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\lhead{\rmfamily \textsf{MAPA3011A\ 1996-97\ }--\ \textbf{\thesection}\ --\ \mylmark\  --} \chead{} \rhead{\rmfamily\thepage}
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\begin{document}

\renewcommand{\mylmark}{CF-Hankel-BMO(A) Approximation.}


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\section{About Constantin Carath\'eodory and Leopold Fej\'er.}

At the request of a kind member of the audience, a word on Carath\'eodory
and Fej\'er's own work [11]: they proceeded to give constructive solutions
to the following problems:

\begin{enumerate}
\item \textsl{Given the complex numbers $\alpha_0$, $\alpha_1,\dotsc,$ how to
 know if} Re$\displaystyle{\left(\Psi(z) = \sum_0^\infty \alpha_k z^k\right)}$
 \textsl{is bounded in} $\mathbb{D}=\{z: |z|<1\}$?

\noindent \hspace*{-35pt}
 Answer: for each $n=1,2,\dotsc,$ construct $\tilde{\alpha}_{n+1}$,
                                            $\tilde{\alpha}_{n+2}$,\dots,
 such that Re$\displaystyle{\left(\Psi_n(z) =
                            \sum_0^n \alpha_k z^k
              + \sum_{n+1}^\infty \tilde{\alpha}_k z^k\right) }$
     has the \emph{\textbf{least}} possible supremum on the unit disk.

   They show that $\sup$ Re$\,\Psi_n$ on the unit disk is $\Lambda_{n,1}$, the largest
   eigenvalue of the Hermitian Toeplitz matrix of order $n+1$
   $\displaystyle{  \boldsymbol{T}_n = \begin{bmatrix}
            (\alpha_0+\overline{\alpha_0})/2  &  \alpha_1/2 & \cdots & \alpha_n/2\\
  \overline{\alpha_1}/2 & (\alpha_0+\overline{\alpha_0})/2  &\cdots& \alpha_{n-1}/2\\
        \vdots          &         \vdots                    &\cdots & \vdots \\
  \overline{\alpha_n}/2 &  \cdots  & \cdots &(\alpha_0+\overline{\alpha_0})/2
                     \end{bmatrix}  }$, that the limit of the
     increasing sequence $\{\Lambda_{n,1}\}_n$ is the answer (for the
     infimum, look at the decrasing sequence $\{\Lambda_{n,n}\}_n$ of
     the lowest eigenvalues), that $\Psi_n$ is a rational function with
     constant real part on the unit circle
      $$ \Psi_n(z) = \dfrac{ w_n + w_{n-1}z + \dotsb + w_0 z^n }
                           { v_n + v_{n-1}z + \dotsb + v_0 z^n }, $$
   where $[v_0, v_1,\dotsc, v_n]^T$ is the eigenvector of
   $\boldsymbol{T}_n$, and where $w_0,\dotsc, w_n$ care for the match of
   the Taylor coefficients of $\Psi_n$.
   One has $\Psi_n(1/\overline{z}) = 2\Lambda_{n,1} -\overline{
     \Psi_n(z) }$.

 \item \textsl{Given $M>$} Re$\;\alpha_0$, \textsl{what is the largest disk
  where $\Psi$ is regular and has a real part $\leqslant M$}?

  Answer: construct the Toeplitz matrix $\boldsymbol{T}_n(r)$  with
  the $\alpha_k r^k$ and increase $r$ until $\Lambda_{n,1}(r)=M$, then
  increase $n$.

\item \textsl{Given the complex numbers $\alpha_0$, $\alpha_1,\dotsc,$ how to
 know if $\displaystyle{\Phi(z) = \sum_0^\infty \alpha_k z^k}$
         is bounded in} $|z|<1$?

