text:
\section{Introduction}

W.R.~Bennett \cite{Bennet} in 1958  introduced cyclostationarity as a phenomenon describing signals in channels of communication.
Studying the statistical characteristics of information transmission, he calls the group of telegraph signals a cyclostationary process, that is the process whose group of statistics changes periodically with time.
W.A.~Gardner and L.~E.~Franks \cite{Gardner1975}  highlighted the similarity of  cyclostationary processes, which form a subclass of nonstationary processes, with stationary processes.
W.A.~Gardner \cite{Gardner1994}, W.~A.~Gardner, A.~Napolitano and L.~Paura \cite{Gardner2006} presented bibliography of works in which properties and applications of cyclostationary processes were studied. Recent developments and applications of cyclostationary signal analysis are reviewed in the papers by A.~Napolitano  \cite{Nap1}, \cite{Nap2}.  Note that in different sources cyclostationary processes are called periodically stationary, periodically nonstationary, periodically correlated. We will use the term periodically correlated processes.

E.G.~Gladyshev \cite{Glad1961} was the first who analysed the spectral properties and representations of periodically correlated sequences based on its connection with the vector valued stationary sequences. He formulated the necessary and sufficient conditions for determining the periodically correlated sequence in terms of the correlation function. A.~Makagon  \cite{Makagon1999}, \cite{Makagon2011} presented a detailed spectral analysis of periodically correlated sequences. The main ideas of the research  of periodically correlated sequences are outlined in the book by H.~L.~Hurd and A.~Miamee~\cite{Hurd}.

The problem of estimation of unobserved values of random processes is one of the very important and topical subsections of the theory of stochastic processes. Processes that are observed can be completely defined by its characteristics (correlation function, spectral density, canonical decomposition) or their characteristics can be defined only by the set of admissible values of characteristics.
The linear extrapolation and interpolation problems for stationary
stochastic sequences under the condition that the spectral densities are
 exactly known were first investigated by A.~N.~Kolmogorov \cite{Kolmogorov}.
Methods of solutions of the extrapolation and filtering problems for stationary
processes and sequences with rational spectral densities were
developed by N.~Wiener \cite{Wiener} and A.~M.~Yaglom \cite{Yaglom-a, Yaglom-b}. Estimation
problems for vector-valued stationary processes were investigated by Yu.~A.~Rozanov~\cite{Rozanov} and E.~J.~Hannan~\cite{Hannan1974}.

 The basic techniques of statistics of stochastic processes are summarized in the books by  D.~Z.~Arov and H.~Dym~\cite{Arov2018}, L.~Aggoun  and R.~J.~Elliott~\cite{Aggoun2004},
 I.~V.~Basawa  and  B.~L.~S.~Prakasa Rao \cite{Basawa1980},  S.~Cohen and   R.~J.~Elliott~\cite{Cohen2015},
 M.~S.~Grewal and A.~P.~Andrews \cite{Grewal2015}, G.~Kallianpur \cite{Kallianpur1980}, Yu.~A.~Kutoyants  \cite{Kutoyants1998}, \cite{Kutoyants2004},
 R.~S.~Liptser  and A.~N.~Shiryaev  \cite{Liptser2001a,Liptser2001b},
 B.~L.~S.~Prakasa Rao \cite{PrakasaRao2010,PrakasaRao2012}, B.~L.~Rozovsky  and S.~V.~Lototsky~\cite{Rozovsky2018},
 M.~B.~Rajarshi~\cite{Rajarshi2012},
W.~A.~Woodward, H.~L.~Gray and A.~C.~Elliott  \cite{Woodward2017}.
The estimation problems occur in different studies. We refer to D.~V.~Koroliouk \cite{Koroliouk1}, D.~V.~Koroliouk et al.~\cite{Koroliouk2}, D.~V.~Koroliouk  and V.~S.~Koroliuk~\cite{Koroliouk3},  where there is investigated the difference stochastic equation
$ \Delta\alpha_{t+1}=-V_0\alpha_t+\sigma_0 \Delta W_{t+1},\, t\geq0,$
which determines a sequence $\alpha_t$, $t\geq0$, for the stochastic component $\Delta W_{t+1}$, $t\geq0$, and studied the problem of filtration of stationary Gaussian statistical experiments considered for the solution $\alpha_t$, $t\geq0$, of the indicated equation.

Since processes often accompanied by undesirable noise it is naturally to assume that the exact value of  spectral density is unknown and the model of process  is given by a set of restrictions on spectral density.
K~S.~Vastola and H~V.~Poor~\cite{VastPoor}  showed for   certain classes of spectral densities that the Wiener filter is very sensitive to minor changes of spectral model unlike the robust Wiener filter.  That is the filter is the least sensitive to the worst case of uncertainty. Thus,  it is reasonable to use the minimax (robust) estimation method, which allows to define the optimal estimate for all densities from a certain given class of the admissible spectral densities simultaneously.
 Ulf Grenander \cite{Grenander} was the first who
proposed the minimax approach to the extrapolation problem for stationary processes.
A survey of results in minimax-robust methods of data processing can be found in the paper by S.~A.~Kassam and H.~V.~Poor \cite{Kassam}.
Formulation and investigation of the problems of extrapolation, interpolation and filtering of linear functionals which depend on the unknown values of stationary sequences and processes  from observations with and without noise are presented by M.~P.~Moklyachuk in the papers \cite{Moklyachuk2000}--\cite{Moklyachuk2015}.
Similar problems of the optimal estimation for the vector-valued stationary sequences and processes were examined by M.~P.~Moklyachuk \cite{Moklyachuk1993}--\cite{Moklyachuk:1995ROSE} and by M.~P.~Moklyachuk and O.~Yu.~Masyutka  \cite{MoklyachukMas2006a}--\cite{MoklyachukMas2012}.
In their papers M.~M.~Luz and M.~P.~Moklyachuk \cite{luz1}--\cite{luz11} investigated the minimax estimation problems for linear functionals which depends on unobserved values of stochastic sequences with stationary increments.
 P.~S.~Kozak and M.~P.~Moklyachuk~\cite{Kozak} study estimates of functionals constructed from random sequences with periodically stationary increments.
In their papers I.~I.~Golichenko(Dubovets'ka) and M.~P.~Moklyachuk \cite{Dubovetska0}--\cite{Dubovetska9}, \cite{MoklyachukD2016} presented results of investigation of the
interpolation, extrapolation and filtering problems for linear functionals from periodically correlated stochastic sequences and processes.

The prediction problem for stationary sequences with missing observations is investigated in the papers by P.~Bondon \cite{Bondon1,Bondon2},
R.~Cheng, A.~G.~Miamee and M.~Pourahmadi \cite{Cheng1}, R.~Cheng and M.~Pourahmadi~\cite{Cheng2},
Y.~Kasahara, M.~Pourahmadi and A.~Inoue~\cite{Kasahara,Pourahmadi}.
The detailed analysis of the estimation problems with missing observations are presented in the
paper by B.~Abraham\cite{Abraham},
books by
M.~J.~Daniels and J.~W.~Hogan \cite{Daniels2008},
 R.~J.~A.~Little and D.~B.~Rubin \cite{Little2002},
 P.~E.~McKnight et al \cite{McKnight2007}, M.~M.~Pelagatti~\cite{Pelagatti}.

In the papers by M.~P.~Moklyachuk and M.~I.~Sidei
\cite{MSidei2015}--\cite{MSidei2017}
results of investigations of the interpolation, extrapolation and filtering problems for stationary stochastic sequences and processes with missing observations are proposed.
The results of the study of the extrapolation, interpolation and filtering problems for linear functionals constructed from
unobserved values of multidimensional stochastic sequences and processes are presented in the papers by O.~Yu.~Masyutka, M.~P.~Moklyachuk and M.~I.~Sidei
\cite{Masyutka:Moklyachuk:Sidei1}--\cite{Masyutka:Moklyachuk:Sidei5}, \cite{Moklyachuk:Mas:Sidei}.
We also refer to the book by M.~P.~Moklyachuk, O.~Yu.~Masyutka and I.~I.~Golichenko \cite{Moklyachuk:Mas:Gol2018}
where results of the investigation of the problem of mean square optimal estimation (forecasting, interpolation, and filtering) of linear functionals constructed from unobserved values
of periodically correlated isotropic random fields are described.

