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Zeros of non-Baxter paraorthogonal polynomials on the unit circle

We provide leading order asymptotics for the size of the gap in the zeros around 1 of paraothogonal polynomials on the unit circle whose Verblunsky coefficients satisfy a slow decay condition and are inside the interval (-1,0). We also include related results that impose less restrictive conditions on the Verblunsky coefficients.


Introduction
In this paper, we will let µ be a probability measure on the unit circle ∂D with infinite support. In the Hilbert Space L 2 (∂D, dµ), the set {1, z, z 2 , z 3 , . . .} is a linearly independent set so by applying Gram-Schmidt orthogonalization, we can obtain the orthonormal polynomials {ϕ n (z)} n≥0 where ϕ n (z) is a polynomial of degree exactly n with positive leading coefficient and we have the relation ϕ n , ϕ m := ∂D ϕ n (z)ϕ m (z)dµ(z) = δ mn .
One useful tool for studying the orthogonal polynomials of a measure µ is the CMV matrix, which is a semi-infinite unitary matrix whose spectral measure is µ and is such that the characteristic polynomial of the upper-left n × n block is exactly Φ n (z) (see Chapter 4 in [9]). However, since the upper-left n × n block is not unitary, one often introduces the so-called paraorthogonal polynomials {Φ (βn) n (z)} n≥1 defined by Φ (βn) n+1 (z) = zΦ n (z) − β n Φ * n (z) where β n ∈ ∂D. These polynomials and their zeros have been studied extensively in recent years (see for example [5,6,11,12,13,14] and references therein).
It is well known (see [14]) that all of the zeros of Φ n } ordered counterclockwise starting from 1. Our investigation is motivated by a picture appearing in Chapter 8 in [9] and also in [8]. It is also closely related to Conjecture D in Section 4 of [11]. In [11], the author points out that if the Verblunsky coefficients have a slow decay property (we will make this more precise later), then the zeros of the orthogonal polynomials appear to exhibit clock spacing on the unit circle away from θ = 0, while there is a large gap around θ = 0. The main result of this note is to give leading order asymptotics for the size of the gap in the zeros of the paraorthogonal polynomials around θ = 0 when β n ∈ (π/2 + , 3π/2 − ) for some > 0 and all n. Along the way we will develop an idea suggested in [8]. In that paper, the authors deal with an explicit measure instead of explicit Verblunsky coefficients but observe the same phenomenon of a gap in the zeros of the orthogonal polynomials. The authors then provide numerical evidence to suggest that the Verblunsky coefficients do satisfy a slow decay condition and attribute the gap in the zeros to the suggestion that the polynomial Φ n "thinks" it belongs to a measure with constant Verblunsky coefficients because the decay is so slow. Our methods will allow us to make this idea more precise and put a condition on the sequence {α n } n≥0 that causes the zeros to exhibit a gap property as in the constant Verblunsky coefficient case.
We will often be interested in situations where the Verblunsky coefficients decay monotonically (in absolute value) to 0. However, this condition is more restrictive than is necessary for our results. We will say that a sequence of Verblunsky coefficients {α n } n≥0 has slow decay controlled by f if there is a non-positive function f (n) so that i. either f (n) = −Cn −b for some b ∈ (0, 1) and C > 0 or lim n→∞ √ n f 2 (n − √ n) = ∞, ii. f (n) < f (n + 1) < 0 for all n > 0 and lim n→∞ f (n) = 0, iii. lim n→∞ when n ≤ m and m is sufficiently large. Of course the two cases in condition (i) are not disjoint, but this will only create some flexibility in how we proceed in proofs. Following the terminology in Section 5 of [11], orthogonal polynomials with 1 Verblunsky coefficients are said to be in the Baxter Class, so we are interested only in the non-Baxter case (results concerning Baxter class Verblunsky coefficients can be found in [11]). Our main result is the following Theorem 1.1. Let k ∈ N,˜ > 0, and ξ ∈ ( π 2 +˜ , 3π 2 −˜ ) be fixed and let β = e iξ . Let {α n } n≥0 be a sequence of real Verblunsky coefficients having slow decay controlled by f . For Similarly, we have In [8], the authors provide numerical and strong heuristic evidence for Theorem 1.1 in the context of a closely related problem. In Section 4 we will prove a generalization of Theorem 1.1 that imposes less restrictive conditions on the Verblunsky coefficients. As a special case of Theorem 1.1, we get the following corollary -conjectured in [11] -which is of special interest. Corollary 1.2. If α n = −(n + 2) −b with 0 < b < 1 then the size of the gap in the zeros of Φ (−1) n (z) around z = 1 is 4 arcsin |α n | to leading order. The neighboring zero spacings are of smaller order.
The latter possibility in condition (i) on f means we can also apply Theorem 1.1 to the sequence of Verblunsky coefficients given for example by α n = −1 log(n+3) . Following the wisdom that an equality is two inequalities, we will prove Theorem 1.1 in two steps; first we prove a lower bound on the size of the gap and then an upper bound. We will prove the lower bound by using the Szegő recursion to control the phases of the Blaschke product Φ n (e iθ )/Φ * n (e iθ ) when θ is sufficiently close to 0. To prove the upper bound, we will find a large collection of mutually orthogonal approximate eigenvectors for a unitary matrix whose characteristic polynomial is Φ (−1) n (z) and apply the variational principle. The result for β = −1 will follow from the interlacing of zeros for distinct values of β (see Theorem 1.3 in [12]).
In the next section we will prove the key lemma for the lower bound and explain how it relates to the ideas put forward in [8]. In Section 3, we will prove Theorem 1.1. In Section 4 we will discuss modifications that we can make to the hypotheses of Theorem 1.1 and state a related theorem for the orthogonal polynomials on the unit circle that can be proved using our techniques. Along the way we will add our result to several previously known results to explain several features of Figure 1.
Acknowledgements. It is a pleasure to thank Barry Simon for introducing me to this problem and for much useful discussion.

