ANALOGS OF THE M-FUNCTION IN THE THEORY OF ORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE

BARRY SIMON^{∗}

To Norrie Everitt, on his 80th birthday, a bouquet to the master of the m-function

Abstract. We show that the multitude of applications of the Weyl-Titchmarsh m-function leads to a multitude of different functions in the theory of orthogonal polynomials on the unit circle that serve as analogs of the m-function.

1. Introduction

Use of the Weyl-Titchmarsh m-function has been a constant theme in Norrie Everitt’s opus, so I decided a discussion of the analogs of these ideas in the theory of orthogonal polynomials on the unit circle (OPUC) was appropriate. Interestingly enough, the uses of the m-functions are so numerous that OPUC has multiple analogs of the m-function!

m-functions are associated to solutions of

−u′′ + qu = zu  (1.1)

with q a real function on [0, ∞) and z a parameter in ℂ_{+} = {z | Im z > 0}. The most fundamental aspect of the m-function is its relation to the spectral measure, ρ, for (1.1) by

m(z) = c + ∫ dρ(x)[\frac{1}{x − z} − \frac{x}{1 + x^{2}} ]  (1.2)

where c is determined by (see Atkinson [3], Gesztesy-Simon [13]):

m(z) = \sqrt{−z} + o(1)  as z → i∞  (1.3)

(1.2) plus (1.3) allow you to compute m given dρ, and dρ is determined by m via

lim_{e↓0} \frac{1}{π} ∫_{a}^{b} m(x + iε) dx = \frac{1}{2}[ρ((a, b)) + ρ([a, b])]  (1.4)

Of course, I haven’t told you what m or ρ is. This is done by defining m, in which case ρ is defined by (1.4). Under weak conditions on q at ∞, for z ∈ ℂ_{+}, (1.1) has a solution u(x, z) which is L^{2} at infinity, and it is unique up to a constant multiple. Then, m is defined by

Date: October 1, 2003.

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

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

For special issue of J. Comp. Appl. Math.

m(z) = \frac{u′(0, z)}{u(0, z)}  (1.5)

With this definition, dρ is a spectral measure for u ↦ −u′′ + qu = Hu in the sense that H is unitarily equivalent to multiplication by λ on L^{2}(ℝ, dρ). (1.5) is often written in the equivalent form,

ψ(x, z) + m(z)φ(x, z) ∈ L^{2}

where φ, ψ solve (1.1) with initial conditions φ(0) = 0, φ′(0) = 1, ψ(0) = 1, ψ′(0) = 0.

Note that if one defines

m(x; z) = \frac{u′(x, z)}{u(x, z)}  (1.6)

the m-function for q_{x}(·) = q( · + x), then m obeys the Riccati equation

m′ = q − z − m^{2}  (1.7)

It could be said that this is backwards: the definition (1.5) should come first, before (1.2). I put it in this order because it is (1.2) that makes m such an important object both in classical results [2, 5, 7, 8, 9, 16, 23, 33] and very recent work [27, 10, 21, 31, 25, 4].

To describe the third role of the m-function, it will pay to switch to the case of Jacobi matrices. We now have, instead of q, two sequences {a_{n}}_{n=1}^{∞}, {b_{n}}_{n=1}^{∞} with a_{n} > 0, b_{n} ∈ ℝ which we will suppose uniformly bounded. Define an infinite 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})  (1.8)

which is a bounded selfadjoint operator. One defines

m(z) = ⟨δ_{1}, (J − z)^{−1}δ_{1}⟩  (1.9)

In terms of the spectral measure, μ, for δ_{1} for J,

m(z) = ∫ \frac{dμ(x)}{x − z}  (1.10)

If u_{n} is the ℓ^{2} solution of a_{n−1}u_{n−1} +(b_{n} −z)u_{n} +a_{n}u_{n+1} = 0 with Im z > 0, one has the analog of (1.5)

m(z) = \frac{u_{1}(z)}{u_{0}(z)}  (1.11)

This process of going from a and b to m and then to μ can be reversed. One way is by iterating (1.15) below, which lets one go from μ to m (by (1.10)) and then gets the a’s and b’s as coefficients in a continued fraction expansion of m. From our point of view, an even more important way of going backwards uses orthogonal polynomials on the real line (OPRL). Given μ (of bounded support), one forms the monic orthogonal polynomials P_{n}(x) for dμ and shows they obey a recursion relation