In this paper we deal with the problem of optimal linear estimation of
the functional $A_s{\zeta}$ which depends on the
unobserved values of a periodically correlated stochastic sequence
${\zeta}(j)$. Estimates are based on observations of the sequence
${\zeta}(j)+{\theta}(j)$ at points  $j\in{\mathbb
Z}\setminus S$,  where $S=\bigcup _{l=0}^{s-1}\{M_l+1,\dots,M_l+N_{l+1}\}$. ${\theta}(j)$ is an uncorrelated
with ${\zeta}(j)$ periodically correlated stochastic sequence.
Formulas for calculation the mean square errors and the spectral
characteristics of the optimal estimates of the functional
$A_{s}\zeta$ are proposed in the case of spectral certainty where the spectral densities are
exactly known. Formulas that determine the least favorable spectral
densities and minimax spectral characteristics are proposed in the case of spectral uncertainty where the spectral densities are
not exactly known while some classes of admissible spectral densities are given.

The paper is organized as follows.
The spectral properties of periodically correlated stochastic sequences and their correlation functions are described in Section 2.
Relations of periodically correlated stochastic sequences with multidimensional stationary sequences are discussed in this section.

In section 3 we consider the problem of mean square optimal linear estimation of the
the functional $$A_{s}\vec {\xi}=\sum_{l=0}^{s-1}\sum_{j=M_l+1}^{M_l+N_{l+1}}\vec {a}^{\top}(j)\vec
{\xi}(j),\,\,M_l=\sum_{k=0}^l(N_k+K_k),\,\,N_0=K_0=0,$$ which depends on the unknown values of a $T$-dimensional
stationary stochastic sequence  $\vec {\xi}(j)$,
based on observations of the sequence $\vec
{\xi}(j)+\vec{\eta}(j)$ at points $j\in {\mathbb Z}
\setminus S$, where $S=\bigcup _{l=0}^{s-1}\{M_l+1,\dots,M_l+N_{l+1}\}$.
Formulas for calculation the mean square error and the spectral
characteristic of the optimal estimate of the functional
$A_{s}\vec \xi$ are proposed in the case where spectral density matrices of the sequences
$\vec {\xi}(j)$ and $\vec{\eta}(j)$ are exactly known.

In section 4 we consider the problem of mean square optimal linear estimation of the
the functional
 $$A_s{\zeta}=\sum_{l=0}^{s-1}\sum_{j=M_l+1}^{M_l+N_{l+1}}{a}(j){\zeta}(j),
\,\,
 M_l=\sum_{k=0}^l(N_k+K_k),\,\,N_0=K_0=0,$$
which depends on the unknown values of T-PC stochastic
sequence $\zeta(j)$, based on
observations of the sequence $\zeta(j)+\theta(j)$ at points $j\in {\mathbb
Z} \setminus S,$ where $S=\bigcup _{l=0}^{s-1}\{M_l+1,\dots,M_l+N_{l+1}\}$.

In section 5 we consider the problem
of optimal estimation for the linear functional
$$A_s{\zeta}=\sum_{l=0}^{s-1}\sum_{j=M_l+1}^{M_l+N_{l+1}}{a}(j){\zeta}(j),\,\,M_l=\sum_{k=0}^l(N_k+K_k),\,\,N_0=K_0=0,$$
which depends on the
unknown values of $T$-PC sequence
${\zeta}(j)$ from observations of the sequence
${\zeta}(j)+{\theta}(j)$ at points $j\in{\mathbb
Z}\setminus S$,
where the number of missed observations at each of the intervals is a multiple of the period $T$.
In sections 4 and 5 the estimation problem is investigated in the case of spectral certainty, where the spectral densities of observed sequences are exactly known.

In section 6 we describe the minimax approach to the problem of  estimation of the linear functionals. In this case we find the
estimate which minimizes the mean square error for all spectral
densities from the given set of admissible densities simultaneously.

In section 7
the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal estimate of ${A_s \vec \zeta}$ are found  for the class $D_{0}^{-}$ of admissible spectral densities.

In section 8
the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal estimate of ${A_s \vec \zeta}$ are found  for the class $D_{G}^{-}$ of admissible spectral densities.

\section{Periodically correlated and multidimensional stationary  sequences}

The term \textit {periodically correlated} process was introduced by E.~G.~Gladyshev \cite{Glad1961} while W.~R.~Bennett \cite{Bennet} called random and periodic processes
\textit {cyclostationary} process.

Periodically correlated sequences are stochastic sequences that have
periodic structure (see the book by H.~L.~Hurd and A.~Miamee~\cite{Hurd}).

\begin{definition} A complex valued stochastic
sequence ${\zeta}(n),n\in{\mathbb Z}$ with zero mean,
$\textsf{E}{\zeta}(n)=0$, and finite variance, $\textsf{E}|{\zeta}(n)|^{2}<+\infty$,
is called cyclostationary or periodically
correlated (PC) with period $T$ ($T$-PC) if for every
$n,m\in{\mathbb Z}$
\begin{equation}\label{2.1}
\textsf{E}{\zeta}(n+T)\overline{\zeta(m+T)}=R(n+T,m+T)=R(n,m)
\end{equation}
and there are no smaller values of $T>0$ for
which (\ref{2.1})  holds true.
\end{definition}

\begin{definition} {A complex valued
T-variate stochastic  sequence $\vec{\xi}(n)=\left\{ \xi_{\nu}(n)
\right\}_{\nu = 1 }^{T},$ $n\in {\mathbb Z}$ with zero mean,
$\textsf{E}\xi_{\nu}(n)=0, \nu=1,\dots,T$, and $\textsf{E}||\vec {\xi}(n)||^{2}<\infty$
is called stationary if for all $n ,m\in {\mathbb Z}$ and
$\nu,\mu\in \{1,\dots,T\}$}
\[
\textsf{E}\xi_{\nu}(n)\overline{\xi_{\mu}(m)}=R_{\nu\mu}(n,m)=R_{\nu\mu}(n-m).
\]
\noindent If this is the case, we denote $R(n)=\left\{ R_{\nu\mu}(n)
\right\}_{\nu,\mu = 1 }^{T}$ and call it the \textit {covariance
matrix} of T-variate stochastic sequence $\vec{\xi}(n)$.
\end{definition}

\begin{Proposition} {\rm{(E. G. Gladyshev \cite{Glad1961})}}.
A stochastic sequence $\zeta(n)$ is PC with period $T$ if and only if there exists
a $T$-variate stationary sequence $\vec {\xi}(n)=\left\{ \xi_{\nu}(n)
\right\}_{\nu = 1 }^{T}$ such that $\zeta(n)$ has the representation
\begin{equation}\label{2.2}
\zeta(n)=\sum_{\nu=1}^{T}{e}^{2\pi in\nu/T}\xi_{\nu}(n),\,\,n\in {\mathbb
Z}.
\end{equation}
\noindent The sequence $\vec \xi(n)$ is called  \textit {generating sequence} of the sequence $\zeta(n)$.
\end{Proposition}

\begin{Proposition} {\rm{(E. G. Gladyshev \cite{Glad1961})}}. A complex valued stochastic
sequence ${\zeta}(n),n\in{\mathbb Z}$ with zero mean and finite variance is PC with period $T$ if and only if the  $T$-variate blocked sequence $\vec \zeta(n)$ of the form
\begin{equation} \label{block}
[\vec{\zeta}(n)]_{p}=\zeta(nT+p),\,\,n\in {\mathbb
Z},p=1,\dots,T
\end{equation}
 is stationary.
 \end{Proposition}

We will denote by $f^{\vec \zeta}(\lambda)=\left\{ f^{\vec \zeta}_{\nu\mu}(\lambda)
\right\}_{\nu,\mu = 1 }^{T}$ the matrix valued
spectral density function of the $T$-variate  stationary
sequence $\vec \zeta(n)=(\zeta_1(n),\dots,\zeta_T(n))^{\top}$ arising from the $T$-blocking (\ref{block}) of a univariate T-PC
sequence $\zeta(n)$.