The Key Lemma
We recall a version of the Szegő recursion used by Khruschev and given in [10] as equation . Although Theorem 1.1 requires the Verblunsky coefficients to be real, we will prove our lower bound under more general hypotheses (see also Corollary 2.2 below).
In order to state the key lemma, we first define the region P α for α ∈ {z : |z + 1 2 | < 1 2 } as all z ∈ D with real part less than or equal to Re(α). Also, following the notation from [9], we define θ α = 2 arcsin |α|. The following lemma is the key to the lower bound.
Proof. The proof is by induction. By equation 10.8 in [6], the phase of the Blaschke product b n (e iθ ) increases with θ. Furthermore, if we prove that arg(b n (e iθα )) < π 2 − arcsin |α| then we will have also proven that arg(b n (e −iθα )) > − π 2 + arcsin |α| by symmetry. Then using the monotonicity of arg(b n (e iθ )) the lemma will follow.
Remark. Corollary 2.2 refines the estimate given by Theorem 3.4.7 in [9], which puts a bound on the number of pure points (under more general hypotheses). A related result can also be found in [7] for the special case when all the α j ∈ R where it is shown that 1 is not a pure point.
Corollary 2.2 makes precise the claim in [8] that the gap in the zeros results from the polynomials "thinking" they are in the constant Verblunsky coefficient case. Put differently, the orthogonal polynomial Φ n is the monic degree n orthogonal polynomial for some measure having exactly the same support as the spectral measure corresponding to constant Verblunsky coefficients (which we described in Section 1). The zeroes of Φ n therefore lie in the convex hull of this measure (this is by Fejer's Theorem, see Theorem 1.7.19 in [9]). However, since e iθ ∈ P α for all θ ∈ ( π 2 + arcsin |α|, 3π 2 − arcsin |α|), we can move α n−1 to such a point on ∂D inside P α and retain this restriction on the location of the zeros of Φ n . Therefore, by the Hurwitz Theorem we have a lower bound on the size of the gap for all β of the form considered in Theorem 1.1 when α is sufficiently small (i.e. M is sufficiently large).
In the next section we will complete the proof of the main theorem.

