One Hat Cyber Team

View File Name : BLkrein.tex

% ------------------------------------------------------------------------
% bmultdoc.tex for birkmult.cls*******************************************
% ------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\documentclass{birkmult}
%
%
% THEOREM Environments (Examples)-----------------------------------------
%
 \newtheorem{thm}{Theorem}[section]
 \newtheorem{cor}[thm]{Corollary}
 \newtheorem{lem}[thm]{Lemma}
 \newtheorem{prop}[thm]{Proposition}
 \theoremstyle{definition}
 \newtheorem{defn}[thm]{Definition}
   \newtheorem{prob}[thm]{Problem}
 \newtheorem{hyp}[thm]{Hypothesis}
 \newtheorem{conv}[thm]{Convention}
 \theoremstyle{remark}
 \newtheorem{rem}[thm]{Remark}
 \newtheorem*{ex}{Example}
 \numberwithin{equation}{section}
 
 
 \usepackage{pdfsync}
 
 
 \usepackage{amsmath,color,graphicx}
\usepackage{amssymb}
\usepackage{euscript}
\newcommand{\script}[1]{\EuScript{#1}}
\newcommand{\Tr}{\operatorname{\!{^{\mbox{\scriptsize\sf T}}}}\!}
\newcommand{\rank}{\operatorname{rank\,}}

\begin{document}
%-------------------------------------------------------------------------
% editorial commands: to be inserted by the editorial office
%
%\firstpage{1}
%\volume{228}
%\Copyrightyear{2004}
%\DOI{003-0001}
%
%
%\seriesextra{Just an add-on}
%\seriesextraline{This is the Concrete Title of this Book\br H.E. R and S.T.C. W, Eds.}
%
% for journals:
%
%\firstpage{1}
%\issuenumber{1}
%\Volumeandyear{1 (2004)}
%\Copyrightyear{2004}
%\DOI{003-xxxx-y}
%\Signet
%\commby{inhouse}
%\submitted{March 14, 2003}
%\received{March 16, 2000}
%\revised{June 1, 2000}
%\accepted{July 22, 2000}
%
%
%
%---------------------------------------------------------------------------
%Insert here the title, affiliations and abstract:
%
\title[The moment problem for rational measures]
 {The Moment Problem for Rational Measures: Convexity in the Spirit of Krein}
%----------Author 1
\author[C. I. Byrnes]{Christopher I. Byrnes}

\address{%
Department of Electrical and Systems Engineering\\
Washington University\\
 St.~Louis, MO 63130\\USA
}

\email{chrisbyrnes@wustl.edu}



\thanks{This research was supported in part by grants from
AFOSR, Swedish Research Council, Swedish Foundation for Strategic Research, and the G\"oran Gustafsson Foundation.}
%----------Author 2
\author[A. Lindquist]{Anders Lindquist}
\address{Department of Mathematics\br
Optimization and Systems Theory\br
Royal Institute of Technology\br
SE-100 44 Stockholm\\Sweden}
\email{alq@math.kth.se}
%----------classification, keywords, date
\subjclass{Primary 30E05; Secondary  44A60}

\keywords{ Moment problems, interpolation, rational positive measures, convex optimization}

\date{January 1, 2004}
%----------additions
\dedicatory{In memory of Mark Grigoryevich Krein on the occasion of the 100th anniversary of his birth}
%%% ----------------------------------------------------------------------

\begin{abstract}
 The moment problem as formulated by Krein and Nude\-l'man  is a beautiful generalization of several important classical moment problems, including the power moment problem, the trigonometric moment problem and the moment problem arising in Nevanlinna-Pick interpolation. Motivated by classical applications and examples, in both finite and infinite dimensions, we recently formulated  a new version of this problem that we call the moment problem for positive rational measures. The formulation reflects the importance of rational functions in signals, systems and control. While this version of the problem is decidedly nonlinear, the basic tools still rely on convexity. In particular, we present a solution to this problem in terms of a nonlinear convex optimization problem that generalizes the maximum entropy approach used in several classical special cases. %For certain special cases, we provide a succinct closed form solution. 
\end{abstract}

%%% ----------------------------------------------------------------------
\maketitle
%%% ----------------------------------------------------------------------
%\tableofcontents


\section{Introduction}
The moment problem for positive measures  is the synthesis, over the course of more than 70 years by Krein and his collaborators  (see \cite{AK,KN} and references therein), of many important  classical problems in pure and applied mathematics. This paper is devoted to the study of a class of moment problems, which we refer to as the moment problem for positive {\em rational\/} measures, whose  formulation reflects the importance of rational functions in signals, systems and control. 
This class of  problems abstracts the recent work of a number of authors \cite{BLGM,SIGEST,BGL1,BLkimura,BLmoments,BLimportantmoments,Gthesis,G1,Georgiou3} who  incorporated various complexity constraints into the refinements of the moment problem for arbitrary  positive measures \cite{KN}. 
 
 We refer to this problem as  the moment problem for positive {\em rational\/} measures. In this paper we develop some basic results for this problem, closely following the approach outlined in \cite{KN}. Indeed, in Section 2 we recall the fundamental result on the generalized moment problem as derived in \cite{KN} using convex cones in finite dimensional function spaces and properties of positive measures.  In Section 3 we derive  similar basic results for the moment problem for positive rational measures using a topological proof that mirrors the steps in the convexity proof in \cite{KN}.
 
 While this version of the problem is decidedly nonlinear, one can still develop an approach based entirely on convexity. In particular, in Section 4 we present a synthesis of our topological approach with a nonlinear convex optimization problem that we discovered in the context of interpolation problems \cite{SIGEST,BGL1} and generalized to  the case of moment problems with complexity constraints \cite{BLkimura,BLmoments,BLimportantmoments,Georgiou3}. In fact, the topological approach developed in Section 3 allows us to significantly streamline our previous proofs concerning the convex functional and its extema. Naturally, the optimization problem itself generalizes the maximum entropy approach. Indeed, for cases where the space of test functions lie in the Hardy space on the unit circle, we provide in Section 5 a succinct closed form for the maximum entropy solution. In Section 6 we describe some amplifications of our basic results using differentiable maps and manifolds, a methodology upon which we based an alternative approach to this problem in \cite{BLmoments,BLimportantmoments} and which is also streamlined by our 
 topological arguments.