\section{Hilbert space projection method of linear interpolation}

Let $\vec {\xi}(j)$ and $\vec {\eta}(j)$ be uncorrelated T-variate
stationary stochastic sequences with the spectral density matrices
$f^{\vec \xi}(\lambda)=\left\{ f^{\vec \xi}_{\nu\mu}(\lambda)
\right\}_{\nu,\mu = 1 }^{T}$ and $f^{\vec \eta}(\lambda)=\left\{
f^{\vec \eta}_{\nu\mu}(\lambda) \right\}_{\nu,\mu=1 }^{T}$,
respectively. Consider the problem of optimal linear estimation of
the functional $$A_{s}\vec {\xi}=\sum_{l=0}^{s-1}\sum_{j=M_l+1}^{M_l+N_{l+1}}\vec {a}^{\top}(j)\vec
{\xi}(j),\,\,M_l=\sum_{k=0}^l(N_k+K_k),\,\,N_0=K_0=0,$$ that depends on the unknown values of the sequence  $\vec {\xi}(j)$,
based on observations of the sequence $\vec
{\xi}(j)+\vec{\eta}(j)$ at points $j\in {\mathbb Z}
\setminus S$, where $S=\bigcup _{l=0}^{s-1}\{M_l+1,\dots,M_l+N_{l+1}\}$.

Let the spectral densities $f^{\vec \xi}(\lambda)$ and $f^{\vec
\eta}(\lambda)$ satisfy the minimality condition
\begin{equation}\label{3.5}
\int_{-\pi}^{\pi}{Tr{\left[ {(f^{\vec \xi}(\lambda)+f^{\vec
\eta}(\lambda))^{-1}} \right]}} d\lambda <+{\infty}.
\end{equation}
\noindent Condition (\ref{3.5}) is necessary and sufficient in order that
the error-free interpolation of the unknown values of the sequence $\vec
{\xi}(j)+\vec {\eta}(j)$ is impossible \cite{Rozanov}.

 Denote by $L_{2}(f)$
the Hilbert space of vector valued functions $\vec  b(\lambda)=\left\{
b_{\nu}(\lambda) \right\}_{\nu = 1 }^{T}$ that are square integrable with
respect to a measure with the density $f(\lambda)=\left\{
f_{\nu\mu}(\lambda) \right\}_{\nu,\mu = 1 }^{T}$:
$$
\int_{-\pi}^{\pi}\vec b^{\top}(\lambda)f(\lambda)\overline{\vec b(\lambda)}d\lambda=\int_{-\pi}^{\pi}\sum_{\nu,\mu=1}^{T}b_{\nu}(\lambda) f_{\nu\mu}(\lambda) \overline{b_{\mu}(\lambda)}d\lambda<+\infty.
$$
\noindent Denote by $L_{2}^{s-}(f)$ the subspace in $L_{2}(f)$
generated by the functions $e^{ij\lambda}\delta_{\nu},\delta_{\nu}=\left\{
\delta_{\nu\mu} \right\}_{\mu = 1 }^{T}$, $\nu=1,\dots,T,\,j\in {\mathbb
Z}\setminus S$, where $\delta_{\nu\nu}=1,\delta_{\nu\mu}=0$
for $\nu\neq\mu$.

Every linear estimate $\widehat{A_s\vec {\xi}}$ of the functional
$A_{s}\vec {\xi}$ from observations of the sequence $\vec
{\xi}(j)+\vec{\eta}(j)$ at points $j\in {\mathbb Z}
\setminus S$ has the form
\begin{equation}\label{est}
\widehat{{A}_{s}\vec
{\xi}}=\int_{-\pi}^{\pi}\vec h^{\top}(e^{i\lambda})(Z^{\vec \xi}(d\lambda)+Z^{\vec \eta}(d\lambda))=\int_{-\pi}^{\pi}\sum_{\nu=1}^{T}h_{\nu}(e^{i\lambda})(Z_{\nu}^{\vec \xi}(d\lambda)+Z_{\nu}^{\vec \eta}(d\lambda)),
\end{equation}
\noindent where $Z^{\vec \xi}(\Delta)=\left\{ Z_{\nu}^{\vec \xi}(\Delta)
\right\}_{\nu = 1}^{T}$ and $Z^{\vec \eta}(\Delta)=\left\{
Z_{ \nu}^{\vec \eta}(\Delta) \right\}_{\nu = 1 }^{T}$ are orthogonal random
measures of the sequences $\vec {\xi}(j)$ and $\vec {\eta}(j)$, and
$\vec h(e^{i\lambda})=\left\{ h_{\nu}(e^{i\lambda}) \right\}_{\nu = 1
}^{T}$ is the spectral characteristic of the estimate
$\widehat{{A}_{s}\vec {\xi}}$. The function $\vec h(e^{i\lambda})\in
L_{2}^{s-}(f^{\vec \xi}+f^{\vec \eta})$.

The mean square error $\Delta(\vec h;f^{\vec \xi},f^{\vec \eta})$ of the
estimate $\widehat{{A}_{s}\vec {\xi}}$ is calculated by the formula
$$
\Delta(\vec h;f^{\vec \xi},f^{\vec \eta})=E|A_{s}\vec
{\xi}-\widehat{{A}_{s}\vec {\xi}}|^{2}=
$$
\begin{equation}\label{3.55}
=\frac{1}{2\pi}\int_{-\pi}^{\pi}{\left[ {
A_{s}(e^{i\lambda})-\vec h(e^{i\lambda})} \right]}^{\top}f^{\vec
\xi}(\lambda){\overline{\left[ {
A_{s}(e^{i\lambda})-\vec h(e^{i\lambda})} \right]}}
d\lambda+\end{equation}
$$+\frac{1}{2\pi}\int_{-\pi}^{\pi} \vec h^{\top}(e^{i\lambda})f^{\vec
\eta}(\lambda)\overline{\vec h(e^{i\lambda})}d\lambda,
$$
$$
A_{s}(e^{i\lambda})=\sum_{l=0}^{s-1}\sum_{j=M_l+1}^{M_l+N_{l+1}}\vec {a}(j)e^{ij\lambda}.
$$
\noindent The spectral characteristic $\vec h(f^{\vec \xi},f^{\vec
\eta})$ of the optimal linear estimate of $A_{s}\vec {\xi}$
minimizes the mean square error
\begin{equation}\label{3.6}
\Delta(f^{\vec \xi},f^{\vec \eta})=\Delta(\vec h(f^{\vec \xi},f^{\vec
\eta});f^{\vec \xi},f^{\vec \eta})=\mathop {\min }\limits_{\vec h \in
L_{2}^{s-}(f^{\vec \xi}+f^{\vec \eta})} \Delta (\vec h;f^{\vec
\xi},f^{\vec \eta})=\mathop {\min }\limits_{\widehat{{A}_{s}\vec
{\xi}}}E|A_{s}\vec {\xi}-\widehat{{A}_{s}\vec {\xi}}|^{2}.
\end{equation}
With the help of the
Hilbert space projection method proposed by A. N. Kolmogorov \cite{Kolmogorov} we can
find a
solution of the optimization problem (\ref{3.6}). The optimal linear estimate $\hat{A}_{s}\vec {\xi}$ is a projection of the functional $A_s \vec \xi$ on the subspace $H^-[\vec \xi+\vec \eta]=H^-[\xi_\nu(j)+\eta_\nu(j),\, j\in \mathbb{Z}\backslash S, \nu=1,\dots,T]$ of the Hilbert space $H=\{\xi: \textsf{E}\xi=0,\, \textsf{E}|\xi|^2<\infty\}$,
 generated by  values $\xi_\nu(j)+\eta_\nu(j),\, j\in \mathbb{Z}\backslash S, \nu=1,\dots,T$. The projection is characterized by the following conditions\\
 1) $ \widehat{A_s \vec \xi} \in H^-[\vec \xi+\vec \eta],$\\
 2) $A_s \vec \xi-\widehat{A_s \vec \xi}\perp H^-[\vec \xi+\vec \eta].$