P_{n+1}(x) = (x − b_{n+1})P_{n}(x) − a_{n}^{2}P_{n−1}(x)  (1.12)

which yields the Jacobi parameters a and b. The orthonormal polynomials, p_{n}(x), are related to P_{n} by

p_{n}(x) = (a_{1} … a_{n})^{−1}P_{n}(x)  (1.13)

and obey

a_{n+1}p_{n+1}(x) = (x − b_{n+1})p_{n}(x) − a_{n}p_{n−1}(x)  (1.14)

(1.7) has the analog

m(z; J) = (b_{1} − z − a_{1}^{2}m(z; J^{(1)}))^{−1}  (1.15)

where J^{(1)} is the Jacobi matrix with parameters ã_{m} = a_{m+1}b̃_{m} = b_{m+1} (i.e., the top row and left column are removed).

If m(x + iε; J) has a limit as ε ↓ 0, (1.15) says that m(x + iε; J^{(1)}) has a limit, and by (1.15),

\frac{Im m(x; J)}{Im m(x; J^{(1)})} = |a_{1}m(x; J)|^{2}  (1.16)

Im m is important because if μ is given by (1.10) then

dμ_{ac} = \frac{1}{π} Im m(x + i0) dx  (1.17)

This property of m, that its energy is the ratio of Im’s, is a critical element of recent work on sum rules for spectral theory [29, 19, 30, 28, 6].

The interesting point is that, for OPUC, the analogs of the functions obeying (1.2), (1.5), and (1.16) are different! In Section 2, we will give a quick summary of OPUC. In Section 3, we discuss (1.2); in Section 4, we discuss (1.16); and finally, in Section 5, the analog of (1.5).

Happy 80th, Norrie. I hope you enjoy this bouquet.

2. Overview of OPUC

We want to discuss here the basics of OPUC, although we will only scratch the surface of a rich and beautiful subject [29]. The theory reverses the usual passage from differential/difference equations to measures, and instead follows the discussion of OPRL in Section 1. μ is now a probability measure on ∂𝔻 = {z | |z| = 1}. We suppose μ is nontrivial, that is, not supported on a finite set. One can then form, by the Gram-Schmidt procedure, the monic orthogonal polynomials Φ_{n}(z) and the orthonormal polynomials, φ_{n}(z) = Φ_{n}(z)/∥Φ_{n}∥ where ∥ · ∥ is the L^{2}(∂𝔻, dμ) norm.

Given fixed n ∈ {0, 1, 2, … }, we define an anti-unitary operator on L^{2}(∂𝔻, dμ) by

f^{∗}(z) = z^{n} \overline{f(z)}  (2.1)

The use of a symbol without “n” is terrible notation, but it is standard! If Q_{n} is a polynomial of degree n, Q_{n}^{∗} is also a polynomial of degree n. Indeed,

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

so if Q_{n}(z) = a_{n}z^{n}+a_{n−1}z^{n−1}+⋯+a_{0}, then Q_{n}^{∗}(z) = ā_{0}z^{n}+ā_{1}z^{n−1}+⋯+ā_{n}.

Since Φ_{n} is monic, Φ_{n}^{∗}(0) = 1, and thus, N(z) ≡ (Φ_{n+1}^{∗}(z) − Φ_{n}^{∗}(z))/z is a polynomial of degree n. Since ^{∗} is anti-unitary,

⟨z^{m}, N(z)⟩ = ⟨z^{m+1}, Φ_{n+1}^{∗} − Φ_{n}^{∗}⟩

= ⟨Φ_{n+1}, z^{n+1−(m+1)}⟩ − ⟨Φ_{n}, z^{n−m−1}⟩ = 0

for m = 0, 1, …, n − 1. Thus N(z) must be a multiple of Φ_{n}(z), that is, for some α_{n} ∈ ℂ,

Φ_{n+1}^{∗}(z) = Φ_{n}^{∗}(z) − α_{n}zΦ_{n}(z)  (2.2)

and its ^{∗},

Φ_{n+1}(z) = zΦ_{n}(z) − ᾱ_{n}Φ_{n}^{∗}(z)  (2.3)

(2.2)/(2.3) are the Szegő recursion formulae ([32]); the α_{n}’s are the Verblunsky coefficients (after [34]). The derivation I’ve just given is that of Atkinson [2].

