One of the motivations for the present body of work is to explore generalizations of Szeg˝o’s Theorem on the unit circle. This is a very profound result, several proofs of which are presented in Chapter 2 of [56]. The result can be stated as follows:
Szeg˝o’s Theorem. If µis a probability measure on the unit circle, then
n→∞lim κ−1n = exp 1
4π Z 2π
0
log(µ0(θ))dθ
=S(∞;µ).
Perhaps one of the deepest consequences of this result is given as Theorem 2.7.14 in [56], where a large list of quantities related to orthogonal polynomials are (perhaps surprisingly) shown to be equal. This theorem relates the limiting behavior of the monic orthogonal polynomial norms to several other properties of the measure such as its entropy (see Section 2.3 in [56]) or the behavior of the associated Christoffel functions (see Section2.5below). This list of equivalences invites one to think about generalizations of Szeg˝o’s Theorem to settings where recursion coefficients do not exist.
One of the earliest generalizations was achieved in [17], where Geronimus generalized Szeg˝o’s Theorem to the case of a sufficiently smooth Jordan curve Γ. His result can be stated as follows:
Theorem 2.3.1 (Geronimus, [17]). Let µbe a finite measure on an analytic Jordan curve Γwhere cap(Γ) = 1. Ifφ∗µis a Szeg˝o measure on the unit circle andq∈(0,∞), then
n→∞lim kPn(·;µ, q)kqLq(µ)= exp Z 2π
0
log((φ∗µ)0(θ))dθ 2π
. (2.3.1)
Before we proceed with the proof of Theorem 2.3.1, we need to make an observation. The upper bound will be obtained using the extremal property, while for the lower bound we will invoke subharmonicity of a particular integrand. This is simple enough whenq≥1 because everyH1(C\D) function is the Poisson integral of its boundary values (see Theorem 17.11 in [47]). However, some care is required when 0< q <1. We simply note here that Theorem 17.11(c) in [47] combined with a well-knownLq inequality (see page 74 in [47]) imply that iff ∈Hq(C\D), then
Z 2π 0
|f(eiθ)|qdθ
2π ≥ |f(∞)|q. (2.3.2)
Now we are ready to prove Theorem2.3.1. The proof we present here is essentially the same as the proof from [17]. We present it here in english for the reader’s convenience.
Proof. We begin with the proof of the lower bound. By definition ofφ∗µ, we have kPn(·;µ, q)kqLq(µ)=
Z
Γ
|Pn(z;µ, q)|qdµ(z) = Z
∂D
|Pn(ψ(w);µ, q)|qd(φ∗µ)(w)
= Z
∂D
Pn(ψ(w);µ, q) wn
q
d(φ∗µ)(w)
≥ Z
∂D
Pn(ψ(w);µ, q)S(w;φ∗µ, q) wn
q d|w|
2π .
Now, notice that the integrand in this last expression is analytic inC\Dand hence we are integrating a subharmonic function around a circle. Therefore, (2.3.2) implies that we can bound this integral from below by the value of the integrand at infinity. Since Γ has capacity 1,Pn(ψ(w);µ, q) grows likewn at∞, so we end up with
kPn(·;µ, q)kqLq(µ)≥ |S(∞;φ∗µ, q)|q = exp Z 2π
0
log((φ∗µ)0(θ))dθ 2π
,
which is the desired lower bound.