Condition 2) gives us the possibility to derive the  formula for the spectral characteristic of the optimal estimate
$$
\vec h^{\top}(f^{\vec \xi},f^{\vec \eta})= \left( A^{\top}_s
(e^{i\lambda} )f^{\vec \xi}(\lambda) - C^{\top}_s (e^{i\lambda
})\right)\, \left[ f^{\vec \xi}(\lambda)+f^{\vec \eta}(\lambda)
\right]^{-1}=
$$
\begin{equation}\label{3.7}
=A_{s}^{\top}(e^{i\lambda})-\left(
{A_{s}^{\top}(e^{i\lambda})f^{\vec
\eta}(\lambda)+C_{s}^{\top}(e^{i\lambda})}\right) \left[ f^{\vec
\xi}(\lambda) +f^{\vec \eta}(\lambda)\right]^{-1},
\end{equation}
\noindent where
$$
C_{s}(e^{i\lambda})=\sum_{l=0}^{s-1} \sum_{k_l=M_l+1}^{M_l+N_{l+1}}\vec
{c}(k_l)e^{ik_l\lambda},$$
$$\vec
{c}(k_l)=\left(
c_{1}(k_l),
\dots,
c_{T}(k_l)
\right)^{\top},$$
$$l=0,\dots,s-1,\, k_l=M_l+1,\dots,M_l+N_{l+1}.
$$

Condition 1) is satisfied when the system of equalities
\begin{equation}\label{syst}
\int_{-\pi}^{\pi}\vec h^{\top}(f^{\vec \xi},f^{\vec
\eta})e^{-ij\lambda}d\lambda=0, \quad j\in S
\end{equation}
holds true.

Denote by $\textbf{D}_s$,
$\textbf{B}_s$ operators that are determined by  $T\rho\times T\rho$, $\rho=N_1+N_2+\dots+N_s$, matrices
 $$
 \textbf{D}_{s}=\begin{pmatrix}
D_{00} & D_{01} &\dots &D_{0,s-1}\\
D_{10} & D_{11} &\dots &D_{1,s-1}\\
\dots&\dots&\dots&\dots\\
D_{s-1,0} & D_{s-1,1} &\dots &D_{s-1,s-1}\\
\end{pmatrix},\,\,\,
\textbf{B}_{s}=\begin{pmatrix}
B_{00} & B_{01} &\dots &B_{0,s-1}\\
B_{10} & B_{11} &\dots &B_{1,s-1}\\
\dots&\dots&\dots&\dots\\
B_{s-1,0} & B_{s-1,1} &\dots &B_{s-1,s-1}\\
\end{pmatrix},
$$
constructed from $TN_{m+1}\times TN_{n+1}$ block-matrices $$ D_{mn}=\left\{D_{mn}(k,j)\right\}_{k=M_{m}+1}^{M_{m}+N_{m+1}} {} _{j=M_{n}+1}^{M_{n}+N_{n+1}},$$ $$B_{mn}=\left\{B_{mn}(k,j)\right\}_{k=M_{m}+1}^{M_{m}+N_{m+1}} {} _{j=M_{n}+1}^{M_{n}+N_{n+1}},$$
\[ m,n=0,\dots,s-1,\]
with elements which are
the Fourier coefficients of  the matrix functions $\left[f^{\vec \xi}(\lambda)(f^{\vec \xi}(\lambda)+f^{\vec
\eta}(\lambda))^{-1}\right]^{\top}$ and $\left[(f^{\vec
\xi}(\lambda)+f^{\vec \eta}(\lambda))^{-1}\right]^{\top}$, correspondingly:
\[
D_{mn}(k,j)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\left[f^{\vec
\xi}(\lambda)(f^{\vec \xi}(\lambda)+f^{\vec
\eta}(\lambda))^{-1}\right]^{\top}e^{i(j-k)\lambda}d\lambda,\]
\[
B_{mn}(k,j)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\left[(f^{\vec
\xi}(\lambda)+f^{\vec
\eta}(\lambda))^{-1}\right]^{\top}e^{i(j-k)\lambda}d\lambda,
\]
\[k=M_{m}+1
,\dots,M_{m}+N_{m+1},\]
\[j=M_{n}+1
,\dots,M_{n}+N_{n+1}.
\]
Making use of the introduced operators, relation (\ref{syst}) can be written in the  form of the equation
\[
\textbf{D}_s \vec a_s=\textbf{B}_s\vec c_s,
\]
\noindent where  $$\vec a_s=\left(\vec a^{\top}(1),\dots,\vec a^{\top}(N_1),\vec a^{\top}(M_1+1)\dots, \vec a^{\top}(M_1+N_2), \dots, \vec a^{\top}(M_{s-1}+1),\dots, \vec a^{\top}(M_{s-1}+N_s)\right)^{\top},$$
$$\vec c_s=\left(\vec c^{\top}(1),\dots,\vec c^{\top}(N_1),\vec c^{\top}(M_1+1)\dots, \vec c^{\top}(M_1+N_2), \dots, \vec c^{\top}(M_{s-1}+1),\dots, \vec c^{\top}(M_{s-1}+N_s)\right)^{\top}$$ are column-vectors.
The unknown coefficients $\vec
{c}(k_l),\,l=0,\dots,s-1,\, k_l=M_l+1,\dots,M_l+N_{l+1}$ are determined from the equation
\begin{equation}\label{c}
\vec c_s=\textbf{B}_s^{-1}\textbf{D}_s \vec a_s,
\end{equation}
where the $k_l$-th component of the vector $\vec c_s$ is calculated by the formula
$$
\vec c(k_l)=\sum _{m=0}^{s-1} \sum_{q=M_{m}+1}^{M_{m}+N_{m+1}}  C_{lm}(k_l,q) \sum _{n=0}^{s-1} \sum_{j=M_{n}+1}^{M_{n}+N_{n+1}} D_{mn}(q,j) \vec a(j),$$
$$ l=0,\dots,s-1,\,k_l=M_{l}+1
,\dots,M_{l}+N_{l+1}.
$$

The operator $\textbf{B}_s^{-1}$ is determined by $T\rho\times T\rho$ matrix
$$\textbf{B}_{s}^{-1}=\begin{pmatrix}
C_{00} & C_{01} &\dots &C_{0,s-1}\\
C_{10} & C_{11} &\dots &C_{1,s-1}\\
\dots&\dots&\dots&\dots\\
C_{s-1,0} & C_{s-1,1} &\dots &C_{s-1,s-1}\\
\end{pmatrix}
$$
that is an inverse matrix for the block-matrix $\textbf{B}_s$. Elements of $\textbf{B}_s^{-1}$ are constructed by dividing $\textbf{B}_s^{-1}$ on  $TN_{m+1}\times TN_{n+1}$ block-matrices  $C_{mn}$ and dividing each  $C_{mn}$  on $T\times T$ matrices  $C_{mn}(k,j),$ $k=M_{m}+1
,\dots,M_{m}+N_{m+1},j=M_{n}+1
,\dots,M_{n}+N_{n+1},$ $m,n=0,\dots,s-1$, in the  such way that
$$C_{mn}=\left\{C_{mn}(k,j)\right\}_{k=M_{m}+1}^{M_{m}+N_{m+1}} {} _{j=M_{n}+1}^{M_{n}+N_{n+1}}.$$