 \section{The moment problem following Krein and Nudel'man}\label{GMP}
The fundamental result on the generalized moment problem derived in \cite{KN} is based on two results, one about properties of convex cones in finite dimensional spaces of continuous functions and the other about properties of postive measures. 
 
The first result concerns a subspace $\mathfrak{P}$ of the Banach space $C[a,b]$ of complex-valued continuous functions defined on the real interval
$[a,b]$ and a choice of basis $(u_0,u_1,\cdots,u_n)$ of $\mathfrak{P}.$  
If  $p \in \frak{P}$ we denote by $P$  its real part $P:=\mbox{\rm Re}(p)$. Following \cite{KN}, we define the subset  $\frak{P}_+$ of those elements $p\in \frak{P}$ such that $P\geq 0.$  The space $\frak{P}_+ \subset \frak{P}$ is  a closed, convex cone. In terms of the basis $(u_i)$, every $\phi \in \mathfrak{P}^\ast$ corresponds to a complex sequence $c = (c_0,c_1,\cdots,c_n)  \in \mathbb{C}^{n+1}.$ Since every $\phi$ is determined by its real part as a linear functional on $\mathfrak{P}$ as a real vector space, we can characterize elements of the dual cone $\mathfrak{P} _+^{\mbox{\scriptsize\sf T}}$ as those sequences $c$ satisfying 
\begin{equation}\label{innerprod} 
\langle c,p\rangle := \mbox{\rm Re}\left\{\sum_{k=0}^n p_k c_k\right\}  \geq 0 
\end{equation}
for all  $p\in\mathfrak{P}_+$. Such a sequence is classically called  {\em positive}, and the space of positive sequences is denoted by $\frak{C}_+$. In particular, $\mathfrak{C}_+ $ is  a  closed, convex cone with $\mathfrak{C}_+^{\mbox{\scriptsize\sf T}}= \mathfrak{P}_+$.  

Following \cite{KN}, consider  the curve
\begin{displaymath}
U(t) =  \begin{pmatrix}
     u_0(t)     \\
      u_1(t)\\\vdots\\u_n(t)  
\end{pmatrix}, \quad a \leq t \leq b.  
\end{displaymath}
%$U(t) = (u_0(t),u_1(t), \dots u_n(t))^{\mbox{\scriptsize\sf T}},$ for $a \leq t \leq b.$ 
We then define the subset $U =\{U(t): t \in [a,b]\}\subset \mathbb{C}^{n+1},$  and let $K(U)$ denote its convex conic hull. Clearly, $K(U)^{\mbox{\scriptsize\sf T}} = \mathfrak{P}_+,$ from which follows:

\begin{thm}[\cite{KN}]\label{TheoremA} $K(U) = \mathfrak{C}_+.$
\end{thm}

We now turn to some results concerning positive measures. Given $c\in \mathbb{C}^{n+1},$ the generalized moment problem (see \cite{KN}) is to find a positive measure $d\mu$ such that 
\begin{equation}\label{moments}
\int_a^b u_k(t) d\mu(t) = c_k, \quad k=0,1,\cdots,n.
\end{equation} 
For the sake of brevity, from now on we shall refer to this problem as simply the moment problem, omitting the adjective ``generalized".
More generally, let 
\begin{equation}
\label{momentmap}
\mathfrak{M}: C[a,b]^\ast \to \mathbb{C}^{n+1}
\end{equation} 
be the continuous mapping defined  via \eqref{moments} for an arbitrary bounded measure \\$d\mu \in C[a,b]^\ast$ and 
consider the subset $\mathcal{M}_+ \subset C[a,b]^\ast$  of positive measures. 

\begin{lem}[\cite{KN}]\label{MinC} 
$\mathfrak{M}(\mathcal{M}_+) \subset \mathfrak{C}_+.$
\end{lem} 
\begin{proof} If $p\in \mathfrak{P}_+ = \mathfrak{C}_+^{\mbox{\scriptsize\sf T}},$ then 
\begin{equation}\label{innerprod2} 
\langle c,p\rangle := \mbox{\rm Re}\left\{\sum_{k=0}^n p_k c_k\right\} = \int_a^b P\, d\mu \geq 0,
\end{equation}
so that $c \in \mathfrak{C}_+.$
\end{proof}

 By Theorem \ref{TheoremA} and Lemma \ref{MinC}, we have that $ \mathfrak{M}(\mathcal{M}_+)$ is a convex subset of $K(U).$ On the other hand, by choosing $d\mu = \delta_{t_{0}}$ for each $t_0 \in [a,b],$ it follows that $U \subset  \mathfrak{M}(\mathcal{M}_+).$ In particular, to say that $\mathfrak{M}(\mathcal{M}_+)$ is closed is to say that $K(U) \subset  \mathfrak{M}(\mathcal{M}_+).$ 
 
 In \cite{KN}, the Helly Selection Theorem is used to show that $\mathfrak{M}(\mathcal{M}_+)$ is closed in $\mathfrak{C}_+$  under the following hypothesis. 

\begin{hyp}\label{Hypothesis1}
There exists $p \in \frak{P}_+$ such that $P>0$ on $[a,b]$.
\end{hyp}

%\medskip 
%{\bf H1.} There exists $p \in \frak{P}_+$ such that $P>0$ on $[a,b]$.

