Content-Type: multipart/mixed; boundary="-------------0402061157661" This is a multi-part message in MIME format. ---------------0402061157661 Content-Type: text/plain; name="04-28.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-28.keywords" Szego's theorem, Toeplitz determinants, orthogonal polynomials on the unit circle ---------------0402061157661 Content-Type: application/x-tex; name="sst.TEX" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="sst.TEX" \documentclass[reqno,centertags, 12pt]{amsart} \usepackage{amsmath,amsthm,amscd,amssymb} \usepackage{latexsym} %\usepackage{showkeys} \sloppy %%%%%%%%%%%%% fonts/sets %%%%%%%%%%%%%%%%%%%%%%% \newcommand{\bbU}{{\mathbb{U}}} \newcommand{\bbR}{{\mathbb{R}}} \newcommand{\bbD}{{\mathbb{D}}} \newcommand{\bbP}{{\mathbb{P}}} \newcommand{\bbZ}{{\mathbb{Z}}} \newcommand{\bbC}{{\mathbb{C}}} \newcommand{\bbH}{{\mathbb{H}}} \newcommand{\calP}{{\mathcal P}} %%%%%%%%%%%%%%%%%% abbreviations %%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\dott}{\,\cdot\,} \newcommand{\no}{\nonumber} \newcommand{\lb}{\label} \newcommand{\f}{\frac} \newcommand{\ul}{\underline} \newcommand{\ol}{\overline} \newcommand{\ti}{\tilde } \newcommand{\wti}{\widetilde } \newcommand{\Oh}{O} \newcommand{\oh}{o} \newcommand{\marginlabel}[1]{\mbox{}\marginpar{\raggedleft\hspace{0pt}#1}} \newcommand{\tr}{\text{\rm{Tr}}} \newcommand{\dist}{\text{\rm{dist}}} \newcommand{\loc}{\text{\rm{loc}}} \newcommand{\spec}{\text{\rm{spec}}} \newcommand{\rank}{\text{\rm{rank}}} \newcommand{\ran}{\text{\rm{ran}}} \newcommand{\dom}{\text{\rm{dom}}} \newcommand{\ess}{\text{\rm{ess}}} \newcommand{\ac}{\text{\rm{ac}}} \newcommand{\s}{\text{\rm{s}}} \newcommand{\sing}{\text{\rm{sc}}} \newcommand{\pp}{\text{\rm{pp}}} \newcommand{\supp}{\text{\rm{supp}}} \newcommand{\AC}{\text{\rm{AC}}} \newcommand{\bi}{\bibitem} \newcommand{\hatt}{\widehat} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\ba}{\begin{align}} \newcommand{\ea}{\end{align}} \newcommand{\veps}{\varepsilon} %\newcommand{\Ima}{\operatorname{Im}} %\newcommand{\Real}{\operatorname{Re}} %\newcommand{\diam}{\operatorname{diam}} % use \hat in subscripts % and upperlimits of int. % % Rowan's unspaced list % \newcounter{smalllist} \newenvironment{SL}{\begin{list}{{\rm\roman{smalllist})}}{% \setlength{\topsep}{0mm}\setlength{\parsep}{0mm}\setlength{\itemsep}{0mm}% \setlength{\labelwidth}{2em}\setlength{\leftmargin}{2em}\usecounter{smalllist}% }}{\end{list}} % %smaller \bigtimes % \newcommand{\bigtimes}{\mathop{\mathchoice% {\smash{\vcenter{\hbox{\LARGE$\times$}}}\vphantom{\prod}}% {\smash{\vcenter{\hbox{\Large$\times$}}}\vphantom{\prod}}% {\times}% {\times}% }\displaylimits} %%%%%%%%%%%%% marginal warnings %%%%%%%%%%%%%%%% % ON: \newcommand{\TK}{{\marginpar{x-ref?}}} % OFF: %\newcommand{\TK}{} %%%%%%%%%%%%%%%%%%%%%% renewed commands %%%%%%%%%%%%%%% %\renewcommand{\Re}{\text{\rm Re}} %\renewcommand{\Im}{\text{\rm Im}} %%%%%%%%%%%%%%%%%%%%%% operators %%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator{\Real}{Re} \DeclareMathOperator{\Ima}{Im} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator*{\slim}{s-lim} \DeclareMathOperator*{\wlim}{w-lim} \DeclareMathOperator*{\simlim}{\sim} \DeclareMathOperator*{\eqlim}{=} \DeclareMathOperator*{\arrow}{\rightarrow} \allowdisplaybreaks \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% end of definitions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] %\newtheorem*{t0}{Theorem} \newtheorem*{t1}{Theorem 1} \newtheorem*{t2}{Theorem 2} \newtheorem*{t3}{Theorem 3} \newtheorem*{t4}{Theorem 4} \newtheorem*{t5}{Theorem 5} %\newtheorem*{c4}{Corollary 4} %\newtheorem*{p2.1}{Proposition 2.1} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} %\newtheorem{hypothesis}[theorem]{Hypothesis} %\theoremstyle{hypothesis} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} % Absolute value notation \newcommand{\abs}[1]{\lvert#1\rvert} \begin{document} \title{The Sharp Form of the Strong Szeg\H{o} Theorem} \author{Barry Simon} \thanks{$^1$ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125. E-mail: bsimon@caltech.edu} \thanks{$^2$ Supported in part by NSF grant DMS-0140592} \thanks{{\it To appear in Proc. Conf. on Geometry and Spectral Theory}} \dedicatory{In memoriam, Robert Brooks {\rm{(}}1952--2002{\rm{)}} } \date{January 21, 2004} \begin{abstract} Let $f$ be a function on the unit circle and $D_n(f)$ be the determinant of the $(n+1)\times (n+1)$ matrix with elements $\{c_{j-i}\}_{0\leq i,j\leq n}$ where $c_m =\hat f_m\equiv \int e^{-im\theta} f(\theta) \f{d\theta}{2\pi}$. The sharp form of the strong Szeg\H{o} theorem says that for any real-valued $L$ on the unit circle with $L,e^L$ in $L^1 (\f{d\theta}{2\pi})$, we have \[ \lim_{n\to\infty}\, D_n(e^L) e^{-(n+1)\hat L_0} = \exp \biggl( \, \sum_{k=1}^\infty k\abs{\hat L_k}^2\biggr) \] where the right side may be finite or infinite. We focus on two issues here: a new proof when $e^{i\theta}\to L(\theta)$ is analytic and known simple arguments that go from the analytic case to the general case. We add background material to make this article self-contained. \end{abstract} \maketitle \section{Introduction} \lb{s1} Let $\{c_m\}_{m=-\infty}^\infty$ be a two-sided sequence of complex numbers. A {\it Toeplitz matrix} is a finite matrix constant along diagonals: \begin{equation} \lb{1.1} T_{n+1} = \begin{pmatrix} c_0 & c_1 & c_2 & \dots & c_n \\ c_{-1} & c_0 & c_1 & \dots & c_{n-1} \\ \vdots & {} & {} & {} & \vdots \\ c_{-n} & c_{-n+1} & {} & \dots & c_0 \end{pmatrix} \end{equation} It turns out that the natural way to label $T$ is in terms of the Fourier transform of $c$, that is, \begin{equation} \lb{1.2} f(\theta) =\sum_{m=-\infty}^\infty c_m e^{im\theta} \end{equation} on $\partial\bbD$ ($\bbD=\{z\mid \abs{z}<1\}$, $\partial\bbD =\{z\mid\abs{z}=1\}$). $f$ is called the {\it symbol} of the Toeplitz matrix. One can define a symbol as a distribution so long as $\abs{c_m}$ is polynomially bounded in $m$, but we will discuss the case where there is a signed measure, $d\mu$, so that \begin{equation} \lb{1.3} c_n =\int e^{-in\theta}\, d\mu(\theta) \equiv \hat\mu_n \end{equation} As usual, for $f\in L^1 (\partial\bbD, \f{d\theta}{2\pi})$, we define $\hat f_n$ to be the Fourier coefficients of the measure $f\f{d\theta}{2\pi}$. We will most often discuss the case where $d\mu$ is absolutely continuous, that is, $d\mu = w(\theta)\f{d\theta}{2\pi}$ and where $w\geq 0$ or even that $w=e^L$. $D_n(d\mu)$ is the determinant of $T_{n+1}$. The strong Szeg\H{o} theorem says that if $L, e^L\in L^1$ with $L$ real, then \begin{equation} \lb{1.4} \log D_n \biggl(e^L \,\f{d\theta}{2\pi}\biggr) \sim (n+1) \hat L_0 + \sum_{k=1}^\infty\, \abs{k} \, \abs{\hat L_k}^2 \end{equation} There are a number of remarkable aspects of \eqref{1.4}. The first term was found in 1915 and the second in 1952. Despite the 37-year break, they were both found by Szeg\H{o} --- the twenty-year old in 1915 \cite{Sz15} and the 57-year old in 1952 \cite{Sz52}! You might wonder about whether \eqref{1.4} is the leading term in a systematic $1/n$ series. In fact, if $L$ is real-valued and if $e^{i\theta}\mapsto L(\theta)$ is analytic in the neighborhood of $\partial\bbD$, then the error in \eqref{1.4} is $O(e^{-Bn})$ --- there are no more terms in the series (this follows from \eqref{2.21new} and \eqref{5.16} below). Lest you be shocked by this, we note that for many models in statistical mechanics, the free energy has a volume term, a surface term, and then, if the interaction is short-range, exponentially small errors. A second remarkable aspect is the subtlety. Why should $\log w$ enter at all, and then in both linear and quadratic terms? There is a fascination with this subject among mathematicians who have extended the result both in the context of function algebras \cite{Werm,HoffActa,Lum} and in the context of pseudodifferential operators on manifolds \cite{W79,GOk2} (see \cite{OPUC} for literally dozens of papers on each aspect). A third aspect is that there are a remarkable number of applications of this result. Szeg\H{o} returned to find the second term because of a question raised by Onsager who ran into Toeplitz determinants in his work on the Ising model (see \cite{McCoy,BoOns} for a discussion of this). They enter in the study of some Coulomb gases \cite{Len64,Len72,FH69} and in electrical engineering applications \cite{Kai74,BakCon}. And they have had a surge of interest recently because of their role in random matrix theory \cite{MehBk}. When Szeg\H{o} \cite{Sz52} proved \eqref{1.4}, he assumed $L$ was $C^{1+\veps}$. There were many papers on this subject which improved this incrementally until Ibragimov \cite{Ib}, fifteen years later, proved the following sharp form: \begin{theorem}[\cite{Ib,GoIb}]\lb{T1.1} Let $L$ be a real-valued function on $\partial\bbD$ so that $L,e^L\in L^1 (\partial\bbD, \f{d\theta}{2\pi})$. Then \begin{equation} \lb{1.5} \lim_{n\to\infty}\, D_n \biggl( e^L \, \f{d\theta}{2\pi}\biggr) e^{-(n+1)\hat L_0} = \exp \biggl(\, \sum_{k=1}^\infty\, \abs{k}\, \abs{\hat L_k}^2\biggr) \end{equation} \end{theorem} Thus, \eqref{1.4} holds whenever the right side makes sense, that is, $L$ in $H^{1/2}$, the Sobolev space of order $\f12$. This should be supplemented with a result of Golinskii-Ibragimov \cite{GoIb}: \begin{theorem}\lb{T1.2} If $d\mu=e^{L}\f{d\theta}{2\pi} + d\mu_\s$ with $d\mu_\s$ singular is a positive measure on $\partial\bbD$ and $\lim_{n\to\infty} D_n (d\mu) e^{-(n+1)\hat L_0} <\infty$, then $d\mu_\s =0$. \end{theorem} The combination of these two theorems has a spectral theory consequence that links it up to the theme of the conference and to Bob Brooks' interests. As we will see in Section~\ref{s2}, probability measures on $\partial\bbD$ have associated parameters $\{\alpha_n (d\mu)\}_{n=0}^\infty$ called Verblunsky coefficients. It