$ denote a
scalar product on the space~$\{ {\cal P}_C\}$ with the property
\begin{equation}
=

\qquad \hbox{\rm for the polynomials}\quad P,Q\in\{ {\cal P}_C\}
\end{equation}
We shall assume that the Toeplitz determinants are nonzero:
$$\Delta_n=\det\left| \, \matrix{
{<1,1>}&{}&\ldots&{}\cr
\overline {}&{<1,1>}&\ldots&{}\cr
\vdots&\vdots&\ddots&\vdots\cr
\overline {}&\overline {}&\ldots&{<1,1>}\cr
}\right|\neq0 ,\qquad n=0,~1,\ldots .$$
It is well known (see for example \cite{Achieser}) that there exists a
$< , >$-orthogonal system of polynomials~$\{P_n(z)\}$ such that
1) $\hbox{\rm deg} P_n(z)=n$
2) the highest-degree coefficient is positive
3) $=0,$ if~$ n\neq m.$
We will consider the monic system of polynomials $P_k(z)$, which are
characterized by the highest degree coefficient being equal to $1$.
The monic polynomials~$\{P_n(z)\}$ satisfy the recurrence relation
for the orthogonal sets of polynomials:
\begin{equation}
P_{n+1}(z)=zP_n(z)+b_{n+1}P_n^*(z),
\label{rec}
\end{equation}
where~$R^{*}(z) \buildrel \rm def \over = z^k \overline R(1/z)$ for any
polynomial~$R(z)$ of degree~$k$; $b_k$ are
some constants for which formal analytic relations in determinant
form can be obtained. Note that $b_0$ does not take part in the
recursion~(\ref{rec}), however, below it will be convenient to define
$$|b_0|^2=1-<1,1>.$$
>From recurrence relations~(\ref{rec}) the following three-term recurrence
relation immediately follows:
\begin{equation}
b_{n+1}P_{n+2}(z)=(zb_{n+1}+b_{n+2})P_{n+1}(z)-
b_{n+2}(1-|b_{n+1}|^2)zP_n(z).
\label{ttrec}
\end{equation}
The following criterion of solvability of the moment problem (\ref{1.c_k})
with the gaps $I_m$ holds \cite{Ugrinovsky}:
\begin{theorem}
The trigonometric moment problem (\ref{1.c_k}) with the gaps
$I_m=\{z:\ z=e^{i\theta},\ \theta\in\cup_{i=1}^m [r_i, t_i],
\ -\pi\le r_1$$
will be called a {\it Chebyshev matrix}.
\end{definition}
Note that $\sigma_{k,l}=0$ for all~$k>l$, i.e. the Chebyshev matrix is
triangular.
The entries of the Chebyshev matrix~$T_{P,\pi}$ and the coefficients of
the three-term recurrence formulas (\ref{ttrec}) and~(\ref{ttrecpi})
are connected by
relations that can be considered as a recursion process to determine
entries of the Chebyshev matrix and unknown elements~$b_k$. Writing
it down in a recursive manner, we have:
{{\it the initial conditions:}~$\sigma_{-1,l}=0,\quad\sigma_{0,l}=<1,\pi_l>,
\quad l=0,~1,\ldots,N ,$ and $b_0=1$
{\it and for all} $k=1,~2,\ldots,~N$
$$b_k={1\over{b_{k-1}}}\bigl[-{\sigma_{k-1,k}\over\sigma_{k-1,k-1}}
+{\overline \beta_k\over\overline \beta_{k-1}}
+(1-|b_{k-1}|^2){\sigma_{k-2,k-1}\over\sigma_{k-1,k-1}}
-(1-|b_{k-1}|^2){\overline \beta_k\over\overline \beta_{k-1}}
(1-|\beta_{k-1}|^2){\sigma_{k-2,k-2}\over\sigma_{k-1,k-1}}
\big]$$
\rightline {{~}~$(m)-T_C$}
\[\sigma_{k,l}=\left\{
\matrix{
0, & \qquad l=0,~1,\ldots,~{k-1}; \cr
{\overline \beta_l\over\overline \beta_{l-1}}\sigma_{k,l-1}+
{{b_k}\over {b_{k-1}}}\sigma_{k-1,l}-
{{b_k}\over {b_{k-1}}}{\overline \beta_l\over\overline \beta_{l-1}}
\sigma_{k,l-1}+\sigma_{k-1,l-1}& \cr
-{\overline \beta_l\over\overline \beta_{l-1}}(1-|\beta_{l-1}|^2)\sigma_{k-1,l-2}-
{{b_k}\over{b_{k-1}}}(1-|b_{k-1}|^2)\sigma_{k-2,l-1} \cr
+{{b_k}\over{b_{k-1}}}(1-|b_{k-1}|^2)
{\overline \beta_l\over\overline \beta_{l-1}}(1-|\beta_{l-1}|^2)\sigma_{k-2,l-2},
& \qquad l=k,~k+1,\ldots,~N \cr
}
\right. \]
}
In the case~$\pi_k(z)=z^k$, i.e. $\beta_k=0$ for~$k=0,~1,\ldots,$ the
$(m)-T_C$-recurrence relations are transformed into the well-known Levinson
algorithm \cite{Levinson}:
{\it step~1:} $P_0(z)=1,\qquad\sigma_{0,0}=<1,1>,$
{\it step~2:} for~$k=1,~2,\ldots,~N$
\begin{eqnarray*}
b_k &=&-{1\over\sigma_{k-1,k-1}},\cr
\sigma_{k,k}&=&(1-|b_k|^2)\sigma_{k-1,k-1},\cr
P_k(z) &=&z P_{k-1}(z)+b_k P_{k-1}^*(z).