Finding The Zeros
To establish an upper bound on the distance from ζ n within a certain proximity of 1, it will suffice to find a unit vector ν n such that ||(C (n) β − 1)ν n || is small. We will define ν n for each n by j ∈ (γ n , n) even i(j − γ n )(n − j), j ∈ (γ n , n) odd 0, otherwise where γ n will depend on f .
Proof of Theorem 1.1. For each n > 0, let us define φ n = 2 arcsin |f (n)|. To obtain a lower bound on the distance, we notice that if θ ∈ (−φ M , φ M ) then by Lemma 2.1 the Blaschke product M has a zero precisely when this Blaschke product is equal toβ, which lies outside this arc for large M . Finally, we notice that since |f (n)| decays monotonically to 0, we have We see from equations (3.1) and (3.4) that A 1 (n)/ ν n 2 = o(n −2b ). We can bound A 2 (n) (from equation (3.2)) by where we evaluated the last sum using equation (3.5). Finally, we can bound A 3 (n) (see equation (3.3)) from above by for some constant K > 0. We can bound this further by eliminating the odd/even subscripts and then evaluate as before using equations (3.5) and (3.6) to see that Putting it all together, we conclude that as n → ∞. Therefore, if M is large enough we can set v(M ) = ν M / ν M and get the desired conclusion.
If f (n) is such that lim n→∞ √ n f 2 (n − √ n) = ∞ then we choose a different trial vector. In this case we set as n → ∞. Therefore, if M is large enough we can set v(M ) = υ M and get the desired conclusion. This completes the proof of the k = 1 case when β n ≡ −1 (the statement concerning ζ (M ) M follows by an obvious symmetry). The conclusion for all β in the desired range will follow from considering k > 1 and using the fact that zeros of Φ For k > 1 we provide the details for the case f (n) = −Cn −b with b < 1 and C > 0 since the other case is a nearly identical calculation. If we let u n = ν n / ν n then the calculation above actually shows that if a natural number p > 1 is fixed then We calculate has cardinality tending to infinity as M → ∞ (since m(M ) → ∞ as M → ∞). By similar reasoning, we have 1 = ω M 2 ≥ t∈X M ( ) | ω M , v t | 2 and so the set also has cardinality tending to infinity as M → ∞. Since > 0 can be chosen arbitrarily, we see that so we have demonstrated the existence of an approximate eigenvector for C It follows that C to the span of the known eigenvectors. By minimizing an expression of the form over all p ≤ m(M ) we get the desired conclusion for arbitrary k > 1 and β = −1. The conclusion for all β in the desired range follows by the interlacing of zeros. 2 If we combine our result with some previously established results, we can understand some prominent features of Figure 1. The picture suggests that for any fixed k ∈ N, as n → ∞ (as was conjectured in [11] when f (n) = −Cn −b ). This was proven in [6] if we order the {ζ (n) k } n k=1 counterclockwise starting from any e iθ = 1, that is, we have uniform clock spacing of the zeros away from 1. Our result proves only that the zero spacings closest to 1 (but outside the large gap) are o(n −b ) as n → ∞.

Extensions and Generalizations
Now we will examine ways in which we can tweak the hypotheses of Theorem 1.1 and obtain similar conclusions. Throughout this section, we will let k ∈ N, β ∈ ∂D, and δ > 0 be fixed as in the statement of Theorem 1.1.
For our first extension, we will relax the condition that α n ∈ R for all n. Notice that a priori the proof of Theorem 1.1 only proves that the k th closest zero of Φ in particular. We can in fact make the same conclusion by imposing a reality condition only on large blocks of the Verblunsky coefficients. The size of these blocks will depend on the decay of the Verblunsky coefficients, as can be seen from the proof of Theorem 1.1. with corresponding orthonormal polynomials p j (z) and let c j (z) = p j (z)/p * j (z). Now we define η n (θ) to be the phase of e iθ b n−1 (e iθ ) so that η n is strictly increasing (by equation 10.8 in [6]) and changes by 2πn as θ runs from 0 to 2π. Similarly, we define τ n (θ) to be the phase of e iθ c n−1 (e iθ ) (note that τ n (0) = 0 for all n). It follows from the formulas in Chapter 10.12 in [10] that η n (θ) = η n−1 (θ) + θ − 2 arg(1 − α n−2 e iη n−1 (θ) ) (4.1) and similarly for τ n (θ) (notice the similarity to Proposition 2.2 in [5]).
Remark. Theorem 4.2 is also related to Conjecture D in [11] and the analysis in [8].

Appendix
Here we present an alternate approach to finding the lower bound for the distance from the closest zero of Φ (−1) n (z) to 1 under the additional hypothesis that {α n } n≥0 is a sequence of real Verblunsky coefficients increasing monotonically to 0. This is a weaker result than what we obtained earlier, but the proof is very different. that is, the signs alternate along the three main diagonals and the first element along the main diagonal has the opposite sign of the first element of the off-diagonals. Then J is invertible.
Proof. We proceed by induction to prove that if n = 2m or 2m + 1 and J 1,1 > 0 then if m is even the determinant of J has the same sign as its (1, 1) element, while if m is odd then then the determinant of J has opposite sign as its (1, 1) element. In either case, this shows the determinant is non-zero so J is invertible. The claim is easily verified for m = 1, 2 so the base case is trivial. If S, T ⊆ {1, 2, 3, . . . , n}, then we denote by J Applying the induction hypothesis to these two smaller matrices gives the desired result.