%\medskip

\begin{thm}[\cite{KN}]\label{TheoremB}   
Whenever Hypothesis~\ref{Hypothesis1} holds, $ K(U)= \mathfrak{M}(\mathcal{M}_+).$ In particular, 
\begin{equation}
\label{GMT}
\mathfrak{C}_+=\mathfrak{M}(\mathcal{M}_+).
\end{equation}
\end{thm}



  Of course, in order for the moment equations to hold it is necessary that
$c_k$ be real whenever $u_k$ is real. Moreover, a purely imaginary moment condition can
always be reduced to a real one. 

\begin{conv}\label{Convention1}
Henceforth we shall assume that $u_0,\dots,u_{r-1}$ are real functions and
$u_r,\dots,u_n$ are complex-valued functions whose real and imaginary parts,
taking together with $u_0,\dots,u_{r-1}$, are linearly independent over
$\Bbb{R}$.
\end{conv}


In particular, we may regard
$\frak{P}$ as the real vector space $\Bbb{R}^{2n-r+2}$   and $\mathfrak{C}_+ \subset \Bbb{R}^{2n-r+2}.$ 
Therefore, it follows \cite{KN} that $\mathfrak{C}_+$ is a closed convex cone of dimension $2n-r+2,$ with interior $\mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;$ consisting of {\em strictly positive}  sequences $c;$ i.e., those sequences $c$ satisfying   
\begin{equation}\label{innerprod2} 
\langle c,p\rangle := \mbox{\rm Re}\left\{\sum_{k=0}^n p_k c_k\right\} > 0 
\end{equation}
for all  $p\in\mathfrak{P}_+\smallsetminus\{0\}$. Assuming Hypothesis \ref{Hypothesis1}, it then follows that 
$\frak{P}_+$ is also a closed convex cone of dimension $2n-r+2,$ with a nonempty interior $\mathfrak{P}_+\kern-.17 in ^{^{^\circ}}\;\;$ consisting of those $p \in \frak{P}_+$ for which ${\rm Re}(p) >0.$




\section{The Main Results}
 
%Throughout the remainder of this paper we assume H1 and that $\mathfrak{P}$ admits a basis satisfying Convention 1.

 In the power and the trigonometric moment problems,  the elements of the subspace $\mathfrak{P}$  are polynomials and trigonometric polynomials, respectively. In part for this reason,  the elements of the subspace $\mathfrak{P}$ in an arbitrary  moment problem are referred to as ``polynomials in $\mathfrak{P}$". Following this precedent, we shall refer to the ratio $p/q$ with $p,q\in\mathfrak{P}$ as a ``rational function". For the classical Nevanlinna-Pick interpolation problem, it turns out that $\mathfrak{P}$ is a coinvariant subspace of $H^2$ so that the ``polynomials" are rational functions $\sigma/\tau$, where $\tau$ is fixed. This of course  implies that the rational functions in $\mathfrak{P}$ are rational in the usual sense. 

\begin{defn} The functions $P:={\rm Re}(p)$, for $p\in \mathfrak{P}$ in the moment problem are referred to as {\em real polynomials for} $\mathfrak{P}$.  We shall refer to the ratio $P/Q$ with $p,q\in\mathfrak{P}$ as a {\em real rational functions for} $\mathfrak{P}$. 
\end{defn}

\begin{rem} Under Convention~\ref{Convention1},
\begin{equation}
p:=\sum_{k=0}^n p_ku_k\in \frak{P}
\end{equation}
corresponds to an $(n+1)$-tuple of points $(p_0.p_1,\dots,p_n)$, where $p_0,p_1,\dots,p_{r-1}$ are real and $p_r,p_{r+1},\dots,p_n$ are complex. Moreover, $p$ is determined by $P$ \cite[p. 165]{BLmoments}. 
\end{rem}

The  moment problem is about measures and combining these two concepts leads us to following definition.

\begin{defn}\label{ratdefn}
 Any measure of the form
\begin{equation} \label{rateqn}
d\mu  =\frac{P(t)}{Q(t)}dt,
\end{equation}
where $P,Q$ are positive real polynomials for $\mathfrak{P}$,  is  a  {\em  rational positive measure}. Let $\mathcal{R}_+\subset \mathcal{M}_+$  denote the subset of  rational positive measures.
\end{defn}

\begin{prob}\label{ratprob}  Given a sequence of complex numbers $c_0,c_1,\cdots,c_n$ and a subspace  $\mathfrak{P} = {\rm span}(u_0, \dots u_n) \subset C[a,b],$ 
the {\em moment problem for rational measures} is to parameterize all positive rational measures $\frac{P(t)}{Q(t)}dt$ such that 
\begin{equation}\label{ratmoments}
\int_a^b u_k(t)\frac{P(t)}{Q(t)}dt = c_k, \quad k=0,1,\cdots,n.
\end{equation} 
\end{prob}

We shall need an additional hypothesis to accomodate the restriction to rational positive measures. 

\begin{hyp}\label{Hypothesis2}
The space $\mathfrak{P}$ consists of Lipschitz continuous functions.
\end{hyp}

%\medskip
%\noindent {\bf Hypothesis H2.} $\mathfrak{P}$ consists of Lipschitz continuous functions. 

\begin{rem} \label{LC}
To the best of our knowledge, all instances of the generalized moment problem that arise in systems and control involve subspaces of $C[a,b]$ consisting of Lipschitz continuous functions. Moreover, we recall the classical result that, if  $\mathfrak{P}$ is spanned by a Chebyshev system (or T-system) and contains a constant function, then after a reparameterization $\mathfrak{P}$ consists of Lipschitz continuous functions \cite[p. 37]{KN}.
\end{rem}

 In the setting  of Section \ref{GMP}, our first result is the following.

\begin{thm}\label{MainTheorem} If Hypotheses \ref{Hypothesis1} and \ref{Hypothesis2} hold, then $$\mathfrak{M}(\mathcal{R}_+) = \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;.$$ In other words,  the moment problem for rational measures is solvable if, and only if, the sequence $c$ is strictly positive.
\end{thm}

For any $d\mu \in \mathcal{R}_+,$  consider the sequence $c$ defined by \eqref{moments} and any $p =\sum_{k=0}^n p_k u_k \in \mathfrak{P}_+\smallsetminus\{0\}.$ Then

\begin{equation}\label{innerprod3} 
\langle c,p\rangle := \mbox{\rm Re}\left\{\sum_{k=0}^n p_k c_k\right\} = \int_a^b P(t)d\mu > 0,
\end{equation}
so that $c \in  \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;.$ This observation yields the rational analogue of Lemma \ref{MinC}.

\begin{lem}\label{inCint} If Hypothesis \ref{Hypothesis1} holds, then $\mathfrak{M}(\mathcal{R}_+) \subset \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\; .$ 
\end{lem}

The following result implies the reverse inclusion. 