The mean-square error of the optimal estimate $\widehat{ A \vec \xi}$
is calculated by the formula (\ref{3.55}) and is of the form
\begin{equation}\label{3.8}
\Delta(f^{\vec \xi},f^{\vec
\eta})=\langle{\vec{a}_{s},\textbf{R}_{s}\vec{a}_{s}}\rangle+\langle{\vec{c}_{s},\textbf{B}_{s}\vec{c}_{s}}\rangle,
\end{equation}
\noindent where
 $\langle{a,b}\rangle$ denotes the scalar product, $\textbf{R}_{s}$ is the linear operator determined by $T\rho\times T\rho$ matrix composed with $TN_{m+1}\times TN_{n+1}$ block-matrices $$ R_{mn}=\left\{R_{mn}(k,j)\right\}_{k=M_{m}+1}^{M_{m}+N_{m+1}} {} _{j=M_{n}+1}^{M_{n}+N_{n+1}},$$
 \[m,n=0,\dots,s-1,\]
 with elements
$$
R_{mn}(k,j)=\frac{1}{2\pi}\int_{-\pi}^{\pi}{\left[ { f^{\vec
\xi}(\lambda)(f^{\vec \xi}(\lambda)+f^{\vec
\eta}(\lambda))^{-1}f^{\vec \eta}(\lambda)}
\right]}^{\top}e^{i(j-k)\lambda} d\lambda,
$$
\[k=M_{m}+1
,\dots,M_{m}+N_{m+1},\]
\[j=M_{n}+1
,\dots,M_{n}+N_{n+1}.
\]
\\
\noindent See \cite{MoklyachukMas2006a} for more details.
\\

The following statement holds true.

\begin{theorem}
\label{theorem3.1}
Let $\vec
{\xi}(j)=\left\{ \xi_{\nu}(j) \right\}_{\nu = 1 }^{T}$ and $\vec
{\eta}(j)=\left\{ \eta_{\nu}(j) \right\}_{\nu = 1}^{T}$ be
uncorrelated T-variate stationary stochastic sequences with the
spectral density matrices $f^{\vec \xi}(\lambda)=\left\{ f^{\vec
\xi}_{\nu\mu}(\lambda) \right\}_{\nu,\mu = 1 }^{T}$ and $f^{\vec
\eta}(\lambda)=\left\{ f^{\vec \eta}_{\nu\mu}(\lambda) \right\}_{\nu,\mu = 1
}^{T}$, respectively. Assume that the matrices $f^{\vec \xi}(\lambda)$
and $f^{\vec \eta}(\lambda)$ satisfy the minimality condition (\ref{3.5}).
The spectral characteristic $\vec h(f^{\vec \xi},f^{\vec \eta})$ and
the mean square error $\Delta(f^{\vec \xi},f^{\vec \eta})$ of the
optimal linear estimate of the functional $A_{s}\vec {\xi}$
based on observations of the sequence $\vec
{\xi}(j)+\vec{\eta}(j)$ at points  $j\in {\mathbb Z}
\setminus S$, are calculated by formulas (\ref{3.7}) and
(\ref{3.8}).
\end{theorem}

In the case of observations  without noise we have the following corollary.

\begin{corollary}
\label{cor3.1}
Let $\vec
{\xi}(j)=\left\{ \xi_{\nu}(j) \right\}_{\nu =1 }^{T}$ be a T-variate
stationary stochastic sequence with the spectral density matrix
$f^{\vec \xi}(\lambda)=\left\{ f^{\vec \xi}_{\nu\mu}(\lambda)
\right\}_{\nu,\mu = 1 }^{T}$, which satisfies the minimality
condition
\begin{equation}\label{3.9}
\int_{-\pi}^{\pi}{Tr{\left[ {(f^{\vec \xi}(\lambda))^{-1}} \right]}}
d\lambda <+{\infty}.\end{equation}
The spectral characteristic $\vec h(f^{\vec
\xi})$ and the mean square error $\Delta(f^{\vec \xi})$ of the
optimal linear estimate of the functional $A_{s}\vec {\xi}$
based on observations of the sequence $\vec {\xi}(j)$ at points  $j\in
{\mathbb Z} \setminus S$, are calculated by formulas
\begin{equation}\label{3.10}
\vec h^{\top}(f^{\vec
\xi})=A_{s}^{\top}(e^{i\lambda})-C_{s}^{\top}(e^{i\lambda}){\left[
{f^{\vec \xi}(\lambda)} \right]}^{-1},\end{equation}
\begin{equation}\label{3.11}
\Delta(f^{\vec
\xi})=\langle{\vec{c}_{s},\vec{a}_{s}}\rangle,\end{equation}
where $$\vec a_s=\left(\vec a^{\top}(1),\dots,\vec a^{\top}(N_1),\vec a^{\top}(M_1+1)\dots, \vec a^{\top}(M_1+N_2), \dots, \vec a^{\top}(M_{s-1}+1),\dots, \vec a^{\top}(M_{s-1}+N_s)\right)^{\top},$$
$$\vec c_s=\left(\vec c^{\top}(1),\dots,\vec c^{\top}(N_1),\vec c^{\top}(M_1+1)\dots, \vec c^{\top}(M_1+N_2), \dots, \vec c^{\top}(M_{s-1}+1),\dots, \vec c^{\top}(M_{s-1}+N_s)\right)^{\top}$$ are column-vectors and $\vec{c}_{s}=\textbf{B}_{s}^{-1}\vec{a}_{s}$. $\textbf{B}_{s}$ is a $T\rho\times T\rho$ matrix composed with $TN_{m}\times TN_n$ block-matrices $ B_{mn}=\left\{B_{mn}(k,j)\right\}_{k=M_{m-1}+1}^{M_{m-1}+N_m} {} _{j=M_{n-1}+1}^{M_{n-1}+N_n}$:
$$
B_{mn}(k,j)=\frac{1}{2\pi}\int_{-\pi}^{\pi}{\left[ { (f^{\vec
\xi}(\lambda))^{-1}} \right]}^{\top}e^{i(j-k)\lambda}
d\lambda,
$$
\[m,n=1,\dots,s,\]
\[k=M_{m-1}+1
,\dots,M_{m-1}+N_m,\]
\[j=M_{n-1}+1
,\dots,M_{n-1}+N_n.
\]
The $k_l$-th component of the vector $\vec c_s$  is calculated by the formula
$$
\vec c(k_l)=\sum _{m=0}^{s-1} \sum_{q=M_{m}+1}^{M_{m}+N_{m+1}} C_{lm}(k_l,q)  \vec a(q),$$
$$ l=0,\dots,s-1,\,k_l=M_{l}
,\dots,M_{l}+N_{l+1}.
$$
The operator $\textbf{B}_s^{-1}$ is determined by $T\rho\times T\rho$ matrix that is the inverse matrix to the block-matrix $\textbf{B}_s$. Elements of $\textbf{B}_s^{-1}$ are obtained by dividing $\textbf{B}_s^{-1}$ on  $TN_{m+1}\times TN_{n+1}$ block-matrices  $C_{mn}$ and dividing each of  $C_{mn}$  on $T\times T$ matrices  $C_{mn}(k,j),$ $k=M_{m}+1
,\dots,M_{m}+N_{m+1},j=M_{n}+1
,\dots,M_{n}+N_{n+1},$ $m,n=0,\dots,s-1$, in the  such way that
$$C_{mn}=\left\{C_{mn}(k,j)\right\}_{k=M_{m}+1}^{M_{m}+N_{m+1}} {} _{j=M_{n}+1}^{M_{n}+N_{n+1}}.$$
\end{corollary}