For the upper bound, we need to first recall a generalization of Szeg˝o’s Theorem. For any measure ν, define the quantity
λn(z;ν, q) = inf Z
C
|Q(w)|qdν(w) : deg(Q)≤n , Q(z) = 1
(2.3.3) and we also defineλ∞(z;µ, q) = limn→∞λn(z;µ, q) (the limit clearly exists as the limit of a non-
increasing sequence of positive real numbers). We will discuss this object in greater detail later (see Section2.5). Its importance for us in this section is derived from Theorem 2.5.4 in [56], which tells us the following:
Theorem 2.3.2 ([56]). If ν is supported on the unit circle, then
λ∞(0;ν, q) =S(∞;ν, q)q. (2.3.4)
We note here that the right-hand side of (2.3.4) is independent of q. The proof of Theorem 2.3.2 is in fact quite elementary and requires only two facts. The first is that for the caseq = 2, the polynomial Pn∗(z;µ,2) = znPn(¯z−1, µ,2) is the unique minimizer of λn(0;ν, q). The second ingredient is the fact thatPn∗(z;µ,2) has all of its zeros outsideD, so (Pn∗(z;µ,2))2/qis an analytic function in a neighborhood of Dand hence can be uniformly approximated by polynomials. These two realizations make proving Theorem2.3.2quite simple, though Theorem 2.5.4 in [56] is, in fact, much more general because it applies toλ∞(z;ν, q) for anyz∈D.
Returning to the proof of the upper bound, recall that since Γ is an analytic Jordan curve, we knowFn(z)−φ(z)n →0 uniformly for z in the closure of the unbounded component ofC\Γ. To see this, we use (1.3.5) to get
|Fn(z)−φ(z)n| ≤ 1 2π
Z
{|z|=r}
tnψ0(t) ψ(t)−z
d|t|.
Analyticity of Γ implies we may taker <1 in this integral and see that forz in the desired set we in fact have convergence to 0 at an exponential rate.
As is often the case, we exploit the extremal property of Pn(·;µ, q) to derive an upper bound on its norm. We know that Fn is a monic polynomial, so for anym ∈Nand any choice of constants γ1, . . . , γm∈Cwe have
kPn(·;µ, q)kqLq(µ)≤ Z
Γ
Fn(z) +
m
X
j=1
γjFn−j(z)
q
dµ(z)
= Z
∂D
Fn(ψ(w)) +
m
X
j=1
γjFn−j(ψ(w))
q
d(φ∗µ)(w)
= Z
∂D
wn+
m
X
j=1
γjwn−j
q
d(φ∗µ)(w) +o(1)
= Z
∂D
1 +
m
X
j=1
γjwj
q
d(φ∗µ)(w) +o(1)
=λm(0;φ∗µ, q) +o(1)
if we choose the constantsγ1, . . . , γm correctly. Therefore,
lim sup
n→∞
kPn(·;µ, q)kqLq(µ)≤λm(0;φ∗µ, q)
and sincem∈Nwas arbitrary, the desired conclusion follows from Theorem2.3.2.
Remark. The same proof works if Γ is any curve for whichFn−φn tends to zero uniformly on the curve Γ and outside it. This is how Geronimus stated his result in [17], where he provides several smoothness conditions on Γ that imply this convergence property holds.
Notice that in deriving the lower bound in Theorem2.3.1, we only used the absolutely continuous part of the measure µ (with respect to arc-length measure). Since this lower bound matches the upper bound, we have proven the following:
Corollary 2.3.3. Under the assumptions of Theorem 2.3.1, ifµsing is the component ofµ that is singular with respect to arc-length measure, then one has
n→∞lim kpn(·;µ, q)kLq(µsing)= 0.
Furthermore, in the proof of Theorem 2.3.1, we showed that
n→∞lim
pn(ψ(z);µ, q)S(z;φ∗µ, q) zn
Hq(
C\D)
= 1, lim
n→∞
pn(ψ(z);µ, q)S(z;φ∗µ, q) zn
z=∞
= 1.
Therefore, the Keldysh Lemma (see Section1.5above) proves the following:
Theorem 2.3.4. Ifµ is as in Theorem2.3.1, then
n→∞lim
pn(ψ(z);µ, q)
zn = 1
S(z;φ∗µ, q), |z|>1 and the convergence is uniform on compact subsets ofC\D.
Theorem 2.3.4 gives a very clear picture of the behavior of the orthonormal polynomials when the measure µ has sufficiently nice properties. The leading order behavior of the polynomials is S(z)φ(z)n for an explicitly computable functionS that is independent of the singular component of the measureµ. This is precisely the kind of statement that we will try to make about orthonormal polynomials whose measure of orthogonality has a more general support.