} > 0, we obtain

|β_{j+1}|^{2}|γ_{j}|^{2} + \frac{1}{2}|β_{j+2}^{2} − β_{j+1}^{2}|^{2} + \frac{1}{2}|β_{j+1}^{2} − β_{j}^{2}|^{2} ≥ \frac{1}{2}ε^{4}|β_{j+1}|^{4}

Indeed, if the third term is smaller than \frac{1}{2}ε^{4}|β_{j+1}|^{4}, then |β_{j} − β_{j+1}| or |β_{j} + β_{j+1}| is less than ε|β_{j+1}|, and similarly for the second term. But then |β_{j+2} + β_{j}|/|β_{j+1}| ∈ [0, 2ε) ∪ (2 − 2ε, 2 + 2ε) and so |γ_{j}|/|β_{j+1}| ≥ min{2|a| − 2ε, 2 − 2ε − 2|a|} ≥ ε, meaning that the first term is at least ε^{2}|β_{j+1}|^{4}. So finally,

(−8)I_{m} ≥ C_{β,m} + ∑ O(|β_{j}|^{6}) + ∑ |γ_{j}|^{2} + \frac{1}{2}ε^{4} ∑ |β_{j}|^{4}

that is (by (5.5) and the definition of β_{j}, γ_{j}),

I_{m} ≤ C_{α,m}+∑ O(|α_{j}|^{6})−\frac{1}{8} ∑ |\{(δ−e^{−iθ_{2}})(δ−e^{−iθ_{1}})α\}_{j}|^{2}−\frac{1}{16}ε^{4} ∑ |α_{j}|^{4} (5.8)

Proof of Theorem 1.3. If the RHS of (1.8) holds, then the RHS of (5.4) for the n^{th} Bernstein-Szegő approximation (with m ≥ n) is bounded (in n), and so

inf_{n} [ ∫ [1 − cos(θ − θ_{1})][1 − cos(θ − θ_{2})] log(w^{(n)}(θ))\frac{dθ}{2π} ] > −∞

By upper semicontinuity of the above integral (which is again an entropy up to a constant), we obtain the LHS of (1.8).

Conversely, assume the LHS of (1.8) holds. Then the essential support of w all of ∂𝔻, and so by Rakhmanov’s theorem [13], |α_{j}| → 0.

Hence, starting from some j, we have O(|α_{j}|^{6}) ≤ \frac{1}{32}ε^{4}|α_{j}|^{4} and so

I_{m} ≤ D_{α,m} − \frac{1}{8} ∑ |\{(δ − e^{−iθ_{2}})(δ − e^{−iθ_{1}})α\}_{j}|^{2} − \frac{1}{32}ε^{4} ∑ |α_{j}|^{4} (5.9)

for large m and some bounded (in m) D_{α,m}. As in the previous sections, ∫ [1 − cos(θ − θ_{1})][1 − cos(θ − θ_{2})] log(w_{m}(θ))\frac{dθ}{2π} is bounded above, and so I_{m} is bounded below by the hypothesis. (5.9) then shows that the RHS of (1.8) holds. □

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