Since Φ_{n}^{∗} ⊥ Φ_{n+1}, (2.3) implies

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

Since ∥Φ_{n}^{∗}∥ = ∥zΦ_{n}∥ = ∥Φ_{n}∥, we have

∥Φ_{n+1}∥ = (1 − |α_{n}|^{2})^{1/2}∥Φ_{n}∥  (2.4)

This implies first of all that

|α_{n}| < 1  (2.5)

and if

ρ_{n} ≡ (1 − |α_{n}|^{2})^{1/2}  (2.6)

then

∥Φ∥_{n} = ρ_{0}ρ_{1} … ρ_{n−1}  (2.7)

so

φ_{n} = (ρ_{0} … ρ_{n−1})^{−1}Φ_{n}  (2.8)

and (2.2), (2.3) becomes

zφ_{n} = ρ_{n}φ_{n+1} + ᾱ_{n}φ_{n}^{∗}  (2.9)
φ_{n}^{∗} = ρ_{n}φ_{n+1}^{∗} + α_{n}zφ_{n}  (2.10)

The α_{n}’s not only lie in 𝔻, but it is a theorem of Verblunsky [34] that as μ runs through all nontrivial measures, the set of α’s runs through all of ×_{n=0}^{∞} 𝔻. The α’s are the analogs of the a’s and b’s in the Jacobi case or of V in the Schrödinger case.

We will later have reason to consider Szegő’s theorem in Verblunsky’s form [35]:

Theorem 2.1. Let

dμ = w \frac{dθ}{2π} + dμ_{s}  (2.11)

Then

∏_{j=0}^{∞}(1 − |α_{j}|^{2}) = exp(∫ log(w(θ))\frac{dθ}{2π} )  (2.12)

Remark. The log integral can diverge to −∞. The theorem says the integral is −∞ if and only if the product on the left is 0, that is, if and only if ∑|α_{j}|^{2} = ∞.

If

∑_{j=0}^{∞} |α_{j}|^{2} < ∞  (2.13)

we say the Szegő condition holds. This happens if and only if

∫ |log(w(θ))| \frac{dθ}{2π} < ∞  (2.14)

In that case, we define the Szegő function on 𝔻 by

D(z) = exp(∫ \frac{e^{iθ} + z}{e^{iθ} − z} log(w(θ)) \frac{dθ}{4π} )  (2.15)

3. The Carathéodory and Schur Functions

Given (1.10) (and (1.2)), the natural “m-function” for OPUC is the Carathéodory function, F (z),

F (z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ)  (3.1)

The Cauchy kernel (e^{iθ} + z)/(e^{iθ} − z) has the Poisson kernel

Re(\frac{e^{iθ} + z}{e^{iθ} − z} )|_{z=re^{iφ}} = \frac{1 − r^{2}}{1 + r^{2} − 2 cos(θ − φ)}  (3.2)

as its real part, and this is positive, so

Re F (z) > 0 for z ∈ 𝔻  F (0) = 1  (3.3)

This replaces Im m > 0 if Im z > 0.

One might think the “correct” analog of m is

R(z) = ∫ \frac{1}{e^{iθ} − z} dμ(θ)  (3.4)

R and F are related by

R(z) = (2z)^{−1}(F (z) − 1)  (3.5)

If one rotates dμ and z (i.e., dμ(θ) → dμ(θ−φ), z → e^{iφ}z), F is unchanged but R is multiplied by e^{−iφ}, so the set of values R can take are essentially arbitrary — which shows F, which obeys Re F (z) > 0, is a nicer object to take. That said, we will see R again in Section 5.

F has some important analogs of m:

(1) lim_{r↑1} F (re^{iθ}) exists for a.e. θ, and if (2.11) defines w, then

w(θ) = Re F (e^{iθ})  (3.6)

(2) θ_{0} is a pure point of μ if and only if lim_{r↑1}(1 − r) Re F (re^{iθ_{0}}) ≠ 0 and,

in general,

lim_{r↑1} (1 − r) Re F (re^{iθ_{0}}) = μ({θ_{0}})

(3) dμ_{s} is supported on {θ | lim_{r↑1} F (re^{iθ}) = ∞}. In fact, the proof of the analogs of these facts for m proceeds by mapping ℂ_{+} to 𝔻 and using these facts for F!