\noindent
 Answer: for each $n=1,2,\dotsc,$ construct $\tilde{\alpha}_{n+1}$,
                                            $\tilde{\alpha}_{n+2}$,\dots,
 such that $\Phi_n(z) =
             \displaystyle{\sum_0^n \alpha_k z^k
                               + \sum_{n+1}^\infty \tilde{\alpha}_k z^k}$
     has the \emph{\textbf{least}} possible $L^\infty$ norm on the unit disk.

 This is the classical CF problem. Their construction of $\Phi_n$ is
 rather clumsy and will be superseded by Schur and Takagi: the harmonic
 function Re$\;\left(\Psi(z) = \dfrac{\Phi(z)-\mu}{\Phi(z)+\mu}\right)$ will be
 negative in the open unit disk with zero as supremum only if $\mu=
 \|\Phi_n\|_\infty$, therefore this $\mu$ leads to $\Lambda_{n,1}=0$
 with the Toeplitz matrix built with the Taylor expansion of $\Psi$\dots
 Then, the sequence $\{\|\Phi_n\|_\infty\}_n$ is increasing and its
 limit is the essential sup of $|\Phi|$.

\item \textsl{Given the complex numbers $\alpha_0$, $\alpha_1,\dotsc,$
    ($\alpha_0 \neq 0$ and $\alpha_0 \neq 1$),
                                                  what is the largest disk
  where $\Phi$ is regular and has \emph{\textbf{never value 0 nor 1}}}?
  (Picard-Landau-type problem).

  {\Small
  Answer: consider the famous \emph{\textbf{modular function}}
   $\mathcal{M}$ which 1) is analytic in any domain which does not contain
   0 nor 1; 2) has a negative real part in any such domain. Therefore,
   consider the expansions of $\Psi(z) := \mathcal{M}(\Phi(rz)) =
   \beta_0 +\beta_1 z+\dotsb $ for increasing values of $r$: while
   $\Phi(r\mathbb{D})$ is still in an admissible domain for $\mathcal{M}$,
   $\Lambda_{n,1}$ is negative. The requested radius is reached when
   $\Lambda_{n,1}=0$.
  } %end Small
\end{enumerate}

               \medskip
               \begin{center}   - - - - - - - - - - - \end{center}
            \smallskip
Of course, direct solution to problem~3 is given by Schur's construction:
one has
$$ \Phi_n(z) = \mu \dfrac{1+\Psi_n(z)}{1-\Psi_n(z)} =
   \mu\dfrac{ v_n+w_n +\dotsb + (v_0+w_0)z^n }
            { v_n-w_n +\dotsb + (v_0-w_0)z^n }
       = \mu\gamma \dfrac{\overline{u_n} + \dotsb +\overline{u_0} z^n }
                         {u_0 + \dotsb +u_n z^n },
$$
with $|\gamma|=1$, leading to
\begin{equation} \label{hankeltri}  \begin{split}
  \begin{bmatrix}
       \alpha_n  &  \alpha_{n-1}  &  \cdots & \alpha_1 & \alpha_0 \\
    \alpha_{n-1} & \alpha_{n-2}  & \cdots  & \alpha_0 &     \\
      \vdots   &    \vdots   & \secddots &        &   \\
      \alpha_0 &             &           &         &
   \end{bmatrix}
  \begin{bmatrix} u_0 \\ u_1 \\ \vdots \\ u_n \end{bmatrix} &=
   \mu\gamma
\begin{bmatrix} \overline{u_0} \\ \overline{u_1} \\ \vdots \\
                                         \overline{u_n} \end{bmatrix} \\
         \text{i.e.   \ }  \boldsymbol{H}_n \boldsymbol{u} &=
         \mu\gamma \overline{\boldsymbol{u}} .
\end{split} \end{equation}


\section{Numerical CF approximation.}

First construct a large number of Chebyshev coefficients of $\displaystyle{
 f(x) = \sideset{}{'}\sum_0^\infty c_k T_k(x) }$, for instance through
 FFT sampling [43].