\begin{remark} Let $s=1,\,N_1=N$. Then
$$
A_s\vec \xi=A_N\vec \xi=\sum_{j=1}
^N \vec a^{\top}(j) \vec \xi(j).$$
The spectral characteristic $\vec h(f^{\vec
\xi},f^{\vec
\eta})$ and the mean square error $\Delta(f^{\vec \xi},f^{\vec
\eta})$ of the
optimal linear estimate of the functional $A_{N}\vec {\xi}$
based on observations of the sequence $\vec {\xi}(j)$ at points $j\in
{\mathbb Z} \setminus \{1,\dots,N\}$ with the noise  $\vec \eta (j)$  are calculated by formulas
$$
h^{\top}(f^{\vec \xi},f^{\vec \eta})= \left( A^{\top}_N
(e^{i\lambda} )f^{\vec \xi}(\lambda) - C^{\top}_N (e^{i\lambda
})\right)\, \left[ f^{\vec \xi}(\lambda)+f^{\vec \eta}(\lambda)
\right]^{-1}=
$$
$$
=A_{N}^{\top}(e^{i\lambda})-\left(
{A_{N}^{\top}(e^{i\lambda})f^{\vec
\eta}(\lambda)+C_{N}^{\top}(e^{i\lambda})}\right) \left[ f^{\vec
\xi}(\lambda) +f^{\vec \eta}(\lambda)\right]^{-1},
$$
\[
\Delta(f^{\vec \xi},f^{\vec
\eta})=\langle{\vec{a}_{N},R_{N}\vec{a}_{N}}\rangle+\langle{\vec{c}_{N},B_{N}\vec{c}_{N}}\rangle.
\]
The spectral characteristic $\vec h(f^{\vec\xi})$ and the mean square error $\Delta(f^{\vec \xi})$ of the
optimal linear estimate of the functional $A_{N}\vec {\xi}$
based on observations of the sequence $\vec {\xi}(j)$ at points $j\in
{\mathbb Z} \setminus \{1,\dots,N\}$ without noise  $\vec \eta (j)$  are calculated by formulas
$$
h^{\top}(f^{\vec
\xi})=A_{N}^{\top}(e^{i\lambda})-C_{N}^{\top}(e^{i\lambda}){\left[
{f^{\vec \xi}(\lambda)} \right]}^{-1},
$$
$$
\Delta(f^{\vec \xi})=\langle{\vec{c}_{N},\vec{a}_{N}}\rangle.
$$
For more details see \cite{Dubovetska0}, \cite{MoklyachukD2016}.
\end{remark}

\begin{example}
Let  $\vec \xi(n)=\begin{pmatrix}
\xi_{1}(n)\\
\xi_{2}(n)
\end{pmatrix}$
be a 2-variate stationary stochastic sequence.
Let $\xi_{1}(n)=\theta(n)$ be a univariate stationary sequence with
the spectral density function $f(\lambda)=\frac{1}{|1-ae^{-i\lambda}|^2},\, |a|<1$, and $\xi_{2}(n)=\theta(n)+\gamma(n)$, where $\gamma(n)$ is an uncorrelated with $\theta(n)$
univariate stationary sequence with the spectral density function
$g(\lambda)=\frac{1}{|1-be^{-i\lambda}|^{2}}, \, |b|<1$.
Consider the problem of estimation of the functional
$$
A_{2}\vec \xi=\vec \xi(1)-\vec \xi(3)=(1,1)
\begin{pmatrix}
\xi_{1}(1)\\
\xi_{2}(1)
\end{pmatrix}+(-1,-1)\begin{pmatrix}
\xi_{1}(3)\\
\xi_{2}(3)
\end{pmatrix}
$$
based on observations of $\vec \xi(n),n\in {\mathbb Z}
\setminus\{{1,3}\}$. Here $\vec a(1)=(1,1),\vec a(3)=(-1,-1)$.

In this case the spectral density matrix of $\vec{\xi}(n)$ is
$$
f^{\vec
\xi}(\lambda)=\begin{pmatrix} f(\lambda) & f(\lambda)\\
f(\lambda) & f(\lambda)+g(\lambda)
\end{pmatrix}
$$
and ${ [{f^{\vec \xi}(\lambda)}]}^{-1}$ satisfies the minimality
condition (\ref{3.9}). The matrix $\textbf{B}_{2}$ and its inverse $\textbf{B}_{2}^{-1}$,
the vector of unknown coefficients $\vec{c}_{2}$ are of the form
$$
\textbf{B}_{2}=\begin{pmatrix}
2+a^2+b^2 & -1-b^2 & 0 & 0\\
-1-b^2 & 1+b^2 & 0 & 0\\
0 & 0 & 2+a^2+b^2 & -1-b^2\\
0 & 0 & -1-b^2 & 1+b^2
\end{pmatrix},$$
$$\textbf{B}_{2}^{-1}=\begin{pmatrix}
\frac{1}{1+a^2} & \frac{1}{1+a^2} & 0 & 0\\
\frac{1}{1+a^2} & \frac{2+a^2+b^2}{(1+b^2)(1+a^2)} & 0 & 0\\
0 & 0 & \frac{1}{1+a^2} & \frac{1}{1+a^2}\\
0 & 0 & \frac{1}{1+a^2} & \frac{2+a^2+b^2}{(1+b^2)(1+a^2)}
\end{pmatrix},$$
$$\vec{c}_{2}=\left(
\frac{2}{1+a^2},
\frac{3+a^2+2b^2}{(1+b^2)(1+a^2)},
-\frac{2}{1+a^2},
-\frac{3+a^2+2b^2}{(1+b^2)(1+a^2)}
\right)^{\top}.
$$
The spectral characteristic can be calculated by (\ref{3.10})
$$
\vec h^{\top}(f^{\vec
\xi})=\left(\left(\frac{2a}{1+a^2}-\frac{b}{1+b^2}\right)+\left(\frac{b}{1+b^2}\right)e^{4i\lambda},\right.$$
$$\left.\frac{b}{1+b^2}-\frac{b}{1+b^2}e^{4i\lambda} \right).
$$
Then the optimal linear estimate of $A_{2}\vec \xi$ determined by (\ref{est}) is of the
form
$$
\widehat{A_2\vec \xi}=\left(\frac{2a}{1+a^2}-\frac{b}{1+b^2}\right)\xi_1(0)+\frac{b}{1+b^2}\xi_1(4)+\frac{b}{1+b^2}\xi_2(0)-\frac{b}{1+b^2}\xi_2(4)).
$$
The mean square error of $\hat{A}_{2}\zeta$ determined by (\ref{3.11}) is
$$\Delta(f^{\vec \xi})=\frac{8}{1+a^2}+\frac{2}{1+b^2}. $$
\end{example}

\section{Interpolation of T-PC stochastic sequences}

Let $\zeta(j)$ and $\theta(j)$ be uncorrelated T-PC stochastic
sequences. Consider the problem of optimal linear estimation  of the
functional $$A_s{\zeta}=\sum_{l=0}^{s-1}\sum_{j=M_l+1}^{M_l+N_{l+1}}{a}(j){\zeta}(j),$$
 $$
 M_l=\sum_{k=0}^l(N_k+K_k),\,\,N_0=K_0=0,$$
 that depends on the unobserved values of T-PC stochastic
sequence $\zeta(j)$, based on
observations of the sequence $\zeta(j)+\theta(j)$ at points $j\in {\mathbb
Z} \setminus S,$ where $S=\bigcup _{l=0}^{s-1}\{M_l+1,\dots,M_l+N_{l+1}\}$.

Using the  Gladyshev relation (\ref{2.2}) of PC and multivariate stationary
sequences  the problem of estimation of the functional
$A_{s}\zeta$ may be reduced to the problem of estimation of the
functional $A_{s}\vec {\xi}$ since
$$
A_{s}\zeta=\sum_{l=0}^{s-1}\sum_{j=M_l+1}^{M_l+N_{l+1}}{a}(j){\zeta}(j)=\sum_{l=0}^{s-1}\sum_{j=M_l+1}^{M_l+N_{l+1}}{a}(j)\sum_{\nu=1}^{T}e^{2\pi
ij\nu/T}\xi_{\nu}(j)=
$$
$$
=\sum_{l=0}^{s-1}\sum_{j=M_l+1}^{M_l+N_{l+1}}\sum_{\nu=1}^{T}{a}(j)e^{2\pi
ij\nu/T}\xi_{\nu}(j)=\sum_{l=0}^{s-1}\sum_{j=M_l+1}^{M_l+N_{l+1}}\vec {a}^{\top}(j)\vec
{\xi}(j)=A_{s}\vec {\xi},
$$
\noindent where $$\vec {a}^{\top}(j)=\left(
a_{1}(j),
\dots,
a_{T}(j)
\right),\,a_{\nu}(j)={a}(j)e^{2\pi
ij\nu/T},\,\nu=1,\dots,T,$$ $\vec {\xi}(j)=\left\{ \xi_{\nu}(j)
\right\}_{\nu = 1}^{T}$ is a T-variate stationary stochastic
sequence that generates the PC sequence  $\zeta(j)$.