These properties provide a strong analogy, but one can note a loss of “symmetry” relative to the ODE case. The m-function maps ℂ_{+} to ℂ_{+}. F though maps 𝔻 to −iℂ_{+}. One might prefer a map of 𝔻 to 𝔻. In fact, one defines the Schur function, f, of μ via

F (z) = \frac{1 + zf(z)}{1 − zf(z)}  (3.7)

then f maps 𝔻 to 𝔻 and (3.7) sets up a one-one correspondence between F ’s with Re F > 0 on 𝔻 and F (0) = 1 and f mapping 𝔻 to 𝔻 (this fact relies on the Schwarz lemma that f maps 𝔻 to 𝔻 with f(0) = 0 if and only if f = zg where g maps 𝔻 to 𝔻).

For at least some purposes, f is a “better” analog of m than F, for example, in regard to its analog of the recursion (1.15). If f is the Schur function associated to Verblunsky coefficients {α_{0}, α_{1}, … } and f_{n} is the Schur function associated to {α_{n}, α_{n+1}, … }, then

f = \frac{α_{0} + zf_{1}}{1 + ᾱ_{0}zf_{1}}  (3.8)

a result of Geronimus (see [29] for lots of proofs of this fact).

Interestingly enough, Schur, not knowing of the connection to OPUC, discussed (3.8) for α_{0} = f(0) as a map of f → (α_{0}, f_{1}) and, by iteration, to a parametrization of functions of 𝔻 to 𝔻 by parameters α_{0}, …, α_{n}, . . .. There is, of course, a formula relating F to F_{1} that can be obtained from (3.7) and (3.8) or directly [22], but it is more complicated than (3.8).

Finally, in discussing f, we note that there is a natural family {dμ_{λ}}_{λ∈∂𝔻} of measures related to dμ (with dμ_{λ=1} = dμ) that corresponds to “varying boundary conditions.” We will discuss those more fully in Section 5, but we note

f(z; dμ_{λ}) = λf(z; dμ)  (3.9)

while the formula for F (dμ_{λ}) is more involved.

The Schur function and Schur iterates, f_{n}, have been used by Khrushchev [17, 18, 14] as a powerful tool in the analysis of OPUC.

4. The Relative Szegő Function

As explained in the introduction, a critical property of m is (1.16), which is the basis of step-by-step sum rules (see [28]). The left side of (1.16) enters as the ratio of a.c. weights of dμ_{J} and dμ_{J^{(1)}}. Thus, we are interested in Im F (e^{iθ}; {α_{j}}_{j=0}^{∞}) divided by Im F (e^{iθ}; {α_{j+1}}_{j=0}^{∞}), that is, Im F/ Im F_{1} in the language of the last section. Neither |F | nor |f| is directly related to this ratio, so we need a different object to get an analog of (1.16). The following was introduced by Simon in [29]:

(δ_{0}D)(z) = \frac{1 − ᾱ_{0}f}{ρ_{0}} \frac{1 − zf_{1}}{1 − zf}  (4.1)

It is called the “relative Szegő function” for reasons that will become clear in a moment.

In (4.1), f_{1} is the Schur function for Verblunsky coefficients

α_{j}^{(1)} = α_{j+1}  (4.2)

Here is the key fact:

Theorem 4.1. Let dμ and dμ^{(1)} be measures on ∂𝔻 with Verblunsky coefficients related by (4.2). Suppose dμ = w(θ)\frac{dθ}{2π} +dμ_{s} and dμ^{(1)} = w^{(1)} \frac{dθ}{2π} +dμ_{s}. Then

(1) For a.e. θ, lim_{r↑1}(δ_{0}D)(re^{iθ}) ≡ δ_{0}D(e^{iθ}) exists.