{\small \begin{verbatim}

% cftref.m
%  Approximation Theory V
% RCF -- REAL RATIONAL CF APPROXIMATION ON THE UNIT INTERVAL
%
% Loyd N. Trefethen, Dept. of Math., M.I.T., March 1986
% Reference: L.N.T. and M.H. Gutknecht,
%            SIAM J. Numer. Anal. 20 (1983), pp. 420-436
%
%     Fx(x) - function to be approximated by R(x)=P(x)/Q(x)
%      m,n  - degree of P,Q
%     nfft  - number of points in FFT (power of 2)
%        K  - degree at which Chebyshev series is truncated
%   F,P,Q,R - functions evaluated on FFT mesh (Chebyshev points)
%    Pc,Qc  - Chebyshev coefficients of P and Q
%
%  If Fx is even, take (m,n)=( odd, even).
%  If Fx is odd , take (m,n)=(even,even ).
%
%  CONTROL PARAMETERS
     m=input('m? ') ; n=input('n? '); np=n+1;
     nfft=input('nfft? ');nfft2=nfft/2;
     K=input('K? '); dim=K+n-m;
%
%  CHEBYSHEV COEFFICIENTS OF Fx
     z = exp(2*pi*sqrt(-1)*(0:nfft-1)/nfft);
     x = real(z); F = Fx(x); Fc = real(fft(F))/nfft2;
%
%  SVD OF HANKEL MATRIX H
     H = toeplitz(Fc(1+rem((dim:-1:1)+nfft+m-n,nfft)));
        H = triu(H); H=H(:,(dim:-1:1));
     [u,s,v] = svd(H);
     s = s(np,np); u = u((dim:-1:1),np)';v = v(:,np)';
%
%  DENOMINATOR POLYNOMIAL Q
     zr = roots(v); qout = []; for i=1:dim-1;
         if abs(zr(i))>1 qout = [qout, zr(i)];end; end;
     qc = real(poly(qout)); qc = qc/qc(np); q = polyval(qc,z);
     Q = q.*conj(q); Qc = real(fft(Q))/nfft2;
     Qc(1)= Qc(1)/2; Q=Q/Qc(1); Qc = Qc(1:np)/Qc(1);
%
%  NUMERATOR POLYNOMIAL P
     b = fft([u zeros(1,nfft-dim)])./fft([v zeros(1,nfft-dim)]);
     Rt = F-real(s*z.^K.*b); Rtc = real(fft(Rt))/nfft2;
     gam = real(fft((1)./Q))/nfft2; gam = toeplitz(gam(1:2*m+1));
     if m==0 Pc = 2*Rtc(1)/gam;
       else Pc = 2*[Rtc(m+1:-1:2) Rtc(1:m+1)]/gam; end;
     Pc = Pc(m+1:2*m+1); Pc(1) = Pc(1)/2;
     P = real(polyval(Pc(m+1:-1:1),z)); R = P./Q;
%
% RESULTS
    plot(x,F-R,'-',x,[s;0;-s]*ones(1,nfft),':'); pause;
    s, err = norm(F-R,'inf'), Pc, Qc

\end{verbatim} } %end small

Then, $f(x)= (f^{(+)}(z)+ f^{(-)}(z))/2,  f^{(+)}(z) =\sideset{}{'}\sum_0^\infty
  c_k z^k;\  f^{(-)}(z) = f^{(+)}(1/z)$. Proceed with a big truncated
  Hankel operator, either
  $\displaystyle{  \mathcal{H}_N =
    \begin{bmatrix}
       c_1 & c_2 & \cdots & \cdots & c_N \\
       c_2 & c_3 & \cdots & c_N   &     \\
       \vdots&\vdots&\secddots&  &  \\
       c_N &    &        &      &
    \end{bmatrix}  }$
   (L.N.~Trefethen's choice, Hankel consistent), or
  $\displaystyle{  \mathcal{H}_N =
    \begin{bmatrix}
       c_1 & c_2 & \cdots & \cdots & c_N \\
       c_2 & c_3 & \cdots & c_N   & c_{N+1}    \\
       \vdots&\vdots&\secddots&\secddots  & \vdots \\
       c_N &c_{N+1} & \cdots &\cdots  & c_{2N-1}
    \end{bmatrix}  }$
  (A.P.~Magnus' choice, operator projection consistent).