can be shown using Theorems~\ref{T1.1}, \ref{T1.2}, and \ref{T2.4} that \begin{theorem}\lb{T1.3} Let $d\mu$ be a probability measure on $\partial\bbD$ and $\{\alpha_n(d\mu)\}_{n=0}^\infty$ its Verblunsky coefficients. Then the following are equivalent: \begin{SL} \item[{\rm{(i)}}] $\sum_{n=0}^\infty n\abs{\alpha_n}^2 <\infty$ \item[{\rm{(ii)}}] $d\mu_\s =0$ and $d\mu =e^L \f{d\theta}{2\pi}$ where $\sum_{k=1}^\infty k \abs{\hat L_k}^2 <\infty$. \end{SL} \end{theorem} This is one of those gems of spectral theory that give necessary and sufficient conditions relating properties of a measure and its inverse spectral parameters. We have two main themes in this article. First, we wish to note that despite it taking fifteen years to go from $L\in C^{1+\veps}$ to $L\in H^{1/2}$, there is an elegant and simple argument to do this jump. This combines arguments of Golinskii-Ibragimov \cite{GoIb} and Johansson \cite{Jo88} whose proofs have ``easy" halves that handle opposite sides of the extension. It does not appear to be widely appreciated that their arguments can be combined in this way. In fact, these results reduce \eqref{1.4} to the case where the $\hat L_k$ decay exponentially, that is, $e^{i\theta} \mapsto L(\theta)$ is real analytic on $\partial\bbD$. Second, we have a new proof of \eqref{1.4} in this real analytic case that is perhaps less mysterious than the elaborate calculation in Szeg\H{o} \cite{Sz52}. From our point of view, the two terms in \eqref{1.4} come from two terms in the Christoffel-Darboux formula for $z=e^{i\theta}$. While the arguments of \cite{GoIb,Jo88} are simple, they depend on considerable general machinery relating Toeplitz determinants to orthogonal polynomials on the unit circle (OPUC) on the one hand, and to the statistical mechanics of Coulomb gases on the other (both themes go back to Szeg\H{o}'s early work: \cite{Sz15} has the Coulomb gas representation and the main point of \cite{Sz20,Sz21} is to discuss the connection to OPUC), so this article, in attempting to be self-contained, provides this background. Sections~\ref{s2} and \ref{s3} discuss the basics of OPUC. In Section~\ref{s4}, we get the leading term in \eqref{1.4}, not only for its own sake, but to define in Section~\ref{s5} the Szeg\H{o} function which will play a critical role in many aspects of the remainder. These preliminaries allow us to present the Golinskii-Ibragimov half of the extension in Section~\ref{s6}. After proving the Coloumb gas representation in Section~\ref{s7}, we can prove the Johansson half of the extension in Section~\ref{s8}. The final three sections provide the proof of \eqref{1.4} in the analytic case: Section~\ref{s9} has a preliminary proving the Christoffel-Darboux formula, and the last two sections finish the proof. I have written a comprehensive book on OPUC \cite{OPUC} and everything in this paper appears there, but it seemed sensible, given the fact that the material is spread through a long book, to pull out exactly what is needed to prove Ibragimov's theorem. While \eqref{1.4} is sharp in one sense, it is not the end of the story by any means. First, there is a simple argument of Johansson \cite{Jo88} that drops the requirement of reality from $L$: if $e^L,L\in L^1$ and $L\in H^{1/2}$, then an extension \eqref{1.4} holds in the sense that \[ e^{-(n+1)\hat L_0} D_n \biggl( e^L \, \f{d\theta}{2\pi}\biggr) \to \exp \biggl(\, \sum_{k=1}^\infty\, \abs{k} \hat L_k \hat L_{-k}\biggr) \] There are also subtleties in extending \eqref{1.4} to allow matrix-valued symbols, to allow complex $w$'s with nonzero winding number, and to determine the leading behavior when $L\notin L^1$. The reader can consult \cite{OPUC} for references on all these issues. \smallskip Over the course of studying asymptotics of Toeplitz determinants, I have learned a lot in discussions with Percy Deift, Rowan Killip, and Irina Nenciu, and I would like to thank them for their insights. Bob Brooks was a substantial mathematician and wonderful person. We lost him too soon. I'm glad to dedicate this article to his memory. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Verblunsky Coefficients and Toeplitz Matrices} \lb{s2} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% If $c_n$ are the moments of a measure, $\mu$, it is natural to form the monic orthogonal polynomials, $\Phi_n (z;d\mu)$, defined by \begin{gather} \Phi_n(z) =z^n + \text{ lower order} \lb{2.1} \\ \langle z^j, \Phi_n\rangle_\mu =0 \qquad \text{if } j=0,1,\dots, n-1 \lb{2.2} \end{gather} where \begin{equation} \lb{2.3} \langle f,g\rangle = \int\, \ol{f(e^{i\theta})}\, g(e^{i\theta})\, d\mu(\theta) \end{equation} is the $L^2 (\partial\bbD, d\mu)$ inner product. In order to form $\Phi_n$ for all $n$, we need the measures $d\mu$ to be nontrivial, that is, not supported on a finite set of points. The matrix elements $c_{k-\ell}=\int e^{i(\ell-k)\theta}\ d\mu =\langle z^k, z^\ell \rangle_\mu$, so $D_n (d\mu)$ is a Gram determinant. Such determinants allow change of basis, that is, if $P_k(z)=z^k + \text{ lower order}$, $\det (\langle P_k,P_\ell \rangle )_{0\leq k,\ell\leq n} =D_n (d\mu)$. We can take $P_k =\Phi_k$, in which case, $\langle P_k,P_\ell\rangle$ is a diagonal matrix (!), and so, \begin{theorem}\lb{T2.1} \begin{equation} \lb{2.3a} D_n (d\mu) =\prod_{j=0}^n \, \|\Phi_j\|_{d\mu}^2 \end{equation} \end{theorem} $\Phi_j$ is the orthogonal projection in $L^2$ of $z^j$ onto the orthogonal complement of $[1,\dots, z^{j-1}]$, the span of $\{1, \dots, z^{j-1}\}$. Since multiplication by $z$ is unitary, $z\Phi_j$ is the projection of $z^{j+1}$ onto $[z,\dots, z^j]^\perp$ while $\Phi_{j+1}$ is the projection of $z^{j+1}$ onto $[1,\dots, z^j]^\perp$, so \begin{equation} \lb{2.3b} \|\Phi_{j+1}\| \leq \|z\Phi_j\| = \|\Phi_j\| \end{equation} Thus, $\|\Phi_j\|$ is decreasing in $j$. It follows that \begin{theorem} \lb{T2.2} \begin{alignat}{2} &\text{\rm{(a)}} \qquad && \lim\, D_n (d\mu)^{1/n+1} = \lim \, \f{D_{n+1}(d\mu)}{D_n (d\mu)} = \lim_{n\to\infty}\, \|\Phi_n\|^2 \lb{2.4} \\ &\text{\rm{(b)}} \qquad && \f{D_{n+1}}{D_n} \leq \f{D_n}{D_{n-1}} \lb{2.5} \end{alignat} \end{theorem} These ideas all go back to Szeg\H{o} \cite{Sz20,Sz21}. The next idea, which is a set of recursion relations of the $\Phi_n$, was first written down by Szeg\H{o} \cite{Szb}, but the basic parameters occurred in a related context in Verblunsky \cite{V35,V36}. To state them, we need to define the reversed polynomials \begin{equation} \lb{2.6} \Phi_n^*(z) = z^n \, \ol{\Phi_n (1/\bar z)} \end{equation} \begin{theorem}\lb{T2.3} For any nontrivial measure, $d\mu$, there exists a sequence of numbers $\{\alpha_n\}_{n=0}^\infty$ so that \begin{equation} \lb{2.7} \Phi_{n+1}(z) =z\Phi_n(z) -\bar\alpha_n \Phi_n^*(z) \end{equation} Moreover, $\alpha_n\in\bbD$ and \begin{equation} \lb{2.8} \|\Phi_{n+1}\|^2 = (1-\abs{\alpha_n}^2) \|\Phi_n\|^2 \end{equation} and \begin{equation} \lb{2.8a} \|\Phi_n\|^2 = c_0 (d\mu) \prod_{j=0}^{n-1} \, (1-\abs{\alpha_j}^2) \end{equation} \end{theorem} {\it Remarks.} 1. In the next section, we will see that $\mu\mapsto \{\alpha_n\}_{n=0}^\infty$ is one-one for $\mu$'s which are normalized. \smallskip 2. There is a converse going back to Verblunsky \cite{V35} (see \cite{OPUC} for many other proofs) that the map from probability measures to $\bigtimes_{n=0}^\infty \bbD$ by $\mu\mapsto \{\alpha_n\}_{n=0}^\infty$ is onto. \smallskip 3. The $\alpha_n$ are called the {\it Verblunsky coefficients} for $d\mu$. \smallskip 4. Applying $Q^*(z)=z^{n+1}\, \ol{Q(1/\bar z)}$ to \eqref{2.7} yields \begin{equation} \lb{2.9} \Phi_{n+1}^*(z) = \Phi_n^*(z) -\alpha_n z\Phi_n(z) \end{equation} \smallskip 5. The proof below is a variant of one of Atkinson \cite{Atk64}. Szeg\H{o}'s proof first proves the Christoffel-Darboux formula (see Section~\ref{s9}) and uses that to prove \eqref{2.7}. \begin{proof} Let $V_ng =z^n \bar g$ on $L^2 (d\mu)$. $V_n$ is anti-unitary, maps $\calP_n$, the polynomials of degree $n$, to themselves, and maps $\Phi_n$ to $\Phi_n^*$. Since $\Phi_n$ is the unique element (up to constants) of $\calP_n$ orthogonal to $\{1,z,\dots, z^{n-1}\}$ and $V_n$ is anti-unitary, $\Phi_n^*$ is the unique element of $\calP_n$ orthogonal to $\{V_n 1, \dots, V_n z^{n-1}\} = \{z^n, z^{n-1}, \dots, z\}$. Now, for $j=1, \dots, n$, \[ \langle z^j, z\Phi_n\rangle = \langle z^{j-1}, \Phi_n\rangle =0 \] and clearly, $\langle z^j, \Phi_{n+1}\rangle =0$. Thus, \[ \langle z^j, \Phi_{n+1} -z\Phi_n\rangle =0 \] for $j=1, \dots, n$. Since $\Phi_n$ and $\Phi_{n+1}$ are monic, $\Phi_{n+1}-z\Phi_n \in\calP_n$. So, by the first part of the proof, \eqref{2.7} holds for a suitable constant $\alpha_n$. Thus \begin{equation} \lb{2.10} \alpha_n = -\ol{\Phi_{n+1}(0)} \end{equation} Since $\Phi_n^* \perp \Phi_{n+1}$, we have \begin{align*} \|\Phi_n\|^2 &= \|z\Phi_n\|^2 = \|\Phi_{n+1} + \bar\alpha_n \Phi_n^*\|^2 \\ &= \|\Phi_{n+1}\|^2 + \abs{\alpha_n}^2 \|\Phi_n\|^2 \end{align*} which implies \eqref{2.8}. \eqref{2.8} in turn implies $\abs{\alpha_n}<1$. \eqref{2.8a} follows by induction and $\|\Phi_0 \|^2 = \|1\|^2 =c_0 (d\mu)$. \end{proof} This leads to \begin{theorem} \lb{T2.4} Suppose $\int d\mu =1$. We have \begin{equation} \lb{2.11} F(d\mu) =\lim_{n\to\infty}\, \f{D_{n+1}(d\mu)}{D_n (d\mu)} = \prod_{j=0}^\infty \, (1 -\abs{\alpha_j}^2) \end{equation} where the product always converges although the limit may be zero. If $F(d\mu) >0$, then \begin{equation} \lb{2.12x} G_n = \f{D_n}{F^{n+1}} \end{equation} obeys \begin{equation} \lb{2.12a} G_{n+1} \geq G_n \end{equation} The limit always exists {\rm{(}}but may be infinite{\rm{)}} and is given by \begin{equation} \lb{2.12} G(d\mu) =\lim_{n\to\infty}\, G_n(d\mu) =\prod_{j=0}^\infty \, (1-\abs{\alpha_j}^2)^{-j-1} \end{equation} \end{theorem} {\it Remarks.