\begin{thm}\label{big} If Hypotheses \ref{Hypothesis1} and \ref{Hypothesis2} hold, then $\mathfrak{M}(\mathcal{R}_+)$ contains a  set which is both open and closed in the convex set  $\mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;.  $ 
\end{thm}

\begin{rem}
The assertions in Theorem \ref{big} are the topological analogue, for the case of rational measures, of the convexity assertions used in the proof of Theorem \ref{TheoremB} for the generalized moment problem, where it was shown that the convex subset $\mathfrak{M}(\mathcal{M}_+) \subset \mathfrak{C}_+ =  K(U)$ both contains $U$ and is closed.
In light of Lemma \ref{inCint},  a point mass $\delta_{t_{0}}$ cannot be realized on $\mathfrak{P}$ by a positive rational measure so that $U \not\subset \mathfrak{M}(\mathcal{R}_+).$
Nonetheless, there exists $\mathcal{P}_+ \subset \mathcal{R}_+$ such that $\mathfrak{M}(\mathcal{P}_+)$ is both open and closed in $\mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;.  $ 
 \end{rem}
Indeed, for a fixed $P \in  \frak{P}_+\kern-.17 in ^{^{^\circ}}\;\;$ consider the set \begin{equation}
\label{Pfixed}
\mathcal{P}_+ = \{d\mu \in  \mathcal{R}_+: d\mu = \dfrac{P}{Q}dt, \;\;Q \in \frak{P}_+\kern-.17 in ^{^{^\circ}}\;\;\}
\end{equation}
and the restriction of the moment mapping $\mathfrak{M}|_{\mathcal{P}_+}: \mathcal{P}_+ \to \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;.$


\begin{prop}\label{open} If Hypothesis \ref{Hypothesis1} holds, then $\mathfrak{M}(\mathcal{P}_+) \subset \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;$ is open.
\end{prop}
%In order to state our result, we first need to compute the dimension of $\frak{P}$ as a real vector space, taking into account the cases where a basis element is real, purely imaginary or neither. 
\begin{proof}  
For simplicity, we view $\mathfrak{P}$ and $\mathfrak{C}$ as real vector spaces, so that $\mathfrak{P}$ is spanned by the real basis $(u_i)$, where we have replaced a complex-valued $(u_k)$ by its real and imaginary parts. We shall also parameterize $d\mu \in \mathcal{P}_+$ by $q \in \frak{P}_+\kern-.17 in ^{^{^\circ}}\;\;\;$. The Jacobian, $\mbox{\rm Jac}(\mathfrak{M}|_{\mathcal{P}_+})_{q_0}$, of $\mathfrak{M}|_{\mathcal{P}_+}$ at a point $q_0 \in \frak{P}_+\kern-.17 in ^{^{^\circ}}\;\;$ is a square matrix 
 $M_q$
 whose $(i,j)$-th entry is
 \begin{equation}\label{Jac}
(M_q)_{(i,j)} = - \int_a^b u_i(t) u_j(t)\frac{P(t)}{Q^{2}(t)}dt
\end{equation}
 evaluated at the point $q_0$.
Thus, $-M_q$ is the gramian matrix of the real basis  $(u_i)$ with respect to the positive definite inner product defined by $P(t)/Q^{2}(t) dt$ 
on $C[a,b]$. Therefore, $\mbox{\rm Jac}(\mathfrak{M}|_{\mathcal{P}_+})_q$ has rank $2n-r+2$ at each point $q\in \frak{P}_+\kern-.17 in ^{^{^\circ}}\;\;$  so that, by the Implicit Function Theorem, $\mathfrak{M}(\mathcal{P}_+)$ is open.
\end{proof}

\begin{prop}\label{closed} If Hypotheses \ref{Hypothesis1} and \ref{Hypothesis2} hold, then $\mathfrak{M}(\mathcal{P}_+) \subset \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;$ is closed.
\end{prop}
 
\begin{proof} Suppose
\begin{equation}
\label{cluster}
\mathfrak{M}(\dfrac{P}{Q_j}dt) = c^j \in \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;
\end{equation} 
and ${\rm lim}_{j \to \infty}c^j = c_0 \in \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;.$  We claim that there exists an $M>0$ such that $\|Q_j\|_{\infty} \leq M.$   
To see this note that \begin{equation}
\label{pdot}
{\rm lim}_{j \to \infty} \int_a^b \dfrac{P^2(t)}{Q_j(t)}dt = {\rm lim}_{j \to \infty}\langle c^j, p \rangle = \langle c, p \rangle >0. 
\end{equation} Setting $\widetilde{Q}_j = Q_j/ \|Q_j\|_{\infty},$ we have 
\begin{equation}
\label{qtilde}
{\rm lim}_{j \to \infty}\|Q_j\|_{\infty} \int_a^b \dfrac{P^2(t)}{\widetilde{Q}_j(t)}dt = \langle c, p \rangle >0. 
\end{equation}
Since $\int_a^b P^2(t)/\widetilde{Q}_j(t)\,dt  \geq \epsilon$ for some $\epsilon >0,$
we must have $\|Q_j\|_{\infty} \leq M < \infty$ for some $M >0.$
Therefore, there is a convergent subsequence  in $\mathfrak{P}\cap C[a,b],$
\begin{equation}
\label{subseq}
 {\rm lim}_{k\to \infty}q_k = q_0, \;\; {\rm with\;} q_0 \in \mathfrak{P}_+
\end{equation} for which
$$ 0 < \int_a^b \dfrac{P^2(t)}{Q_0(t)}dt = \langle c, p \rangle < \infty $$
%In particular,   \begin{equation}
%\label{c0}
%\mathfrak{M}(\dfrac{P_0}{Q_0}dt) = c_0 
%\end{equation} 
We claim that $q_0 \in \mathfrak{P}_+\kern-.17 in ^{^{^\circ}}\;\;\;$. 