  Compute the eigenvalues $\lambda_1,\dotsc, \lambda_N$ of $\mathcal{H}_N$
  (with $\sigma_1=|\lambda_1| \geqslant \sigma_2=|\lambda_2| \geqslant
   \dotsb \geqslant \sigma_N = |\lambda_N|)$. No need to keep 2 subscripts,
   as the reductions $N\to N-1$, $N-1\to N-2 \dotsb n+1 \to n$
   keep $\lambda_1,\dotsc, \lambda_n$.

  Nor shall we reach $r_n^{(+)}$ through exhausting multiple one step
  reductions from $N$ to $n$: as far as near-best approximations are
  considered, we may as well work directly on the eigenvectors of
  $\mathcal{H}_N$. Take the lemma p.~5, with $N$ instead of $n+1$, and
  $n+1$ instead of $k$ ($n\geqslant 0$): \\
  \textsl{$\tilde{r}_{N,n} := n_{N-1,n+1}/d_{N-1,n+1}$ is the best approximation
    to $f^{(+)}$ on the unit circle, among all rational functions of
    degree $N-1$ with $\geqslant N-n-1$ poles in the unit disk}, \\
  or:
  \textsl{$\tilde{r}_{N,n} := n_{N-1,n+1}/d_{N-1,n+1}$ is the best approximation
    to $f^{(+)}$ on the unit circle, among all rational functions of
    degree $N-1$ with $\leqslant n$ poles \textbf{outside} the unit disk}, \\
  the number of these poles is the right one ($r_n^{(+)}$ must have its $n$
   poles outside the unit disk), we simply keep them to make a reasonable
   denominator $\tilde{d}_n$ of $r_n^{(+)}$.

  $\displaystyle{ \dfrac{\tilde{d}_n(z) }{z^n} \tilde{r}_{N,n}(z) }$
  is therefore a rational function of high degree without pole in
  $|z|>1$, and so has a Laurent expansion with nonpositive
  powers:
  $$ \tilde{r}_{N,n}(z) = \dfrac{\tilde{\nu}_0 z^n +
                                 \tilde{\nu}_1 z^{n-1} +  \dotsb
                               +  \tilde{\nu}_n
                               +  \tilde{\nu}_{n+1} z^{-1} +\dotsb }
                           {\tilde{d}_n (z) },
  $$
  and we check again (p.~9)
  $$ f^{(+)}(z) -\tilde{r}_{N,n} (z) =
    \sum_{k=0}^\infty c_k z^k  -\tilde{r}_{N,n} (z) =
                          \lambda_{n+1}
      \dfrac{ v_0 +v_1 z +v_2 z^2 +\dotsb }
            {v_0 z^{-1} +v_1 z^{-2} +v_2 z^{-3} +\dotsb }.
  $$
  Finally,
   we construct the CF numerator
   by keeping only the nonnegative powers of $z$ in
   $$ \tilde{d}_n(z) \tilde{r}_{N,n}(z) =
    \tilde{d}_n(z) \left( f^{(+)}(z) -\lambda_{n+1}
      \dfrac{ v_0 +v_1 z +v_2 z^2 +\dotsb }
            {v_0 z^{-1} +v_1 z^{-2} +v_2 z^{-3} +\dotsb } \right).
   $$

 \medskip
 We compare the two techniques (CF in [24-25, 39-44] sense, and successive
 one-step degree reductions) for $f(x) = (1+2x)/(6+2x+x^3)$,
 looking for $r_1$.