} 1. In particular, \begin{align} F>0 &\Leftrightarrow \sum_{n=0}^\infty \, \abs{\alpha_n}^2 <\infty \lb{2.13} \\ G<\infty &\Leftrightarrow \sum_{n=0}^\infty \, (n+1) \abs{\alpha_n}^2 < \infty \lb{2.14} \end{align} \smallskip 2. \eqref{2.11} in a sense goes back to Verblunsky \cite{V36}. \eqref{2.12} seems to have only been noted by Baxter \cite{Bax} many years later. \smallskip 3. If $c_0\neq 1$, $F(d\mu) =c_0 \prod_{j=0}^\infty (1-\abs{\alpha_j}^2)$ while $G(d\mu)$ is still given by \eqref{2.12}. Indeed, $G_n (d\mu) = G_n (d\mu/\int d\mu)$. \smallskip 4. \eqref{2.5} says $\log D_n$ is concave in $n$. The monotonicity of $G_n$ is a standard fact about concave functions with finite asymptotics. \begin{proof} By \eqref{2.3a} and then \eqref{2.8a}, \begin{equation} \lb{2.15} \f{D_{n+1}}{D_n} = \|\Phi_{n+1}\| = \prod_{j=0}^n \, (1-\abs{\alpha_j}^2) \end{equation} (if $\|\Phi_1\|^2 =c_0 =1$). From this and $\abs{\alpha_j}<1$, \eqref{2.11} is immediate. If $F$ is nonzero, \[ \f{D_{n+1}}{D_n}\, \f{1}{F} = \prod_{j=n+1}^\infty\, (1 -\abs{\alpha_j}^2)^{-1} \] so that \begin{align} G_n &= \f{c_0}{F}\, \prod_{k=1}^n \, \biggl[ \f{D_k}{D_{k-1}} \, \f{1}{F} \biggr] \notag \\ &= \prod_{j=0}^\infty\, (1 -\abs{\alpha_j}^2)^{-\min (n,j)-1} \lb{2.21new} \end{align} from which $G_{n+1}\geq G_n$ and \eqref{2.12} are immediate. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Bernstein-Szeg\H{o} Approximations} \lb{s3} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Given a nontrivial probability measure, $d\mu$, with Verblunsky coefficients $\{\alpha_n\}_{n=0}^\infty$, we will identify the measures, $d\mu^{(N)}$, with \begin{equation} \lb{3.1} \alpha_j (d\mu^{(N)}) = \begin{cases} \alpha_j (d\mu) & j=0,1,\dots, N-1 \\ 0 & j\geq N \end{cases} \end{equation} and see $d\mu^{(N)} \to d\mu$ weakly. In many ways, the general proof of the strong Szeg\H{o} theorem will play off this approximation and the distinct approximation obtained by truncating the Fourier series for $L$ in $e^L \f{d\theta}{2\pi}$. As a preliminary, we need \begin{theorem} \lb{T3.1} $\Phi_n (z)$ has all its zeros in $\bbD$. $\Phi_n^*(z)$ has all its zeros in $\bbC\backslash\bar\bbD$. \end{theorem} {\it Remark.} This proof is due to Landau \cite{Land}. See \cite{OPUC} for many other proofs of this fact. \begin{proof} Let $\Phi_n(z_0) =0$. Then $\pi_{n-1} =\Phi_n(z)/(z-z_0)$ is a polynomial of degree $n-1$, and so $\langle\pi_{n-1},\Phi_n\rangle =0$. Since $(z-z_0) \pi_{n-1} =\Phi_n$, \begin{align*} \|\pi_{n-1}\|^2 &= \|z\pi_{n-1}\|^2 = \|z_0 \pi_{n-1} + \Phi_n\|^2 \\ &= \abs{z_0}^2 \|\pi_{n-1}\|^2 + \|\Phi_n\|^2 \end{align*} or \begin{equation} \lb{3.2} (1-\abs{z_0}^2) \|\pi_{n-1}\|^2 = \|\Phi_n\|^2 \end{equation} from which we conclude $\abs{z_0} <1$, that is, the zeros of $\Phi_n$ lie in $\bbD$. Since $\Phi_n^* (z_0) =0$ if and only if $\Phi_n (1/\bar z_0) =0$, the zeros of $\Phi_n^*$ lie in $\bbC\backslash\bar\bbD$. \end{proof} Given $\mu$, define a measure, $\ti\mu^{(N)}$, by \begin{equation} \lb{3.3} d\tilde\mu^{(N)} = \f{d\theta}{2\pi \abs{\Phi_N (e^{i\theta};d\mu)}^2} \end{equation} which can be defined since $\Phi_N (e^{i\theta})\neq 0$ for all $\theta$ by Theorem~\ref{T3.1}. We have \begin{lemma} \lb{L3.2} For $j\geq 0$, \begin{equation} \lb{3.4} \Phi_{N+j} (z;d\ti\mu^{(N)}) = z^j \Phi_N (z;d\mu) \end{equation} Moreover, \begin{equation} \lb{3.4a} \alpha_\ell (d\ti\mu^{(N)}) =0 \end{equation} for $\ell \geq N$. \end{lemma} \begin{proof} Since \begin{equation} \lb{3.5} \abs{\Phi_N(e^{i\theta})}^2 = e^{-iN\theta} \Phi_N (e^{i\theta}) \Phi_N^* (e^{i\theta}) \end{equation} we have for $k\in\bbZ$, $k\leq N-1$ (includes $k<0)$, \begin{align*} \int_0^{2\pi} e^{-ik\theta}\, \f{\Phi_N (e^{i\theta})}{\abs{\Phi_N (e^{i\theta})}^2}\, \f{d\theta}{2\pi} &= \f{1}{2\pi i}\, \oint_{\abs{z}=1} z^{N-k} \, \f{dz}{z\Phi_N^*(z)} \\ &= 0 \end{align*} since $N-k-1 \geq 0$ and $\Phi_N^*(z)^{-1}$ is analytic in a neighborhood of $\bar\bbD$ by Theorem~\ref{T3.1}. This says \begin{equation} \lb{3.6} \langle z^\ell, z^j \Phi_N\rangle_{\ti\mu^{(N)}} =0 \end{equation} for $\ell=0,1,\dots, N+j-1$. Since $z^j \Phi_N$ is monic, \eqref{3.4} holds. By \eqref{2.10}, if $\Phi_{k+1}(0)=0$, then $\alpha_k =0$, so \eqref{3.4} implies $\Phi_{N+j} (0;d\ti\mu^{(N)})=0$, which in turn implies \eqref{3.4a}. \end{proof} \begin{theorem}[Geronimus \cite{Ger46}]\lb{T3.3} Let $d\mu,d\nu$ be two nontrivial measures on $\partial\bbD$. Suppose that for some fixed $N$\!, \begin{equation} \lb{3.7} \Phi_N(z;d\mu) =\Phi_N (z;d\nu) \end{equation} Then \begin{alignat}{2} &\Phi_j (z;d\mu) = \Phi_j (z;d\nu) \qquad && j=0,1,\dots, N-1 \lb{3.8} \\ &\alpha_j (d\mu) = \alpha_j (d\nu) \qquad && j=0,1,\dots, N-1 \lb{3.9} \\ &\f{c_j (d\mu)}{c_0 (d\mu)} = \f{c_j(d\nu)}{c_0(d\nu)} \qquad && j=0,1,\dots, N \lb{3.10} \\ &\|\Phi_j\|_{d\mu}^2 = c_0 (d\mu) c_0 (d\nu)^{-1} \|\Phi_j\|_{d\nu}^2 \qquad && j=0,1,\dots, N \lb{3.11x} \end{alignat} \end{theorem} \begin{proof} \eqref{2.7} and \eqref{2.9} can be written in