Suppose, on the contrary, that  $Q_0(t_0) = 0$ for some $t_0\in [a,b],$ Then, since  $Q_0$ is Lipschitz continuous at $t_0$, there exists an $\varepsilon >0$ and an $L>0$ such that $Q_0(t)\leq L|t-t_0|$ whenever $|t-t_0|<\varepsilon$ and $t\in [a,b]$. In particular, if $t_0\in (a,b)$, 
\begin{displaymath}
\int_a^b\frac{P^2}{Q_0}\,dt\geq \frac{1}{L}\int_{t_0-\varepsilon}^{t_0+\varepsilon}\frac{P^2}{|t-t_0|}\,dt =+\infty,
\end{displaymath}
contrary to assumption. If $t_0=a$ or $t_0=b$, a similar estimate holds. Hence, $q_0 \in \mathfrak{P}_+\kern-.17 in ^{^{^\circ}}\;\;,$ as claimed.
\end{proof}

\begin{cor}\label{surjectioncor}
If Hypotheses \ref{Hypothesis1} and \ref{Hypothesis2} hold, the moment mapping $\mathfrak{M}|_{\mathcal{P}_+}: \mathcal{P}_+ \to \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;$ is surjective.
\end{cor}



\section{A Dirichlet Principle for the  moment problem with rational positive measures}\label{dualsec}


In the course of proving Theorem \ref{MainTheorem} we showed that, for any $c\in\mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\,\,\,$ and any choice of $P \in \frak{P}_+\kern-.17 in ^{^{^\circ}}\;\;\,,$ the moment problem for rational positive  
measures always has a solution in the set 
\begin{equation}
\label{Pfixed}
\mathcal{P}_+ = \{d\mu \in  \mathcal{R}_+: d\mu = \dfrac{P}{Q}dt, \;\;Q \in \frak{P}_+\kern-.17 in ^{^{^\circ}}\;\;\}.
\end{equation}
In this section, using  a convex optimization argument, we show that the surjection  $\mathfrak{M}|_{\mathcal{P}_+}$ is injective, and we characterize the unique rational measure as the solution of a variational problem. In fact, we derive both a primal optimization problem and its dual.  Remarkably,  the moment problem for rational  positive measures {\em is\/} the set of critical point equations for the dual variational problem. In this classical sense, a nonlinear convex optimization provides an illustration of the Dirchlet Principle for this class of  moment problems.

Let $ \mathbb{I}_p:\, C_+[a,b]\to \mathbb{R}\cup\{ -\infty\}$ be the relative entropy functional
\begin{equation}\label{primalfunctional}
 \mathbb{I}_p(\Phi)= \int_a^b P(t)\log\Phi(t)dt, 
\end{equation}
which is a generalization of the  entropy functional obtained by setting $P=1$. 
>From Jensen's inequality we see that $ \mathbb{I}_p(\Phi)\leq \log\left(\int_a^b P\Phi dt\right)\leq \int_a^b P\Phi dt<\infty$. 

\begin{thm}\label{primalthm} Assume that Hypotheses \ref{Hypothesis1} and \ref{Hypothesis2} hold, and 
let $c\in\mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\,\,\,$. Then, for any $P\in\mathfrak{P}_+\kern-.17 in ^{^{^\circ}}\;\;$, the constrained optimization problem to maximize \eqref{primalfunctional}
%\begin{equation}\label{primalfunctional}
% \mathbb{I}_p(\Phi)= \int_a^b P(t)\log\Phi(t)dt, 
%\end{equation}
over $C_+[a,b]$ subject to the moment constraints 
\begin{equation}\label{momentPhi}
\int_a^bu_k(t)\Phi(t)dt =c_k, \quad k=0,1,\dots,n,
\end{equation}
has a unique solution, and it has the form
\begin{equation}
\label{Phi=P_0/Q_0}
\Phi =\frac{P}{Q},\quad Q:=\mbox{\rm Re}\{ q\},
\end{equation}
where $q\in\frak{P}_+\kern-.17 in ^{^{^\circ}}\;\;$.
\end{thm}

The optimization problem of Theorem~\ref{primalthm}, to which we shall refer as the {\em primal problem},  can be solved by Lagrange relaxation. In fact, we have the Lagragian
\[
L(\Phi,q)=\mathbb{I}(\Phi) + \mbox{\rm Re}\sum_{k=0}^n q_k \left[ c_k
-\int_a^bu_k\Phi dt\right],
\]
where $(q_0,q_1,\dots,q_n)\in \mathbb{R}^r\times \mathbb{C}^{n-r+1}$ are Lagrange
multipliers. Then,
\[
L(\Phi,q)= \int_a^b P\log\Phi\; dt +\langle c,q\rangle
-\int_a^b Q\Phi dt,
\]
where $Q=\mbox{\rm Re}\{ q\}$ with $q:=\sum_{k=0}^n q_ku_k\in\mathfrak{P}$.
Clearly, comparing linear and logarithmic growth, we see that the dual functional
\[
\psi(q)=\sup_{\Phi\in C_+[a,b]} L(\Phi,q)
\]
takes finite values only if $q\in \mathfrak{P}_+$, so we may restrict our attention to such Lagrange multipliers.
For any  $q\in \mathfrak{P}_+$ and any  $\Phi\in C_+[a,b]$ such that $P/\Phi$ is integrable,  the directional derivative
\[
d_{(\Phi,q)}L(h)=\int_a^b \left[\frac{P}{\Phi}-Q\right] h\, dt =0
\]
for all $h\in C[a,b]$  if and only if $\Phi =\frac{P}{Q}\in C_+[a,b]$,
which inserted into the dual functional yields 
\begin{equation}
\label{psi}
\psi(q) = \mathbb{J}_p(q) + \int_a^b P(\log P -1) dt, 
\end{equation}
where $\mathbb{J}_p:\,  \mathfrak{P}_+\to  \mathbb{R}\cup\{\infty\}$ is the strictly convex functional
\begin{equation}
\label{dualfunctional}
\mathbb{J}_p(q)=\langle c,q\rangle -\int_a^b P\log Q dt.
\end{equation}

As the last term in \eqref{psi} is constant, the dual problem to minimize $\psi(q)$ over
$\mathfrak{P}_+$ is equivalent to the convex optimization problem 
\begin{equation}
\label{dual}
\min_{q\in\mathfrak{P}_+}\mathbb{J}(q).
\end{equation}
Since 
\begin{displaymath}
\frac{\partial\mathbb{J}_p}{\partial q_k}= c_k -\int_a^bu_k \frac{P}{Q}\, dt,\quad k=0,1,\dots,n,
\end{displaymath}
it follows from Corollary~\ref{surjectioncor} that the optimization problem \eqref{dual} has an optimal solution $\hat{q}\in\frak{P}_+\kern-.17 in ^{^{^\circ}}\;\;\,$ satisfying the moment equations \eqref{ratmoments}. Moreover, since the functional \eqref{dualfunctional} is strictly convex, this optimum is unique. 