For the interpolation problem for PC sequences the following results
hold true.

\begin{theorem}
\label{theorem3.2}
Let $\zeta(j)$ and
$\theta(j)$ be uncorrelated T-PC stochastic sequences. Then the
optimal linear estimate of the functional $A_{s}\zeta$ based on
observations of the sequence $\zeta(j)+\theta(j)$ at points $j\in {\mathbb Z}
\setminus S$, is given by the formula
\smallskip
\begin{equation}\label{est1}
\widehat{{A}_{s}\zeta}=\int_{-\pi}^{\pi}\vec h^{\top}
(f^{\vec \xi},f^{\vec \eta})
(Z^{\xi}(d\lambda)+Z^{\eta}(d\lambda))=\int_{-\pi}^{\pi}\sum_{\nu=1}^{T}h_{\nu}(f^{\vec
\xi},f^{\vec \eta})(Z_{\nu}^{\xi}(d\lambda)+Z_{\nu}^{\eta}(d\lambda)),
\end{equation}
where $\vec{\xi}(j)$ and $\vec{\eta}(j)$ are
generating sequences of the sequences $\zeta(j)$ and $\theta(j)$, correspondingly.
The spectral characteristic $\vec h(f^{\vec \xi},f^{\vec \eta})$ and the
mean square error $\Delta(f^{\vec \xi},f^{\vec \eta})$ of
$\widehat{{A}_{s}\zeta}$ are calculated by formulas (\ref{3.7}) and (\ref{3.8}), where
$\vec
{a}(j)=(a_{1}(j),\dots,a_{T}(j))^{\top},a_{\nu}(j)={a}(j)e^{2\pi
ij\nu/T},\nu=1,\dots,T$.
\end{theorem}

\begin{corollary}
\label{cor3.2}
The optimal linear estimate $ \widehat{\zeta(1)}$ of the unknown value $\zeta(1)$, based on observations of the sequence $\zeta(j)+\theta(j)$ at points  $j\in {\mathbb Z}
\setminus S$ is defined by the formula (\ref{est1}). The spectral characteristic $\vec h(f^{\vec \xi},f^{\vec \eta})$ and the
mean square error $\Delta(f^{\vec \xi},f^{\vec \eta})$  of the
optimal linear estimate $\widehat{\zeta(1)}$ are calculated by formulas (\ref{3.7}) and (\ref{3.8}), where the unknown coefficients $\vec c(k_l), \, l=0,\dots,s-1,\,k_l=M_{l}+1
,\dots,M_{l}+N_{l+1}$ are defined by formulas
$$
\vec c(k_l)=\sum _{m=0}^{s-1} \sum_{q=M_{m}+1}^{M_{m}+N_{m+1}}  C_{lm}(k_l,q) D_{00}(q,1)\vec a(1),$$
where elements $C_{lm}(k_l,q),\,l,m=0,\dots,s-1,\,k_l=M_{l}+1
,\dots,M_{l}+N_{l+1}, q=M_{m}+1,\dots,M_{m}+N_{m+1}$ are determined by the same way as in Theorem \ref{theorem3.1}.
\end{corollary}

In the case  of observations without noise we have the following corollary.

\begin{corollary}
\label{cor3.3}
 Let $\zeta(j)$ be a T-PC
stochastic sequence. Then the optimal linear estimate of the
functional $A_{s}\zeta$ based on observations of the sequence
$\zeta(j)$ at points $j\in {\mathbb Z} \setminus S$, is given
by
\smallskip
\begin{equation}\label{3.12}
\widehat{{A}_{s}\zeta}=\int_{-\pi}^{\pi}\vec h^{\top}(f^{\vec
\xi})Z^{\xi}(d\lambda)=\int_{-\pi}^{\pi}\sum_{\nu=1}^{T}h_{\nu}(f^{\vec
\xi})Z_{\nu}^{\xi}(d\lambda),
\end{equation}
where $\vec{\xi}(j)$ is generating sequence of
$\zeta(j)$. The spectral characteristic $\vec h(f^{\vec \xi})$ and the
mean square error $\Delta(f^{\vec \xi})$ of $\widehat{{A}_{s}\zeta}$ are
calculated by formulas (\ref{3.10}) and (\ref{3.11}), where  $\vec
{a}(j)=(a_{1}(j),\dots,$ $a_{T}(j))^{\top}, a_{\nu}(j)={a}(j)e^{2\pi
ij\nu/T},\nu=1,\dots,T$.
\end{corollary}

\begin{corollary}
\label{cor3.4}
The optimal linear estimate $ \widehat{\zeta(1)}$ of the unknown value $\zeta(1)$, based on observations of the sequence $\zeta(j)$ at points $j\in {\mathbb Z}
\setminus S$ is defined by the formula (\ref{3.12}). The spectral characteristic $\vec h(f^{\vec \xi})$ and the
mean square error $\Delta(f^{\vec \xi})$  of the
optimal linear estimate $ \widehat{\zeta(1)}$ are calculated by formulas (\ref{3.10}) and (\ref{3.11}), where the unknown coefficients $\vec c(k_l), \, l=0,\dots,s-1,\,k_l=M_{l}+1
,\dots,M_{l}+N_{l+1}$ are defined by formulas
$$
\vec c(k_l)= C_{00}(k_l,1)\vec a(1),$$
where elements $C_{00}(k_l,1),\, l=0,\dots,s-1,\,k_l=M_{l}+1
,\dots,M_{l}+N_{l+1}$ are determined by the same way as in Corollary \ref{cor3.1}.
\end{corollary}

\section{Interpolation of T-PC stochastic sequences with special
 sets of missed observations}

Consider the problem of optimal estimation for the linear functional
$$A_s{\zeta}=\sum_{l=0}^{s-1}\sum_{j=M_l+1}^{M_l+N_{l+1}}{a}(j){\zeta}(j),\,\,M_l=\sum_{k=0}^l(N_k+K_k),\,\,N_0=K_0=0,$$
which depends on the
unobserved values of $T$-PC sequence
${\zeta}(j)$ from observations of the sequence
${\zeta}(j)+{\theta}(j)$ at points $j\in{\mathbb
Z}\setminus S$,
where the number of missed observations at each of the intervals is a multiple of the period $T$, what means that
 $$K_1=T \cdot K_1^T, K_2=T\cdot K_2^T,\dots, K_{s-1}=T \cdot K_{s-1}^T,$$
and the number of observations at each of the intervals is a multiple of $T$ $$N_1=T \cdot N_1^T, N_2=T \cdot N_2^T,\dots, N_{s}=T \cdot N_s^T,$$
and coefficients $a(j), j\in S$ are of the form
\begin{equation}\label{aj}
a(j)=a\left(\left(j-\left[\frac{j}{T}\right]T\right)+\left[\frac{j}{T}\right]T\right)=a(\nu+\tilde{j}T)=a(\tilde{j})e^{2\pi i\tilde{j}\nu/T},\,
\end{equation}
$$
\nu=1,\dots,T,\, \tilde{j}\in \tilde{S},$$
$$
\tilde{S}=\bigcup_{l=0}^{s-1}\left\{M_l^T,\dots,M_l^T+N_{l+1}^T-1\right\},
$$
where  $\nu=T$ and $\tilde{j}=\lambda-1$, if $j=T\cdot \lambda, \,\lambda \in \mathbb{Z},$ or
$$
 a(j)=a(T\cdot \lambda)=a(T+(\lambda-1)T)=a(\lambda-1)e^{2\pi i(\lambda-1)T/T},
 $$
and $M_l=T\cdot M_l^T,\, l=0,\dots,s-1.$