(2) If w(θ) ≠ 0, then (for a.e. θ w.r.t. \frac{dθ}{2π}), w_{1}(θ) ≠ 0 and

\frac{w(θ)}{w_{1}(θ)} = |(δ_{0}D)(e^{iθ})|^{2}  (4.3)

Sketch of Proof. Each of the functions 1 − ᾱ_{0}f, 1 − zf_{1}, and 1 − zf takes values in \{w | |w − 1| < 1\} on 𝔻, so their arguments lie in [−\frac{π}{2} , \frac{π}{2} ], so their logs are in all H^{p}, 1 < p < ∞. That is, they are outer functions, and so δ_{0}D is an outer function, which means that assertion (1) holds (see Rudin [24] for a pedagogic discussion of outer functions).

To get (4.3), we note that (3.7) implies

Re F (z) = \frac{1 − |f|^{2}|z|^{2}}{|1 − zf|^{2}}

so

\frac{Re F (z)}{Re F_{1}(z)} = |\frac{1 − zf_{1}}{1 − zf} |^{2} \frac{1 − |f|^{2}|z|^{2}}{1 − |f_{1}|^{2}|z|^{2}}  (4.4)

On the other hand, (3.8) implies

zf_{1} = \frac{f − α_{0}}{1 − ᾱ_{0}f}  (4.5)

which implies

1 − |zf_{1}|^{2} = \frac{ρ_{0}^{2}(1 − |f|^{2})}{|1 − ᾱ_{0}f|^{2}}  (4.6)

so, putting these formulae together,

\frac{Re F (z)}{Re F_{1}(z)} = |(δ_{0}D)(z)|^{2}(\frac{1 − |z|^{2}|f|^{2}}{1 − |f|^{2}} )  (4.7)

which, as |z| → 1, yields (4.3). □

In particular, one has the nonlocal step-by-step sum rule that if w(θ) ≠ 0 for a.e. θ, then

(δ_{0}D)(z) = exp(∫_{0}^{2π} \frac{e^{iθ} + z}{e^{iθ} − z} log(\frac{w(θ)}{w_{1}(θ)}) \frac{dθ}{4π} )  (4.8)

and, in particular, setting z = 0,

ρ_{0}^{2} = exp(∫_{0}^{2π} log(\frac{w(θ)}{w_{1}(θ)}) \frac{dθ}{2π} )  (4.9)

which is not only consistent with Szegő’s theorem (2.12) but, using semicontinuity of the entropy, can be used to prove it (see [19, 29]) as follows:

(1) Iterating (4.9) yields

(ρ_{0} … ρ_{n−1})^{2} = exp(∫_{0}^{2π} log(\frac{w(θ)}{w_{n}(θ)}) \frac{dθ}{2π} )  (4.10)

(2) Since exp(∫_{0}^{2π} log(w_{n}(θ)\frac{dθ}{2π}) ≤ ∫_{0}^{2π} w_{n}(θ)\frac{dθ}{2π} ≤ 1, (4.10) implies

(ρ_{0} … ρ_{n−1})^{2} ≥ exp(∫_{0}^{2π} log(w(θ)) \frac{dθ}{2π} )  (4.11)

(3) If w^{(n)} is the weight associated to the measure with

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

(4.10) proves

(ρ_{0} … ρ_{n−1})^{2} = exp ∫_{0}^{2π} log(w^{(n)}(θ)) \frac{dθ}{2π}  (4.12)

(4) dμ → ∫_{0}^{2π} log(w(θ))\frac{dθ}{2π} is an entropy, hence, weakly upper semicontinuous. Since w^{(n)} \frac{dθ}{2π} → dμ weakly as n → ∞, this semicontinuity shows

lim_{n→∞} (ρ_{n} … ρ_{n−1})^{2} ≤ exp(∫_{0}^{2π} log(w(θ)) \frac{dθ}{2π} )  (4.13)

(4.11) and (4.13) is Szegő’s theorem. Two other properties of δ_{0}D that we should mention are:

(A) If ∑_{n=0}^{∞}|α_{n}|^{2} < ∞, then

(δ_{0}D)(z) = \frac{D(z; α_{0}, α_{1}, α_{2}, …)}{D(z; α_{1}, α_{2}, α_{3}, …)}  (4.14)

(B) In general, one has

δ_{0}D(z) = lim_{n→∞} \frac{φ_{n−1}^{∗}(z; α_{1}, α_{2}, …)}{φ_{n}^{∗}(z; α_{0}, α_{1}, …)}  (4.15)

5. Eigenfunction Ratios

Finally, we look at the analogs of m as a function ratio, its initial definition by Weyl and Titchmarsh. The key papers on this point of view are by Geronimo-Teplyaev [11] and Golinskii-Nevai [15]. We will see from one point of view [15] that F (z) plays this role, but from other points of view [11] that other functions are more natural.