We split $f(x)= (f^{(+)}(z) + f^{(+)}(z^{-1}))/2$:

$$ f^{(+)}(z) = \dfrac{ -0.0952 z^3 -0.2871 z^2 +15.1044z +4.3088}
                      {         z^3 -0.2444 z^2 +11.0651 z +45.2721},
$$
with poles at $z= 1.3795\pm 4.0125i$ and $-2.5147$.

$\mathcal{H}_3$ has the three eigenvalues
$$ \lambda_1 = 0.333359\;,\qquad
   \lambda_2 = -0.007973\;,\qquad
   \lambda_3 = 0.007947\;.
$$



\begin{minipage}[t]{3in}
\  \par

%\hspace*{-5mm}

  \begin{center} \textbf{CFGT} \end{center}

Take $\lambda_{n+1} = \lambda_2$,

poles are $-3.8188$ and $-0.9985$.

\vspace*{58pt}

Keep only the pole $-3.8188$, so:\\
 $\tilde{d}_1(z) = z+3.8188$.

Final $z-$fraction is
 $$ \dfrac{1.282 z+0.363}{z+3.8188},$$
and final $x-$fraction is
$$ \dfrac{0.3493 +0.6884 x}{2.0404 +x}.$$

\end{minipage}
\begin{minipage}[t]{0.2in}
\   \par

\end{minipage}
 \begin{minipage}[t]{1mm}

 \ \par %\rule{0.2mm}{250pt}
 \end{minipage}
\begin{minipage}[t]{3in}
\  \par

  \begin{center} \textbf{One-step} \end{center}

Take $\lambda_3 = 0.007947$, poles are
$-3.821$ and $1.0016$, one finds

$$ r_2^{(+)}(z) = \dfrac{1.2818 z^2 -0.9200 z -0.3644}
                        {    z^2  +2.8194 z -3.8273}.$$

Restart with $r_2^{(+)}$, now, $\lambda_2$ is the
least modulus (non vanishing) eigenvalue, pole is $-3.5227$,
final $z-$fraction is
$$ r_1^{(+)}(z) = \dfrac{1.1743 z+0.3334}{z+3.5227},$$
and final $x-$fraction is
$$ r_1(x) = \dfrac{0.3334 +0.6345 x}{1.9033 +x}.$$

\end{minipage}

\smallskip

CFGT fraction is found to have $0.0083$ as error norm,
one-step fraction shows $0.0147$, almost twice as bad for this
particular experiment. There must be some hidden tricks in
L.N.T.'s program.


\section{BMO, etc.}

A fascinating topic of function theory is the connection between
function classes and approximation properties.

Best known examples are the Jackson-Bernstein theorems on polynomial
errors and Lipschitz classes:
 $$ f\in\mathscr{C}^r(\mathbb{T}) \Rightarrow E_n(f)\leqslant
    C_r n^{-r} \omega\left( f^{(r)},n^{-1}\right)\;, \quad
  E_n(f)\leqslant C_r n^{-r-\alpha}, 0<\alpha<1 \Rightarrow
     f^{(r)}\in \text{Lip}_\alpha .$$

The subject has been thoroughly amplified\dots

\subsection{BMO, BMOA, and VMO} \  \\

Ref: [7, p.~40-42]

You probably did not suspect any problem in dealing with $f^{(+)}$ in
$$ f(x) = f(\cos\theta) = f\left(\dfrac{z+z^{-1}}{2}\right) =
  \sideset{}{'}\sum_0^\infty c_k \cos(k\theta)
  = \dfrac{1}{2} \sum_{-\infty}^\infty c_k z^k
 \Rightarrow
   f^{(+)}(z) = \sideset{}{'}\sum_0^\infty c_k z^k.$$
Well, there could be one: $f$ could be bounded on the unit circle
$\mathbb{T}$, and $f^{(+)}$ could be unbounded on the unit disk
$\mathbb{D}$.