\end{eqnarray*}
Let us note now that if the orthogonal polynomial $P_n(z)$ of the first type
is known the orthogonal polynomial $Q_n(z)$ of the second type can be found
by means of the relation
\begin{equation}
\left[ \, \matrix{
q_{n,0} \cr
q_{n,1} \cr
q_{n,2} \cr
\vdots \cr
q_{n,n} \cr
}\right] =-{1\over{c_0}} \left[ \, \matrix{
0 & c_1 & c_2 & c_3 & \ldots & c_n \cr
0 & c_0 & 2c_1 & 2c_2 & \ldots & 2c_{n-1} \cr
0 & 0 & c_0 & 2c_1 & \ldots & 2c_{n-2} \cr
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \cr
0 & 0 & 0 & 0 & \ldots & c_0 \cr
}\right]
\left[ \, \matrix{
p_{n,0} \cr
p_{n,1} \cr
p_{n,2} \cr
\vdots \cr
p_{n,n} \cr
}\right] \ ,
\label{P_k-to-Q_k}
\end{equation}
where $[q_{n,0},~q_{n,1},\ldots,~q_{n,n}]^T$ is the vector of coefficients
of the polynomial $Q_n(z)=\sum\limits_{m=0}^n q_{n,m} z^m $,
$[p_{n,0},~p_{n,1},\ldots,~p_{n,n}]^T$ is the vector of coefficients
of the polynomial $P_n(z)=\sum\limits_{m=0}^n p_{n,m} z^m $ and
$c_0,~c_1,\ldots,~c_N$ are the given moments.
\section{Example}
In this section we give an illustrative example and consider the PSDE
problem with one gap $[r,t]$, i.e., further we will suppose $m=1$. In this
case the above described numerical algorithm is transformed as follows:
\begin{description}
\item[Step 1] We find the system of monic polynomials $\{P_k(z)\}_{k=0}^N$
orthogonal with respect to given moment sequence by means of the
$(m)-T_C$-recursion or Levinson algorithm.
\item[Step 2] From relation (\ref{P_k-to-Q_k}) we find the polynomials
$Q_{N-1}(z)$ and $Q_N(z)$.
\item[Step 3] We find the functions
\begin{eqnarray*}
{\cal A}(z,\zeta)&=&Q_N(z)P_N(z,\zeta)-P_N(z)Q_N(z,\zeta) , \\
{\cal B}(z,\zeta)&=&P_N^*(z)Q_N(z,\zeta)+Q_N^*(z)P_N(z,\zeta) ,
\end{eqnarray*}
where
\begin{eqnarray*}
P_N(z,\zeta)&=&P_N(z)P_{N-1}(\zeta)-P_N(\zeta)P_{N-1}(z) , \\
Q_N(z,\zeta)&=&Q_N(z)P_{N-1}(\zeta)-Q_{N-1}(z)P_N(\zeta)
\end{eqnarray*}
for $\zeta=\zeta_r$ and $\zeta=\zeta_t$ ($\zeta_r=e^{-i r}$, $\zeta_t=e^{-i t}$)
and the functions
\begin{eqnarray*}
\alpha(z)&=&|z|^2 [|{z-\zeta_t}|^2 |{\cal A}(z,\zeta_r)|^2-
|{z-\zeta_r}|^2|{\cal A}(z,\zeta_t)|^2] , \\
\beta(z)&=&\overline z [|{z-\zeta_t}|^2
\overline{{\cal A}(z,\zeta_r)}{{\cal B}(z,\zeta_r)}
-|z-\zeta_r|^2
\overline{{\cal A}(z,\zeta_t)} {{\cal B}(z,\zeta_t)}] , \\
\gamma(z)&=&|{z-\zeta_t}|^2 |{\cal B}(z,\zeta_r)|^2
-|z-\zeta_r|^2 |{\cal B}(z,\zeta_t)|^2 .