The recursion relations (2.9)/(2.10) can be rewritten as

\binom{φ_{n+1}}{φ_{n+1}^{∗}} = A(α_{n}, z)\binom{φ_{n}}{φ_{n}^{∗}}  (5.1)

where

A(α, z) = ρ^{−1} (\begin{matrix}z & −ᾱ_{n} \\ −α_{n}z & 1 \end{matrix})  (5.2)

(with ρ = (1 − |α|^{2})^{1/2}). From this point of view, the analog of the fundamental differential/difference equation in the real case is

Ξ_{n} = T_{n}(z)Ξ_{0}  (5.3)

with

T_{n}(z) = A(α_{n−1}, z) … A(α_{0}, z)  (5.4)

The correct boundary conditions for the usual OPUC are Ξ_{0} = \binom{1}{1}.

One can ask for what other initial conditions the polynomials associated with the top component of T_{n}(z)Ξ_{0} are OPUC for some measure. Note that

\binom{1}{λ} = U(λ)\binom{1}{1}  (5.5)

with

U(λ) = (\begin{matrix}1 & 0 \\ 0 & λ \end{matrix})  (5.6)

and that

U(λ)^{−1}A(α, z)U(λ) = ρ^{−1} (\begin{matrix}z & −ᾱ_{n}λ \\ −α_{n}λ^{−1}z & 1 \end{matrix})  (5.7)

We see from this that λ̄ = λ^{−1}, that is, |λ| = 1 will yield U(λ)^{−1}A(α_{1}, z)U(λ) = A(λ̄α, z). Changing λ to λ̄, we see that Proposition 5.1. Let |λ| = 1. If φ_{n}^{(λ)}(z) are the OPUC for Verblunsky coefficients α_{n}^{(λ)} = λα_{n}, then

\binom{φ_{n}^{(λ)}(z)}{λ̄φ_{n}^{(λ)∗}(z)} = T_{n}(z; \{α_{j}\}_{j=1}^{∞})\binom{1}{λ̄}  (5.8)

This suggests that one look at the family dμ_{λ} or measures with

α_{j}(dμ_{λ}) = λα_{j}(dμ)  (5.9)

called the family of Aleksandrov measures associated to {α_{j}}_{j=0}^{∞} after [1]. The special case λ = −1 goes back to Verblunsky [35] and Geronimus [12], and are called the second kind polynomials, denoted ψ_{n}(z). The following goes back to Verblunsky [35]:

Theorem 5.2. For z ∈ 𝔻, uniformly on compact subsets of 𝔻,

lim_{n→∞} \frac{ψ_{n}^{∗}(z)}{φ_{n}^{∗}(z)} = F (z)  (5.10)

Clearly related to this is the following result of Golinskii-Nevai [15]:

Theorem 5.3. Let z ∈ 𝔻. Then

∑_{n=0}^{∞} |\binom{ψ_{n}(z)}{−ψ_{n}^{∗}(z)} + β\binom{φ_{n}(z)}{φ_{n}^{∗}(z)}|^{2} < ∞  (5.11)

if and only if

β = F (z)  (5.12)

From this point of view, F is again the “correct” analog of m! Indeed, the Golinskii-Nevai [15] proof uses Weyl limiting circles to prove the theorem (one is always in limit point case!).

But this is not the end of the story. Define

u_{k} = ψ_{k} + F (z)φ_{k}  u_{k}^{∗} = −ψ_{k}^{∗} + F (z)φ_{k}^{∗}  (5.13)

so \binom{u_{k}}{u_{k}^{∗}} is the unique solution of Ξ_{n} = T_{n}(z)Ξ_{0} which is in ℓ^{2}. In the OPRL case, the basic vector solution is of the form \binom{u_{n}}{u_{n+1}}, so we have the analog of (1.11),

m̃(z) = \frac{u_{0}^{∗}}{u_{0}} = \frac{−1 + F}{1 + F} = zf  (5.14)

So one analog of the m-function is zf.