\noindent \hspace*{-15pt}
For instance, $\displaystyle{ f(\cos\theta) = \text{sign}(\pi/2-|\theta|)
  = \dfrac{4}{\pi} \sum_{\substack{k\geqslant 1\\k \text{ odd}}}
       (-1)^{(k-1)/2} \dfrac{\cos(k\theta)}{k}
  \text{ in } (-\pi,\pi)
 \Rightarrow
     f^{(+)}(z) = \dfrac{2}{\pi i}\log\dfrac{1+iz}{1-iz}  }$.

BMO is the space where the $f^{(+)}$'s and the $f^{(-)}$'s of bounded
functions live:

\begin{equation} \label{BMOdef1}
\text{BMO } = \{ f^{(+)} +g^{(-)}, f \text{ and } g\in L^\infty(\mathbb{T})\}.
\end{equation}

BMO is a Banach space containing $L^\infty$, but contained in any
other $L^p$. A valuable norm is
$$ h\in \text{ BMO } \Rightarrow \|h\| =
      \inf_{\substack{ f,g\in L^\infty \\ f^{(+)} +g^{(-)} =h }}
        \|f\|_\infty + \|g\|_\infty .
$$

BMOA is the analytic part of BMO, just ignore $g$ above:

BMOA = $\{ f^{(+)} , f \in L^\infty(\mathbb{T})\}$.

\begin{equation} \label{normBMOA1}
 h\in \text{ BMOA } \Rightarrow \|h\| =
      \inf_{\substack{ f\in L^\infty \\ f^{(+)}  =h }}
        \|f\|_\infty  ,
\end{equation}
which is of course the Hankel norm, i.e., $\sigma_1 =
 |\lambda_1|$. For the BMO meaning of $\lambda_n$, see a later
section on AAK.


\medskip

BMO means ``Bounded Mean Oscillation'', referring to the initial
definition \\ \  BMO $=\displaystyle{ \{h\in L^1(\mathbb{T}):
  \sup_{0<\delta<2\pi} M_\delta(h) <\infty }$, where
 $\displaystyle{ M_\delta(h) = \sup_{|I|\leqslant \delta}
      \int_I |h(z) -h_I|\; |dz| }$, with $h_I =\displaystyle{
       \dfrac{1}{|I|}\int_I h(z) |dz| }$, mean value of $h$ on
   the arc $I\subset \mathbb{T}$.

Equivalence with the definition \eqref{BMOdef1} above: Fefferman.

\smallskip

VMO, set of functions of vanishing mean oscillation, is the subset of
BMO with $M_0=0$. VMO is exactly to the continuous functions what BMO
is for $L^\infty$ (Sarason). Winding numbers can still be computed with
functions of VMO.

\subsection{Approximation spaces} \  \\

Ref: [32-34]

            Let $f$ be in a space $X$ of functions
 defined on $\mathbb{T}$, and consider various norms
      of best rational approximation $\rho_n(f,X)$:
  $$ \rho_n(f,p)=\inf \|f-r\|_p, \quad
     \rho_n(f) = \inf \|f-r\|_\infty, \quad
     \rho_n(f,*) = \inf \|f-r\|_{\text{BMO}} ,
  $$

The problem is: what is the smoothness of the functions
with a given rate of decrease of $\rho_n(f,X)$?

\smallskip

Some results [32]:

$f^{(r)}$ of bounded variation $\Rightarrow \rho_n =O(n^{-r-1})$  (Popov)

$f\in \text{ Lip}_1 \Rightarrow \rho_n = o(1/n)$  (small $o$!)
       (Popov: Newman's conjecture)

$f$ piecewise analytic $\Rightarrow \rho_n =O(e^{-c\sqrt{n}})$
    (Sz\H usz \& Turan (after Newman))

\smallskip

Let $\kappa(f;n) = \sup \sum_1^{n-1} |f(z_{i+1})- f(z_i)|$ be the
modulus of variation of $f$. Then,