matrix form \begin{equation} \lb{3.11} \begin{pmatrix} \Phi_{j+1}(z) \\ \Phi_{j+1}^*(z) \end{pmatrix} = \begin{pmatrix} z & -\bar\alpha_j \\ -z\alpha_j & 1 \end{pmatrix} \begin{pmatrix} \Phi_j(z) \\ \Phi_j^*(z) \end{pmatrix} \end{equation} The $2\times 2$ matrix in \eqref{3.11} has an inverse $z^{-1} \rho_j^{-2} \left(\begin{smallmatrix} 1 & \bar\alpha_j \\ \alpha_j z & z\end{smallmatrix}\right)$ where \begin{equation} \lb{3.12} \rho_j = (1-\abs{\alpha_j}^2)^{1/2} \end{equation} Thus \eqref{3.12} implies the inverse Szeg\H{o} recursions: \begin{align} \Phi_j(z) &= \rho_j^{-2} \f{[\Phi_{j+1}(z) + \bar\alpha_j \Phi_{j+1}^*]}{z} \lb{3.13} \\ \Phi_j^*(z) &= \rho_j^{-2} [\Phi_{j+1}^*(z) + \alpha_j \Phi_{j+1}(z)] \lb{3.14} \end{align} \eqref{3.7} implies \[ \alpha_N (d\mu) = \ol{-\Phi_N(0;d\mu)} = \ol{-\Phi_N(0;d\nu)} = \alpha_N(d\nu) \] and thus, by \eqref{3.13} with $j=N-1$, we have \eqref{3.8} for $j=N-1$. By iterating this argument, we conclude that \eqref{3.8} and \eqref{3.9} hold. We need only prove \eqref{3.10} and \eqref{3.11x}, assuming $c_0 (d\mu) = c_0 (d\nu) =1$ since with $d\ti\mu = d\mu/c_0 (d\mu)$, we have $c_n (d\mu) =c_0 (d\mu)c_n (d\ti\mu)$ and $\|\Phi_j\|_{d\mu}^2 =c_0(d\mu) \|\Phi_j\|_{d\ti\mu}^2$. \eqref{3.11x} is immediate from \eqref{2.8a}. We prove \eqref{3.10} when $c_0(d\mu)=c_0 (d\nu)$ by induction and noting $\langle 1, \Phi_k\rangle =0$ yields a formula for $c_k$ in terms of the coefficients of $\Phi_k$ and $c_0, c_1, \dots, c_{k-1}$. \end{proof} {\it Remark.} The last paragraph of the proof shows that the $\alpha$'s determine the $c$'s and proves that the map of $\mu$ to $\alpha$ is one-one. \smallskip To succinctly state this section's final result, we introduce \begin{equation} \lb{3.15} \varphi_N (z;d\mu) = \f{\Phi_N (z;d\mu)}{\|\Phi_N\|} \end{equation} the orthonormal polynomials. By \eqref{2.8a}, \begin{equation} \lb{3.16} \varphi_N (z;d\mu) =\prod_{j=0}^{N-1} \rho_j^{-1} c_0 (d\mu)^{-1/2} \Phi_N (z;d\mu) \end{equation} We define \begin{equation} \lb{3.16a} \kappa_n = \biggl(\, \prod_{j=0}^{n-1} \rho_j^{-1}\biggr) (c_0 (d\mu))^{-1/2} = \|\Phi_n\|^{-1} \end{equation} and \begin{equation} \lb{3.16b} \kappa_\infty = \lim_{n\to\infty}\, \|\Phi_n\|^{-1} \end{equation} which exists (but it may be $+\infty$) by $\rho_j <1$. The infinite product in \eqref{3.16a}, and so $\kappa_\infty$, is finite if and only if \begin{equation} \lb{3.17c} \kappa_\infty < \infty \Leftrightarrow \sum_{j=0}^\infty \, \abs{\alpha_j}^2 <\infty \end{equation} We also note the translation of \eqref{2.7}/\eqref{2.9} from $\Phi$ to $\varphi$: \begin{align} \varphi_{n+1}(z) &= \rho_n^{-1} (z\varphi_n (z) - \bar\alpha_n \varphi_n^*(z)) \lb{3.17d} \\ \varphi_{n+1}^* (z) &= \rho_n^{-1} (\varphi_n^*(z) - \alpha_n z \varphi_n^*(z)) \lb{3.17e} \end{align} \begin{theorem}\lb{T3.4} Let $d\mu$ be a nontrivial probability measure on $\partial\bbD$. Define \begin{equation} \lb{3.17} d\mu^{(N)} =\f{d\theta}{2\pi\abs{\varphi_N (e^{i\theta})}^2} \end{equation} Then $d\mu^{(N)}$ is a probability measure on $\partial\bbD$ for which \eqref{3.1} holds. As $N\to\infty$, $d\mu^{(N)}\to d\mu$ weakly. \end{theorem} {\it Remark.} We call measures of the form \eqref{3.17} {\it BS measures} and $d\mu^{(N)}$ the {\it BS approximation}. \begin{proof} Since $d\mu^{(N)}$ is a multiple of $d\ti\mu^{(N)}$, we have the bottom half of \eqref{3.1} by \eqref{3.4a}. Since \eqref{3.4} holds for $j=0$, Theorem~\ref{T3.3} and \eqref{3.9} imply the top half of \eqref{3.1}. Since $\Phi_N =\|\Phi_N\|_\mu \varphi_N$, we clearly have \[ \|\Phi_N\|_{\mu^{(N)}}^2 = \|\Phi_N\|_\mu^2 \] so, by \eqref{3.11}, $c_0 (d\mu^{(N)}) =c_0 (d\mu) =1$, that is, $d\mu_N$ is a probability measure. By the above and \eqref{3.10}, we have \begin{equation} \lb{3.18} c_j (d\mu^{(N)}) = c_j (d\mu) \qquad j=0,1,\dots, N \end{equation} This and its complex conjugate implies that for any Laurent polynomial, $f$ (polynomial in $z$ and $z^{-1}$), \begin{equation} \lb{3.19} \lim_{N\to\infty}\, \int f(e^{i\theta})\, d\mu^{(N)} = \int f(e^{i\theta})\, d\mu \end{equation} since the left side is equal to the right for $N$ large. Since Laurent polynomials are dense in $C(\partial\bbD)$, \eqref{3.19} holds for all $f$, that is, we have weak convergence. \end{proof} We note we have proven that \begin{equation} \lb{3.20} \alpha_j (d\mu^{(N)}) = \begin{cases} \alpha_j & j\leq N-1 \\ 0 & j\geq N \end{cases} \end{equation} The ideas of this section go back to Geronimus \cite{Ger46} and were rediscovered in \cite{ENZG91} and \cite{DGK78}. In particular, the use of inverse recursion to prove Geronimus' theorem is taken from \cite{DGK78}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Szeg\H{o}'s Theorem} \lb{s4} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section, our main goal is to prove \begin{theorem}\lb{T4.1} For any $w\in L^1 (\f{d\theta}{2\pi})$, \begin{equation} \lb{4.1} \lim_{N\to\infty}\, \f{1}{N} \, \log D_N(w) = \int \log w(\theta) \, \f{d\theta}{2\pi} \end{equation} \end{theorem} {\it Remarks.