\end{eqnarray*}
\item[Step 4] We calculate the function
$$\varepsilon_0(z)=\left({{|\beta(z)|}\over{\alpha(z)}}-R(z)\right)
e^{i {{\Im m\beta(z)}\over{\Re e\beta(z)}}} , $$
where
$$O(z)={{\beta(z)}\over{\alpha(z)}}, \qquad
R(z)={{\sqrt{|\beta(z)|^2-\alpha(z)\gamma(z)}}\over{\alpha(z)}} ,$$
for all $z=e^{i\theta}$, $\theta\in [-\pi,\pi]$.
\item[Step 5] We find the PSDE by
$$
P_{\varepsilon_0}(z)={{c_0}\over{h_N}}\cdot
\Re e {
{zQ_N(z)\varepsilon_0(z)+Q^*_N(z)}
\over
{zP_N(z)\varepsilon_0(z)-P^*_N(z)}
}, \quad z=e^{i\theta}, \quad \theta\in [-\pi,\pi] .
$$
\end{description}
Let us note that the functions $P_N(z,\zeta)$, $Q_N(z,\zeta)$,
${\cal A}(z,\zeta)$, ${\cal B}(z,\zeta)$, $\alpha(z)$, $\beta(z)$ and
$\gamma(z)$ from Steps~1--3 are polynomials and, therefore, it is sufficient
to perform all calculations only with the coefficients of these polynomials,
but not with their values. The computation of values of the functions
$\alpha(z)$, $\beta(z)$ and $\gamma(z)$ for $z=e^{i\theta}$ is necessary
only in Step~4 to calculate the function $\varepsilon_0(z)$ for $z=e^{i\theta}$
and then the PSDE $P_{\varepsilon_0}(z)$ corresponding to the function
$\varepsilon_0(z)$.
Now we will consider the elementary numerical example for correlation sequence
$$c_k=\sum_{l=1}^L q_l {{\alpha_l}\over{4\alpha_l^2-k^2}}
{{\sin{{\pi k}\over{2\alpha_l}}}\over{{\pi k}\over{2\alpha_l}}}
e^{-ika_l}+\epsilon\delta_k ,$$
where $q_l, a_l$ and $\alpha_l$, $l= 1,~2,~\ldots,~L$ and $L$ are given values,
$\epsilon$ is a parameter of regularization. Let us note, that the original
spectral power density has the gaps
$[a_l+{\pi\over{2\alpha_l}},a_{l+1}-{\pi\over{2\alpha_l}} ]$, $l=0,\ldots,~L-1$
with the widths
$\Delta\theta=a_{l+1}-a_l-{\pi\over 2}
\left( {1\over{\alpha_l}}+{1\over{\alpha_{l+1}}}\right)$.
The original power spectral density for $L=2$, $a_1=-\pi /7, a_2=\pi /9,
q_1=10, q_2=100, \alpha_1=16.1, \alpha_2=2.37$ is shown in Fig.~1%\ref{fig1}.
We assume that finite number of moments is known, for example $N=21$.
The PSDE obtained by means of proposed technique for the pointed out parameters
and different values of the regularization parameter $\epsilon$ is presented
in Fig.~2.%\ref{fig2}.
We used the algorithm for $m=1$ and $r=a_1+{\pi\over{2\alpha_1}}+0.15$,
$t=a_2-{\pi\over{2\alpha_2}}-0.15$.
As one can see from the figures the proposed algorithm
helps to get the PSDE with good resolution and to keep gaps in the PSDE.
\section{Conclusion}
We considered the spectral power density estimation problem with gaps.
The maximum entropy (ME) approach developed in the paper to solve the problem
is universal and can be applied to the solution of other problems.
The specificity of PSDE problem prompts us to consider trigonometric
moment problem and construct the ME algorithm to find the extreme solution
of trigonometric moment problem with gaps. In complete analogy the classic
moment problem can be considered and the ME algorithm to find the extreme
solution of moment problem with gaps can be obtained.
\section*{Acknowledgements}
I am grateful to I.E.~Ov\^carenko for his direct as well as indirect
influence and very useful discussions.
This work was supported by the Russian Foundation of Fundamental Researches,
Grant No. 93-02-16043.
%\begin{figure}[s]
%\vspace{14cm}
%\caption{\label{fig1}}
%\end{figure}
%\begin{figure}[s]
%\vspace{14cm}
%\caption{\label{fig2}}
%\end{figure}
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\end{document}