$\kappa(f;n)=O(n^\beta), f\in\text{ Lip}_\alpha,
  \alpha>0,  \Rightarrow,
   \rho_n(f)=O(n^{\beta+\varepsilon-1})$ (Petrushev)

$\rho_n(f)= O(n^{\beta-1}) , 0<\beta<1 \Rightarrow
        \kappa(f;n) =O(n^\beta)$  (Popov)

\medskip

$f$ is in the \emph{\textbf{approximation space}} $R_{X,q}^\alpha$ if
$$ \left[ \sum_\nu \left( 2^{\nu\alpha} \rho_{2^\nu}(f,X)\right)^q
  \right]^{1/q} <\infty,\qquad\qquad \alpha>0, 0<q\leqslant\infty. $$
$q=\infty$: $f\in R_{X,\infty}^\alpha \iff \rho_n(f,X) =O(n^{-\alpha})$.

\smallskip
How do the functions in
$R_{X,q}^\alpha$ behave?

\smallskip

Answer: $R_{p,\sigma}^\alpha = B_\sigma^\alpha$, where
  $1/\sigma= 1/p+\alpha$, $1<p<\infty$, $\alpha>0$
  (\emph{\textbf{Pekarskii's main theorem}}).

\smallskip
For $p=\infty$, use the, yes, BMO metric: $R_{*,1/\alpha}^\alpha =
 B_{1/\alpha}^\alpha$ (Peller, Semmes, Pekarskii),

\smallskip

where $B_p^\alpha$ is the \emph{Besov} space $\displaystyle{
 \left\{ f\in L^p(\mathbb{T}): \int_{-\pi}^\pi |t|^{-1-\alpha p}
      \|\Delta^n_t f\|_p^p\, dt <\infty \right\} }$, with
   any integer $n>\alpha$, and where $\Delta_t^n$ is the $n^{\text{th}}$
   power of $\Delta_t: (\Delta_t f)(e^{i\theta}) =
     f(e^{i(\theta+t)} -f(e^{i\theta})$.

\smallskip

Analytic Besov: $\displaystyle{A_p^\alpha = \left\{ f^{(+)}\in H^p:
    \int\!\!\int_{\mathbb{D}} (1-|z|)^{(n-\alpha)p-1}
        \left|\left(f^{(+)}\right)^{(n)(z)}\right|^p\,dxdy
   < \infty \right\}}$.

\subsection{Hankel: AAK and Peller} \  \\


Main result of AAK [2-5], from [29]:

Let $f^{(+)}(z) = \sum_0^\infty c_k z^k$ such that
$\displaystyle{ \mathcal{H}(f^{(+)} = \begin{bmatrix}
                                  c_1 & c_2 & c_3 & \cdots  \\
                                  c_2 & c_3 & c_4 & \cdots \\
                                  c_3 & c_4 & c_5 & \cdots \\
                                  \vdots&\vdots&\vdots&   \\
                                  \end{bmatrix}  }$
 is compact on $\ell^2$, true iff there is a $g^{(-)}(z)=\sum_1^\infty
  d_k z^{-k}$ such $f^{(+)}+g^{(-)}$ continuous on $\mathbb{T}$.

\smallskip

Then,

\begin{enumerate}
\item $\sigma_{n+1} = \min \|\mathcal{H} -C\|_{\ell^2}$ on all the
 bounded $C$'s of rank $\leqslant n$ (a general property of singular
 values of compact operators).