} 1. Since $\log(x) \leq x-1$ and $w(\theta)\in L^1$, $\int \max (0,\log w(x)) \f{d\theta}{2\pi} <\infty$, so the integral on the right side of \eqref{4.1} is either convergent or diverges to $-\infty$, in which case \eqref{4.1} says $D_n(w)^{1/n} \to 0$. \smallskip 2. This was conjectured by P\'olya \cite{Pol} and proven by the twenty-year old Szeg\H{o} in 1915 \cite{Sz15}. Our proof here is essentially the one Szeg\H{o} presented in \cite{Sz20,Sz21}. \smallskip 3. The same result is true for the symbol $d\mu = w(\theta)\f{d\theta}{2\pi} + d\mu_\s$, that is, the limit is independent of $d\mu_\s$. This extension was first proven by Verblunsky \cite{V36}. We will not prove this more general result here (\cite{OPUC} has four different proofs in Chapter~2) since it is peripheral to our main result. \smallskip The first half of the theorem is a simple use of Jensen's inequality: \begin{proposition}\lb{P4.2} Let $w=e^L$ with $w,L\in L^1$. Then \begin{equation} \lb{4.2} \|\Phi_n\|_{w\f{d\theta}{2\pi}}^2 \geq \exp\biggl( \int L(\theta)\, \f{d\theta}{2\pi} \biggr) \end{equation} In particular, \begin{equation} \lb{4.3} D_n(w) \geq \exp\biggl( \int (n+1)L(\theta)\, \f{d\theta}{2\pi}\biggr) \end{equation} \end{proposition} \begin{proof} \eqref{4.3} follows from \eqref{4.2} and \eqref{2.3a}. To prove \eqref{4.2}, we write \begin{align} \|\Phi_n\|^2 = \|\Phi_n^*\|^2 &= \int \exp (2\log \abs{\Phi_n^*(e^{i\theta})} + L(e^{i\theta}))\, \f{d\theta}{2\pi} \notag \\ &\geq \exp\biggl( \int [2\log \abs{\Phi_n^*(e^{i\theta})} + L(e^{i\theta})]\biggr)\, \f{d\theta}{2\pi} \lb{4.4x} \end{align} by Jensen's inequality. By Theorem~\ref{T3.1}, $\log (\Phi_n^*(z))$ is analytic in $\bbD$, so \begin{align*} \int \log \abs{\Phi_n^* (e^{i\theta})}\, \f{d\theta}{2\pi} &= \Real \int \log (\Phi_n^*(e^{i\theta}))\, \f{d\theta}{2\pi} \\ &= \log \abs{\Phi_n^* (0)} =0 \end{align*} since $\Phi_n$ monic implies $\Phi_n^*(0)=1$. \end{proof} The other half of the theorem depends on a variational principle noted by Szeg\H{o}: \begin{proposition}\lb{P4.3} We have for any $w\in L^1 (\partial\bbD, \f{d\theta}{2\pi})$, \begin{equation} \lb{4.4} \lim_{n\to\infty} \, [D_n (w)]^{1/n} = \inf \biggl\{\int \abs{f(e^{i\theta})}^2 w(\theta)\, \f{d\theta}{2\pi} \biggm| f\in H^\infty (\bbD);\, f(0)=1 \biggr\} \end{equation} \end{proposition} {\it Remark.} $H^\infty (\bbD)$ is the Hardy space of bounded analytic functions on $\bbD$. By general principles \cite{Duren,Rudin}, for $\f{d\theta}{2\pi}$ a.e.~$e^{i\theta} \in\partial\bbD$, $\lim_{r\uparrow 1} f(re^{i\theta})$ exists, and that is what we mean by $f(e^{i\theta})$ in \eqref{4.4}. \begin{proof} Since $\|\Phi_n^*\| = \|\Phi_n\|$, by \eqref{2.4}, \begin{align} \text{LHS of \eqref{4.4}} &= \lim_{n\to\infty}\, \|\Phi_n^*\|^2 \notag \\ &= \inf_n \, \|\Phi_n^*\|^2 \lb{4.5} \end{align} by \eqref{2.3b}. By the argument at the start of the proof of Theorem~\ref{T2.3}, \[ \Phi_n^* = \pi_n 1 \] where $\pi_n$ is the projection in the space of polynomials, $\calP_n$, of degree $n$ onto the orthogonal component of the span of $z,z^2,z^3, \dots, z^n$. This span is $\{P\in\calP_n \mid P(0) =0\}$, so by standard geometry, \begin{equation} \lb{4.6} \|\Phi_n^*\|^2 = \inf \{\|P\|^2 \mid P\in\calP_n;\, P(0)=1\} \end{equation} proving again that $\|\Phi_n^*\|^2$ is decreasing in $n$ and proving \eqref{4.4} if $H^\infty$ is replaced by the set of all polynomials. To complete the proof, we need only show that for any $f\in H^\infty$ with $f(0)=1$, there are polynomials $P_\ell(z)$ so that $P_\ell (0)=1$ and \begin{equation} \lb{4.7} \int\, \abs{P_\ell (e^{i\theta})}^2 w(\theta) \, \f{d\theta}{2\pi} \to \int\, \abs{f(e^{i\theta})}^2 w(\theta)\, \f{d\theta}{2\pi} \end{equation} If $f$ is analytic in a neighborhood of $\bbD$, the Taylor approximations converge uniformly on $\bar\bbD$, so \eqref{4.7} holds. For general $f$, by the dominated convergence theorem, \[ \lim_{r\uparrow 1} \, \int \, \abs{f(re^{i\theta})}^2 w(\theta)\, \f{d\theta}{2\pi} = \int \, \abs{f(e^{i\theta})}^2 w(\theta)\, \f{d\theta}{2\pi} \] so, by a two-step approximation, we find $P_\ell$'s so \eqref{4.7} holds. \end{proof} \begin{proof}[Proof of Theorem~\ref{T4.1}] We will prove that for any $\veps >0$, there is an $f$ in $H^\infty$ with $f(0)=1$ and \begin{equation} \lb{4.7a} \int\, \abs{f(e^{i\theta})}^2 w(\theta) \, \f{d\theta}{2\pi} \leq \exp \biggl( \int \log (w(\theta)+\veps)\, \f{d\theta}{2\pi}\biggr) \end{equation} so taking $\veps\downarrow 0$ yields the opposite inequality to \eqref{4.3}. Define \begin{equation} \lb{4.8} g(z) = \exp \biggl( -\int \log (w(\theta)+\veps) \biggl( \f{e^{i\theta}+z}{e^{i\theta}-z} \biggr) \, \f{d\theta}{4\pi}\biggr) \end{equation} and $f(z) = g(z)/g(0)$, so $f(0)=1$. Moreover, $\abs{g(z)}\leq \veps^{-1/2}$ by the fact that the Poisson kernel \[ P_r (\theta, \varphi) = \Real \biggl( \f{e^{i\theta} + re^{i\varphi}} {e^{i\theta} - re^{i\varphi}}\biggr) \] is nonnegative and $\int \f{d\theta}{2\pi} P_r (\theta,\varphi)=1$. By standard maximal function arguments (see \cite{Rudin}), $\abs{g(e^{i\theta})}=\abs{w(\theta) +\veps}^{-1/2}$, so \[ \int \, \abs{f(e^{i\theta})}^2 w(\theta) \, \f{d\theta}{2\pi} \leq g(0)^{-2} = \text{RHS of \eqref{4.7a}} \] \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The Szeg\H{o} Function} \lb{s5} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% When $w(\theta) =e^{L(\theta)}$ with $L\in L^1$, Szeg\H{o} \cite{Sz20,Sz21} introduced a natural function, $D(z)$ on $\bbD$, which will play a critical role in several places below: \begin{equation} \lb{5.1} D(z) =\exp\biggl( \int \biggl( \f{e^{i\theta} +z}{e^{i\theta}-z}\biggr) L(\theta)\, \f{d\theta}{4\pi}\biggr) \end{equation} Do not confuse $D_n$ and $D(z)$. Both symbols are standard, but the objects are very different. \begin{theorem}\lb{T5.1} \begin{SL} \item[{\rm{(a)}}] If \eqref{2.13} holds, then for $\abs{z}<1$, \begin{align} D(z) &= D(0) \exp\biggl( \, \sum_{k=1}^\infty \hat L_k z^k \biggr) \lb{5.2} \\ D(0) &= \biggl[ c_0 \biggl( w\, \f{d\theta}{2\pi}\biggr)\biggr]^{1/2} \, \prod_{n=0}^\infty \, (1-\abs{\alpha_n}^2)^{1/2} \lb{5.3} \end{align} \item[{\rm{(b)}}] $D(z)$ lies in $H^2 (\bbD)$. \item[{\rm{(c)}}] $\lim_{r\uparrow 1} D(re^{i\theta})\equiv D(e^{i\theta})$ exist for a.e.~$\theta$ and \begin{equation} \lb{5.4} \abs{D(e^{i\theta})}^2 =w(\theta) \end{equation} \item[{\rm{(d)}}] $D$ is nonvanishing on $\bbD$. \end{SL} \end{theorem} \begin{proof} (a) We get \eqref{5.2} and \begin{equation} \lb{5.5} D(0) =\exp (\tfrac12\, L_0) \end{equation} from \eqref{5.1} and \[ \f{e^{i\theta} + z}{e^{i\theta} -z} = 1+2\sum_{j=0}^\infty \, (e^{-i\theta} z)^n \] uniformly in $e^{i\theta}\in\partial\bbD$ and $z\in\{\abs{z}0$. \end{theorem} \begin{proof} Analyticity says $\abs{\hat L_k}\leq Ce^{-A\abs{k}}$ for some $A>0$. So, by \eqref{5.2}, $D(z)$ is analytic and nonvanishing in a disk of radius $e^A$. In particular, if \begin{equation} \lb{5.13} D(z)^{-1} =\sum_{j=0}^\infty d_{j,-1} z^j \end{equation} then \begin{equation} \lb{5.14} \abs{d_{j,-1}}\leq C_1 e^{-A\abs{j}/2} \end{equation} Plug \eqref{5.13} into \eqref{5.10} and note that \[ \int \ol{\Phi_{n+1} (e^{i\theta})}\, e^{ik\theta}\, d\mu(\theta) =0 \] for $k=0,1,\dots, n$. Thus \begin{align} \abs{\alpha_n} &\leq \kappa_\infty \sum_{k=n+1}^\infty \, \abs{d_{k,-1}} \, \biggl| \int \ol{\Phi_{n+1}(e^{i\theta})}\, e^{ik\theta}\, d\mu \biggr| \notag \\ &\leq \kappa_\infty \|\Phi_{n+1}\| \sum_{k=n+1}^\infty\, \abs{d_{k,-1}} \notag \\ &\leq \kappa_\infty \sum_{k=n+1}^\infty \, \abs{d_{k,-1}} \lb{5.15} \end{align} since $\|\Phi_{n+1}\|\leq 1$. So, by \eqref{5.14}, \begin{equation} \lb{5.16} \abs{\alpha_n} \leq C_2 e^{-A\abs{n}/2} \end{equation} By \eqref{2.9} and $\abs{\Phi_n^* (e^{i\theta})} = \abs{\Phi_n (e^{i\theta})}$, we have \begin{align} \sup_\theta\, \abs{\Phi_{n+1}^* (e^{i\theta})} &\leq (1+\abs{\alpha_n}) \sup_\theta \, \abs{\Phi_n^* (e^{i\theta})} \notag \\ &\leq \prod_{j=0}^n \, (1+\abs{\alpha_j}) \notag \\ &\leq \exp \biggl( \, \sum_{j=0}^\infty \, \abs{\alpha_j}\biggr) = C_4 <\infty \lb{5.17} \end{align} by iteration. $C_4<\infty$ follows from \eqref{5.16}. Since $\Phi_n^*$ is analytic, we get \begin{equation} \lb{5.18} \sup_{z\in\bar\bbD}\, \abs{\Phi_n^*(z)} \leq C_4 \end{equation} and thus, since $\Phi_n^*(z) = z^n \, \ol{\Phi_n (1/\bar z)}$, we get \begin{equation} \lb{5.19} z\in\bbC\backslash\bbD \Rightarrow \abs{\Phi_n (z)} \leq C_4 \abs{z}^n \end{equation} Returning to \eqref{2.9}, \begin{align*} \sum_{n=0}^\infty\, \abs{\Phi_{n+1}^*(z) - \Phi_n^*(z)} &\leq \sum_{n=0}^\infty\, \abs{\alpha_n} \, \abs{\Phi_n(z)} \\ &\leq C_4 C_2 \sum_{n=0}^\infty\, \abs{ze^{-A/2}}^n \end{align*} showing $\Phi_n^*$, and so $\varphi_n^*$, converges uniformly in $\{z\mid \abs{z} \leq e^{A/4}\}$. Since the limit is $D^{-1}$ in $\bbD$, it is $D^{-1}$ in this larger disk. \end{proof} Finally, in terms of $D$, we want to rewrite the second term in the Szeg\H{o} asymptotic formula \eqref{1.4}: \begin{theorem}\lb{T5.5} Let $d\mu =e^{L(\theta)} \f{d\theta}{2\pi}$ with $L\in L^1$. Let $\hat L_k$ be given by \eqref{1.3}. Then \begin{equation} \lb{5.20} \sum_{k=1}^\infty k \abs{\hat L_k}^2 = \f{1}{\pi} \int_{\abs{z}\leq 1} \, \abs{D(z)}^{-2} \biggl| \f{\partial D}{\partial z}\biggr|^2 \, d^2 z \end{equation} where both sides can be infinite. \end{theorem} \begin{proof} \eqref{5.20} follows by taking $r\uparrow 1$ in \begin{equation} \lb{5.21} \sum_{k=1}^\infty k\, \abs{\hat L_k}^2 r^{2k} =\f{1}{\pi} \int_{\abs{z} \leq r} \, \abs{D(z)}^{-2} \biggl| \f{\partial D}{\partial z}\biggr|^2 \, d^2 z \end{equation} (by using monotone convergence). To prove \eqref{5.21}, note that by \eqref{5.2}, \[ \log \biggl[ \f{D(z)}{D(0)}\biggr] = \sum_{k=1}^\infty \hat L_k z^k \] converges uniformly in $\abs{z}