Consequently,  
\begin{equation}\label{Phihat}
\hat{\Phi}:=\frac{P}{\hat{Q}}\in C_+[a,b]
\end{equation}
is the unique optimal solution of the primal problem. To see this, observe that 
 $\Phi\mapsto L(\Phi,\hat{q})$ is strictly concave and that $dL_{(\hat{\Phi},\hat{q})}(h)=0$ 
%\begin{equation}
%\label{stationarity}
%dL_{(\hat{\Phi},\hat{q})}(h)=\int_a^b\left[\frac{P}{\hat{\Phi}}-\hat{Q}\right] h\; dt=0,
%\end{equation}
for all $h\in C_+[a,b]$. Therefore,  
\begin{equation}\label{L1}
L(\Phi,\hat{q})\leq L(\hat{\Phi},\hat{q}), \quad \mbox{\rm for all $\Phi\in C_+[a,b]$}
\end{equation}
with equality if and only if $\Phi=\hat{\Phi}$.
However, $L(\Phi,\hat{q})=\mathbb{I}_p(\Phi)$ for all $\Phi$ satisfying the
moment conditions \eqref{momentPhi}. In particular, since \eqref{momentPhi}
holds with $\Phi =\hat{\Phi}$, $L(\hat{\Phi},\hat{q})=\mathbb{I}_p(\hat{\Phi})$.
Consequently, \eqref{L1} implies that $\mathbb{I}_p(\Phi)\leq \mathbb{I}_p(\hat{\Phi})$ for all $\Phi\in  C_+[a,b]$ satisfying the moment conditions,
%\begin{equation}
%\label{Iinequality}
%\mathbb{I}_p(\Phi)\leq \mathbb{I}_p(\hat{\Phi}),
%\end{equation}
 with equality if and only if $\Phi=\hat{\Phi}$.
Hence, $\mathbb{I}_p$ has a unique maximum in the space of  all $\Phi\in C_+[a,b]$ satisfying the constraints \eqref{momentPhi}, and it is given
by \eqref{Phihat}. 

This concludes the proof of Theorem~\ref{primalthm}, but we have also proven the following theorem.

\begin{thm}\label{dualthm}
Assume that Hypotheses \ref{Hypothesis1} and \ref{Hypothesis2} hold. Let
$(c,p)\in\mathfrak{C}_+\kern-.17 in ^{^{^\circ}}\;\;\times\mathfrak{P}_+\kern-.15 in ^{^{^\circ}}\;\,$, and set $P:=\mbox{\rm Re}\{ p\}$. Then the functional \eqref{dualfunctional}
has a unique minimizer $\hat{q}\in\mathfrak{P}_+\kern-.17 in ^{^{^\circ}}\;\,$, and $\hat{Q}:=\mbox{\rm Re}\{ \hat{q}\}$ is the unique solution to the moment equations
\begin{equation}\label{momentP/Qhat}
\int_a^bu_k\frac{P}{Q}dt =c_k, \quad k=0,1,\dots,n.
\end{equation}
\end{thm}

\begin{cor}\label{bijective}
If Hypotheses \ref{Hypothesis1} and \ref{Hypothesis2} hold, the moment mapping $\mathfrak{M}|_{\mathcal{P}_+}: \mathcal{P}_+ \to \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;$ is a bijection.
\end{cor}

\section{Moment problems in a Hardy space setting}


Some important special cases of the  moment problem is when 
\begin{equation}
\label{g2u}
u_k(t)=g_k(e^{it})  \quad \mbox{\rm where $g_k\in H^2(\mathbb{D})$},  \quad k=0,1,\dots,n,
\end{equation}
and $[a,b]=[-\pi,\pi]$.
A case in point is the trigonometric moment problem when $g_k(z)=\frac{1}{2\pi}z^k$; another is Nevanlinna-Pick interpolaton when $g_k(z)=\frac{1}{2\pi}\frac{z+z_k}{z-z_k}$, where $z_0,z_1,\dots, z_n$ are the (distinct) interpolation points. In both of these cases, $g:=(g_0, g_1,\dots,g_n)^{\mbox{\scriptsize\sf T}}$ can be represented as 
\begin{equation}
\label{g}
g(z)=(I-zA)^{-1}B,
\end{equation}
where $A$ is a $n\times n$ stability matrix and $B$ an $n$-vector such that $(A,B)$ is a reachable pair; i.e., 
\begin{equation}
\label{H}
G = \int_{-\pi}^\pi g(e^{it})g(e^{it})^*dt >0.
\end{equation}
Indeed, positive definiteness of $G$ follows readily from the fact that  the basis functions are linearly independent. This condition also insures that there is a unique function of the form 
\begin{displaymath}
w(z)=\sum_{k=0}^n w_kg_k(e^{it})^*
\end{displaymath}
that satisfies
\begin{displaymath}
\int_{-\pi}^\pi g_k(e^{it})w(e^{it})dt =c_k, \quad k=0,1,\dots,n,
\end{displaymath}
namely the one provided by the unique solution of the system of linear equations
 \begin{displaymath}
\sum_{k=0}^nG_{kj}w_j=c_k, \quad k=0,1,\dots,n.
\end{displaymath}
Consequently, for any $q\in\mathfrak{P}$,
\begin{equation}
\label{(c,q)}
\langle c,q\rangle =\int_{-\pi}^\pi Q(t)w(e^{it})dt.
\end{equation}