In particular, (5.14) implies

|u_{k}^{∗}| < |u_{k}|  (5.15)

for z ∈ 𝔻, and thus the rate of exponential decay of |\binom{u_{k}}{u_{k}^{∗}}| is that of u_{k}. If there is such exponential decay in the sense that

γ_{2} = lim_{n→∞} [ ∥\binom{u_{n}}{u_{n}^{∗}}∥^{1/n} ]  (5.16)

exists, then, by (5.15),

γ_{2} = lim_{n→∞} \frac{1}{n} ∑_{j=0}^{n−1} log|m_{n}^{+}|  (5.17)

where

m_{n}^{+} = \frac{u_{n+1}}{u_{n}}  (5.18)

For n = 0, u_{1} = ψ_{1} + F φ_{1}, u_{0} = 1 + F, ψ_{1} = ρ_{0}^{−1}(z + ᾱ_{0}), φ_{1} = ρ_{0}^{−1}(z − ᾱ_{0}), so by a direct calculation,

m_{0}^{+}(z) = ρ_{0}^{−1}z(1 − ᾱ_{0}f)  (5.19)

yet another reasonable choice for an m-function.

Indeed, if γ(z) = lim_{n→∞} \frac{1}{n} log ∥T_{n}(z)∥ exists, the fact that det(T_{n}) = z^{n} implies that γ = log|z|−γ_{2}, and one finds in the case of stochastic Verblunsky coefficients that [11, 29]

𝔼(log|m_{ω}^{+}(z)|) = log|z| − γ(z)  (5.20)

an analog of a fundamental formula of Kotani [20, 26] that in his case uses m!

Finally, we turn to the connection of m to whole-line Green’s functions. Given V on (−∞, ∞) and z ∈ ℂ_{+}, it is natural to look at the two solutions of (1.1), u_{±}(x, z), which are ℓ^{2} on ±(0, ∞) and the m-functions,

m_{±}(z) = ± \frac{u′_{±}(0, z)}{u_{±}(0, z)}  (5.21)

m_{±} are the m-functions for V (±x) ↾ [0, ∞). Standard Green’s function formulae show that the integral kernel, G(x, y; z) of (−\frac{d^{2}}{dx^{2}} + V − z)^{−1} is

G(x, y; z) = \frac{u_{−}(x_{<})u_{+}(x_{>})}{(u_{+}(0)u′_{−}(0) − u′_{+}(0)u_{−}(0))}

where x_{<} = min(x, y) and x_{>} = max(x, y). In particular,

G(0, 0; z) = −(m_{+}(z) + m_{−}(z))^{−1}  (5.22)

A complete description of the OPUC analog would require too much space, so we sketch the ideas, leaving the details to [29]. Just as the difference equation is associated to a triagonal selfadjoint matrix whose spectral measure is the one generating the OPRL, any set of α’s is associated to a five-diagonal unitary matrix, called the CMV matrix, whose spectral measure is the dμ with α_{j}(dμ) = α_{j}.

The CMV matrix is one-sided, but given {α_{j}}_{j=−∞}^{∞}, one can define a twosided CMV matrix, E, in a natural way. If G(z) is the 00 matrix element of (E − z)^{−1}, then (see [11, 17, 29])

G(z) = \frac{f_{+}(z)f_{−}(z)}{1 − zf_{+}(z)f_{−}(z)}  (5.23)

where f_{+} is the Schur function for (α_{0}, α_{1}, α_{2}, …) and f_{−} the Schur function for (−ᾱ_{−1}, −ᾱ_{−2}, …). On the basis of the analogy between (5.23) and (5.22), Geronimo-Teplyaev [11] called f_{+} and zf_{−} the m_{+} and m_{−} functions.

6. Summary

We have thus seen that there are many analogs of the m-function in the theory of OPUC:

(1) The Carathéodory function, F (z), given by (3.1), an analog of (1.2) and also related to the classic Weyl definition (5.11)/(5.12).

(2) The Schur function, f(z), given by (3.7) with a recursion, (3.8), closer to the recursion (1.15) for the m-function of OPRL. f also enters via (5.23).

(3) zf(z), the m̃-function of (5.14).

(4) The relative Szegő function, (4.1), which, via (4.3) and (1.16), is an analog of a_{1}m(z).

(5) The m^{+}-function, (5.19), which plays the role that m does in Kotani theory.

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