\item $\sigma_{n+1} = \min \|\mathcal{H} -C\|_{\ell^2}$ on all the
 bounded \emph{\textbf{Hankel operators}} $C$'s of rank $\leqslant n$
   (first big AAK result).
\end{enumerate}

\smallskip

{\Small
 An excerpt from [29], pp.~241-242:


For the applications we have in mind (which essentially involve finite
rank Hankel matrices), the most profitable point of view is usually
\emph{sequential} rather than \emph{functional}; that is why we prefer
here, unlike the authors of [4], to speak of \emph{singular value decomposition
of matrices} (whose exceptionally fruitful idea apparently dates back to
E.~Beltrami and C.~Jordan, 1873-1874, see [C.C. MacDuffee: \textsl{The
Theory of Matrices}, Chelsea, New York, 1946, p.78],
            rather than of
\emph{Schmidt expansion of operators} (which was introduced by E.~Schmidt in
the quite different setting of integral equations with non-Hermitian kernels).
Now, unlike the best approximation property 1. and the reminder complementing
its proof, the following is conditioned by the property of
 $\mathcal{H}$ to be a Hankel
matrix.

}

\begin{enumerate}  \setcounter{enumi}{2}
\item $\sigma_{n+1} = \min \|\mathcal{H}(B_n f^{(+)})\|_{\ell^2}$ on the
 Blaschke products $\displaystyle{
   B_n(z) =\prod_{j=1}^n \dfrac{z-\zeta_j}{1-\overline{\zeta_j}\; z} }$
   with $|\zeta_j|>1$.  ($\mathcal{H}(B_n f^{(+)})$ means the Hankel
   operator built with the coefficients of $z$, $z^2$\dots of the Laurent
   expansion on $\mathbb{T}$ of $B_n f^{(+)}$).
\item $\displaystyle{\sigma_{n+1} = \min \sup_\theta \left|
   f^{(+)}(e^{i\theta}) -g^{(-)}(e^{i\theta})
     -\sum_{j=1}^n \dfrac{\rho_j}{e^{i\theta} -\zeta_j}\right|  }$
     on $g^{(-)}$, the $\rho$'s and the $|\zeta_j|>1$'s.
\end{enumerate}

\medskip

Rate of decrease of the $\sigma$'s:

$\mathcal{H}(f^{(+)})$ is in the Schatten-von Neumann class
$\mathfrak{S}_p$, i.e., $\sum_1^\infty \sigma_n^p <\infty \iff
  f^{(+)}\in A_p^{1/p}$ ($1\leqslant p<\infty$) (Peller [34]).

\emph{\textbf{Exercise}}: check when $p=2$: $n=1$ in the definition of
$A_p^{1/p}$ above, and the trace of $\mathcal{H}^2$ is recovered.

\smallskip

Other: Chui, Glover

\subsection{Exponential decrease of the $\sigma$'s for analytic functions}
\  \\

Let $f^{(+)}$ be analytic in a domaining containing $\overline{\mathbb{D}}$,
so

$\sigma_{n+1} =$ norm of $\mathcal{H}\left(\left( B_n f^{(+)} \right)^{(+)}\right)
 \leqslant $
 $\sum_\theta \left|\left(B_n (e^{i\theta}) f^{(+)}(e^{i\theta}) \right)^{(+)}\right|$.

 $\left(B_n (e^{i\theta}) f^{(+)}(e^{i\theta}) \right)^{(+)} =
   \displaystyle{ \dfrac{1}{2\pi i}\int_C
        B_n (z) f^{(+)}(z) \,  \dfrac{dz}{z-e^{i\theta}}
  }$, where $C$ is any contour outside the unit disk.

We build $C$ and $B_n$ such that: 1. $f^{(+)}$ is still analytic in the
interior of $C$, 2. $\max_z |B_n(z) f^{(+)}(z)|$ is as small as possible
on $C$: weighted Zolotarev problem.

\smallskip

Example:  $C= \{z: |z|=R>1\}$, $\zeta_j=\infty \Rightarrow B_n(z)=z^{-n}$:
 $\sigma_{n+1} \leqslant$ const.$/(R-\varepsilon)^n$.

\smallskip
And: 3. numerator of $B_n$ orthogonal with respect to complex weight
function $f^{(+)}/$ square of denominator of $B_n$: connection with
Stahl's theory.

\end{document}