It can be shown that $g_0,g_1,\dots, g_n$ span the coinvariant subspace $\script{K} :=H^2\ominus\phi H^2$, where $\phi$ is the inner function
\begin{displaymath}
\phi(z)=\frac{\det(zI-A^*)}{\det(I-zA)}.
\end{displaymath}
In view of \eqref{g2u}, $\script{K}$ is a Hardy space model of $\mathfrak{P}$. Moreover, for any $\psi\in\script{K}$, there is a $v\in\script{K}$ such that $\Psi:=\mbox{\rm Re}\{\psi\}=vv^*$ \cite[Proposition 9]{BGLM}. Therefore, for any $q\in\mathfrak{P}_+$, there is an $\mathbf{a}\in\mathbb{C}^n$ such that $Q(t)=a(e^{it})^*a(e^{it})$ where $a(z):=g(z)^*\mathbf{a}$. Then, by \eqref{(c,q)}, 
\begin{equation}
\label{ }
\langle c,q\rangle = \mathbf{a}^*\int_{-\pi}^\pi w(e^{it})g(e^{it})g(e^{it})^*dt\; \mathbf{a}=
 \mathbf{a}^*\mathbf{P} \mathbf{a},
\end{equation}
where
\begin{equation}
\label{Pick}
\mathbf{P} := \frac12\int_{-\pi}^\pi g(e^{it})[w(e^{it})+w(e^{it})^*]g(e^{it})^*dt.
\end{equation}
Consequently, $c\in\mathfrak{C}_+$ if and only if $\mathbf{P}\geq 0$, and $c\in\mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;$ if and only if $\mathbf{P}> 0$. In the trigonometric moment problem $\mathbf{P}$ is the Toeplitz matrix, and  in the Nevanlinna-Pick case $\mathbf{P}$ is the  the Pick matrix. 

If $\mathfrak{P}$ contains constants, then we may determine the maximum-entropy solution, corresponding to setting $P=1$ in \eqref{primalfunctional}, in closed form. 

\begin{prop}
Suppose that the basis functions in  $\mathfrak{P}$ satisfy \eqref{g2u} and $\mathfrak{P}$ contains constants. Then the maximum-entropy solution is 
\begin{equation}
\label{Qhat}
\hat{\Phi}(t)=\frac{g(0)^*\mathbf{P}^{-1}g(0)}{|g(e^{it})^*\mathbf{P}^{-1}g(0)|^2},
\end{equation} 
where $\mathbf{P}$ is given by \eqref{Pick}.
\end{prop} 

\begin{proof}
We proceed as in \cite{GL1,Lpicci}.
Since, by Jensen's formula \cite[p.184]{Ahlfors}, the last term in the dual functional \eqref{dualfunctional} (with $P=1$) can be written $2\log |a(0)|$, \eqref{dualfunctional} becomes
\begin{equation*}
\label{modJfunctional}
J(\mathbf{a}):=\mathbb{J}_p(a^*a)=\mathbf{a}^*\mathbf{P}\mathbf{a}- 2\log |\mathbf{a}^*g(0)|.
\end{equation*}
Setting the gradient of $J(\mathbf{a})$ equal to zero, we obtain $\mathbf{a}=\mathbf{P}^{-1}g(0)/|a(0)|$ and hence 
$a(z)=g(z)^*\mathbf{P}^{-1}g(0)/|a(0)|$. Then  $|a(0)|^2=g(0)^*\mathbf{P}^{-1}g(0)$, and therefore the optimal $\bold{a}$ becomes
\begin{equation}\label{gPinv2a}
a(z)=\frac{g(z)^*\mathbf{P}^{-1}g(0)}{\sqrt{g(0)^*\mathbf{P}^{-1}g(0)}}.
\end{equation}
Moreover, in view of Theorems \ref{primalthm} and \ref{dualthm}, 
\begin{displaymath}
\hat{\Phi}(t)=\frac{1}{Q(t)}=\frac{1}{|a(e^{it})|^2},
\end{displaymath}
and therefore \eqref{Qhat} follows from \eqref{gPinv2a}. 
\end{proof}

In the trigonometric moment problem, modulo normalization, $$\varphi_n(z):=g(z)^*\mathbf{P}^{-1}g(0)$$ reduces to the Szeg\"o polynomial orthogonal on the unit circle of degree $n$ (cf \cite{PE}). 

\section{Amplifications and conclusions}
In this paper we showed  that the moment problem for rational positive measures is solvable for all strictly positve sequences, provide Hypotheses \ref{Hypothesis1} and \ref{Hypothesis2} hold for $\mathfrak{P}.$ In the language of functions and spaces, we showed that the moment map $\mathfrak{M}$ defined by 
\eqref{momentmap} restricts to a surjection
\begin{equation}
\label{mainsurj}
\mathfrak{M}|_{\mathcal{R}_+}: \mathcal{R}_+ \to \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;
\end{equation} 
by proving that the restriction 

\begin{equation}
\label{mainbij}
\mathfrak{M}|_{\mathcal{P}_+}: \mathcal{P}_+ \to \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;
\end{equation} is surjective. Indeed, using the strict convexity of the dual functional, we were able to conclude in Corollary \ref{bijective} that \eqref{mainbij} is a bijection. 

In this section we briefly discuss these maps in more detail. Following Hada\-mard, the problem of solving, ${\rm for} \;c \in \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;$, the equations 
\begin{equation}
\label{Had}
\mathfrak{M}|_{\mathcal{P}_+}(d\mu(q)) =c, \; {\rm for} \;q \in  \mathfrak{P}_+\kern-.17 in ^{^{^\circ}}\;\;\,,
\end{equation}
is {\em well-posed} provided a solution $q$ exists, is unique and varies continuously 
with $c$ (in some reasonable topology). As elements of open convex subsets of Euclidean space, the choice of topology is clear. Existence and uniqueness is the essence of Corollary \ref{bijective}. Moreover, our proof of Proposition \ref{open} reposed on the observation that ${\rm Jac}(\mathfrak{M}|_{\mathcal{P}_+})_{q}$ is nonsingular at each $
q \in   \mathfrak{P}_+\kern-.17 in ^{^{^\circ}}\;\;$ so that, by the Inverse Function Theorem,  $\mathfrak{M}|_{\mathcal{P}_+}$ is a smooth bijection with a smooth inverse. Since $\mathfrak{M}|_{\mathcal{P}_+}^{-1}$ is differentiable, it is continuous, so that $q$ is a continuous function of $c$ and this restricted moment problem is well-posed. 

Our second amplification concerns the map \eqref{mainsurj}. Here, $\mathfrak{M}|_{\mathcal{R}_+}$ is not injective and one would instead like a continuous or smooth parameterization of the solutions, ${\rm for} \;c \in \mathfrak{C}_+\kern-.15 in ^{^{^\circ}}\;\;$, to the equations 
\begin{equation}
\label{Had}
\mathfrak{M}|_{\mathcal{R}_+}(d\mu) =c, \; {\rm for} \;d\mu \in  \mathcal{R}_+.\end{equation}
As before, one can compute the Jacobian ${\rm Jac}(\mathfrak{M}|_{\mathcal{R}_+})_{d\mu}$ and show \cite{BLimportantmoments} that $$\rank {\rm Jac}(\mathfrak{M}|_{\mathcal{R}_+})_{d\mu} = 2n -r +2,$$ for all $d\mu \in \mathcal{R}_+$. In fact, in \cite{BLimportantmoments} we prove that the solution space $\mathfrak{M}|_{\mathcal{R}_+}^{-1}(c)$ is a smooth manifold, smoothly parameterized by $p \in \mathfrak{P}_+\kern-.17 in ^{^{^\circ}}\;\;$ as described in Theorem \ref{primalthm}. 





%\begin{enumerate}
%  \item if your proof ends with a\ \ s i n g l e\ \ displayed line, the end-of-proof sign would
%be placed in the line below; if you want to avoid this, write your line in the form
%\begin{verbatim}$$displayed math line \eqno\qedhere$$\end{verbatim}
%which results in

%\begin{proof}
%$$displayed math line \eqno\qedhere$$
%\end{proof}
%\item if your proof ends with a larger displayed environment, you either may leave the
%end-of-proof sign where it is located or you could add some
%\begin{verbatim}\vskip-1em\end{verbatim}
%one line above the
%\begin{verbatim}\end{proof}\end{verbatim}
%command, in order to place the symbol a bit above.
%\end{enumerate}


%\begin{rem}
%Additional comments are being typeset without boldfaced entrance
%word as they may be minor important.
%\end{rem}

%\begin{ex}
%For some constructs, even no number is required.
%\end{ex}


\begin{thebibliography}{1}
\bibitem{AK}
N.I. Ahiezer and M. Krein,
{\em Some Questions in the Theory of Moments.} American Mathematical
Society, Providence, Rhode Island, 1962.

\bibitem{Ahlfors}
L. V. Ahlfors, {\em Complex Analysis.} McGraw-Hill, 1953.

\bibitem{BLGM}
C.~I. Byrnes, A.~Lindquist, S.~V. Gusev, and A.~S. Matveev,
\textit{A complete parameterization of all positive rational extensions of a covariance sequence.} IEEE Trans. Automat. Control {\bf 40} (1995), 1841--1857.

\bibitem{SIGEST} C. I. Byrnes,  S. V. Gusev,  and A. Lindquist,
\textit{From finite covariance windows to modeling filters: A convex optimization
approach.} SIAM Review {\bf 43} (2001) 645--675.

 \bibitem{BGL1}
C.I. Byrnes, T.T. Georgiou and A. Lindquist, 
\textit{A generalized entropy criterion for Nevanlinna-Pick interpolation with degree constraint.} IEEE Trans. Automatic Control {\bf AC-46} (2001) 822--839.

\bibitem{BLkimura}     
C.~I. Byrnes and A.~Lindquist,
\textit{A convex optimization approach to generalized moment problems.} In  ``Control and Modeling of Complex Systems: Cybernetics in the 21st Century: Festschrift in Honor of Hidenori Kimura on the Occasion of his 60th Birthday", K. Hashimoto, Y. Oishi  and Y. Yamamoto, Editors, Birkh{\" a}user, 2003, 3--21.

\bibitem{BGLM} C. I. Byrnes, T. T.  Georgiou, A.  Lindquist, and A. Megretski, \textit{Generalized interpolation in $H^\infty$ with a complexity constraint.}
Trans.  American Mathematical Society  \textbf{358} (2006),  965--987. 

\bibitem{BLmoments}
C. I. Byrnes and A. Lindquist,
\textit{The generalized moment problem with complexity constraint.} Integral Equations and Operator Theory. \textbf{56} (2006) 163--180.
\bibitem{BLimportantmoments} C. I. Byrnes and A. Lindquist, \textit{Important moments in systems and control.} SIAM J. Control and Optimization, to be published.
%\textbf{88} (2000), 100--120.

\bibitem{PE}
P. Enqvist,
{\em A homotopy approach to rational covariance extension with degree
constraint.} Intern. J. Applied Mathematics and Computer Science 
{\bf 11(5)} (2001), pp. 1173--1201.

\bibitem{Gthesis} 
T.T. Georgiou, {\em Partial Realization of Covariance Sequences}, Ph.D. thesis, CMST, University of Florida, Gainesville 1983.

\bibitem{G1}
T.~T. Georgiou,
\textit{Realization of power spectra from partial covariance sequences.}
IEEE Trans. Acoustics, Speech and Signal Processing
{\bf  35}  (1987), 438--449.

\bibitem{Georgiou3} T.T. Georgiou, Solution of the general moment problem via a one-parameter imbedding,  {\em IEEE Trans. Automatic Control} {\bf AC-50} (2005) 811-826.

\bibitem{GL1}
T.T. Georgiou and A. Lindquist, \textit{Kullback-Leibler approximation of spectral
density functions.} IEEE Trans. on Information Theory {\bf 49(11)},
November 2003.

\bibitem{KN}
M. G. Krein and A. A. Nudelman,
 \textit{The Markov Moment Problem and Extremal Problems.}  American Mathematical
Society, Providence, Rhode Island, 1977.

\bibitem{Lpicci} A. Lindquist,
 \textit{Prediction-error approximation by convex optimization, in Modeling, Estimation and Control: Festschrift in honor of Giorgio Picci on the occasion of his sixty-fifth Birthday}, A. Chiuso, A.  Ferrante and S. Pinzoni (eds), Springer-Verlag, 2007, 265--275.
\end{thebibliography}

% ------------------------------------------------------------------------

%\subsection*{Acknowledgment}
%Many thanks to our \TeX-pert for developing this class file.
% ------------------------------------------------------------------------
\end{document}
% ------------------------------------------------------------------------