Using Proposition 2.3  the linear functional $A_s \zeta$ can be written as follows
$$A_s{\zeta}=\sum_{l=0}^{s-1}\sum_{j=M_l+1}^{M_l+N_{l+1}}{a}(j){\zeta}(j)=
$$
$$=\sum_{l=0}^{s-1}\sum_{\tilde{j}={M_l^T}}^{M_l^T+N_{l+1}^T-1}\sum_{\nu=1}^{T}{a}(\nu+\tilde{j}T){\zeta}(\nu+\tilde{j}T)=\sum_{l=0}^{s-1}\sum_{\tilde{j}={M_l^T}}^{M_l^T+N_{l+1}^T-1}\sum_{\nu=1}^{T}{a}_\nu(\tilde{j}){\zeta}_\nu(\tilde{j})=
$$
\begin{equation}\label{veczeta}
=\sum_{l=0}^{s-1}\sum_{\tilde{j}={M_l^T}}^{M_l^T+N_{l+1}^T-1}{\vec a}^{\top}(\tilde{j}){\vec \zeta}(\tilde{j})=A_s{\vec\zeta},
\end{equation}
where $$\vec a(\tilde{j})=(a_{1}(\tilde{j}),\dots,a_{T}(\tilde{j}))^{\top},\,\, a_{\nu}(\tilde{j})=a(\nu+\tilde{j}T)={a}(\tilde{j})e^{2\pi
i\tilde{j}\nu/T},$$
$$
\vec \zeta(\tilde{j})=(\zeta_{1}(\tilde{j}),\dots,\zeta_{T}(\tilde{j}))^{\top},\,\, \zeta_{\nu}(\tilde{j})=\zeta(\nu+\tilde{j}T),$$
$$\nu=1,\dots,T,\, \tilde{j}\in \tilde{S}
$$
and $\vec \zeta(\tilde{j}),\,\tilde{j}\in \tilde{S}$ is $T$-variate stationary sequence, obtained by the $T$-blocking (\ref{block}) of univariate $T$-PC sequence $\zeta(j), \, j \in S$.

Let  $f^{\vec \zeta}(\lambda)$ and $f^{\vec
\theta}(\lambda)$ be the
spectral density matrices of T-variate stationary sequences $\vec \zeta(j)$
and $\vec \theta(j)$, obtained from the $T$-blocking (\ref{block}) of univariate $T$-PC sequences
$\zeta(j)$ and $\theta(j)$, respectively.
\\

Taking into account the definition of the functional $A_s{\vec\zeta}$ and Theorem \ref{theorem3.1} we can verify that the following statements hold true.

\begin{theorem}
\label{theorem3.3}
Let $\zeta(j)$ and
$\theta(j)$ be uncorrelated T-PC stochastic sequences with the
spectral density matrices $f^{\vec \zeta}(\lambda)$ and $f^{\vec
\theta}(\lambda)$ of T-variate stationary sequences $\vec \zeta(j)$
and $\vec \theta(j)$, respectively. Assume that  $f^{\vec
\zeta}(\lambda)$ and $f^{\vec \theta}(\lambda)$ satisfy the
minimality condition (\ref{3.5}). Then the optimal linear estimate of
$A_s\vec {\zeta}$ based on observations of $\vec\zeta(j)+\vec\theta(j)$ at points  $j\in
{\mathbb Z} \setminus \tilde{S}$, is given by
$$
\widehat{{A}_{s}\vec \zeta}=\int_{-\pi}^{\pi}\vec h^{\top}(f^{\vec \zeta},f^{\vec
\theta})(Z^{\vec\zeta}(d\lambda)+Z^{\vec\theta}(d\lambda))=\int_{-\pi}^{\pi}\sum_{\nu=1}^{T}h_{\nu}(f^{\vec
\zeta},f^{\vec
\theta})(Z_{\nu}^{\vec \zeta}(d\lambda)+Z_{\nu}^{\vec \theta}(d\lambda)),
$$
where $Z^{\vec \zeta}(\Delta)=\left\{ Z_{\nu}^{\vec \zeta}(\Delta)
\right\}_{\nu = 1}^{T}$ and $Z^{\vec \theta}(\Delta)=\left\{
Z_{ \nu}^{\vec \theta}(\Delta) \right\}_{\nu = 1 }^{T}$ are orthogonal random
measures of the sequences $\vec \zeta(\tilde{j})$ and $\vec \theta(\tilde{j})$. The spectral
characteristic $\vec h(f^{\vec \zeta},f^{\vec \theta})$ and the mean
square error $\Delta(f^{\vec \zeta},f^{\vec \theta})$ of $\widehat{{A}_s \vec
\zeta}$ are calculated by formulas
$$
\vec h^{\top}(f^{\vec \zeta},f^{\vec \theta})= \left( \tilde{A}^{\top}_s
(e^{i\lambda} )f^{\vec \zeta}(\lambda)
-\tilde{C}^{\top}_s (e^{i\lambda })\right)\left[ f^{\vec \zeta}(\lambda)+f^{\vec \theta}(\lambda)
\right]^{-1}
$$
\begin{equation}\label{3.13}
=\tilde{A}_{s}^{\top}(e^{i\lambda})-\left( \tilde{A}_{s}^{\top}(e^{i\lambda})f^{\vec
\theta}(\lambda)+
\tilde{C}_{s}^{\top}(e^{i\lambda})\right) \left[ f^{\vec
\zeta}(\lambda)+f^{\vec \theta}(\lambda)
\right]^{-1},
\end{equation}
\begin{equation}\label{3.14}
\Delta(f^{\vec \zeta},f^{\vec
\theta})=\langle{\vec{a}^{\zeta}_{s},\textbf{R}^\zeta_{s}\vec{a}^{ \zeta}_{s}}\rangle+\langle{\vec{c}^\zeta_{s},\textbf{B}^\zeta_{s}\vec{c}^\zeta_{s}}\rangle,
\end{equation}
where
$$\vec a^{ \zeta}_s=\left(\vec a^{\top}(0),\dots,\vec a^{\top}(N_1^T-1), \dots, \vec a^{\top}(M_{s-1}^T),\dots, \vec a^{\top}(M_{s-1}^T+N_s^T-1)\right)^{\top},$$
$$\vec c^{ \zeta}_s=\left(\vec c^{\top}(0),\dots,\vec c^{\top}(N_1^T-1), \dots, \vec c^{\top}(M_{s-1}^T),\dots, \vec c^{\top}(M_{s-1}^T+N_s^T-1)\right)^{\top},$$
$\tilde{A}_s(e^{i\lambda})=\sum_{\tilde{j}\in \tilde{S}}\vec a(\tilde{j})e^{i\tilde{j}\lambda},$ $\tilde{C}_s(e^{i\lambda})=\sum_{\tilde{j}\in \tilde{S}}\vec c(\tilde{j})e^{i\tilde{j}\lambda},$
the unknown coefficients $\vec c(\tilde{j}),\, \tilde{j}\in \tilde{S}$ are determined from the relation $$\vec{c}^\zeta_{s}=(\textbf{B}^{\zeta}_{s})^{-1}\textbf{D}^\zeta_{s}\vec{a}^{\zeta }_{s},$$
operators $\textbf{B}^\zeta_{s}, \textbf{D}^\zeta_{s}, \textbf{R}_{s}^\zeta$ are determined by $\rho\times \rho$ matrices,
constructed from $T\cdot N_{m+1}^T\times T\cdot N_{n+1}^T$ block-matrices $$B^\zeta_{mn}=\left\{B^\zeta_{mn}(k,j)\right\}_{k={M_{m}^T}}^{M_{m}^T+N_{m+1}^T-1} {} _{j=M_{n}^T}^{M_{n}^T+N_{n+1}^T-1},$$    $$ D^\zeta_{mn}=\left\{D^\zeta_{mn}(k,j)\right\}_{k={M_{m}^T}}^{M_{m}^T+N_{m+1}^T-1} {} _{j=M_{n}^T}^{M_{n}^T+N_{n+1}^T-1},$$ $$R^\zeta_{mn}=\left\{R^\zeta_{mn}(k,j)\right\}_{k={M_{m}^T}}^{M_{m}^T+N_{m+1}^T-1} {} _{j=M_{n}^T}^{M_{n}^T+N_{n+1}^T-1},\, m,n=0,\dots,s-1$$ with elements:
\[
B^\zeta_{mn}(k,j)=\frac{1}{2\pi}\int_{-\pi}^{\pi}{\left[
{( f^{\vec \zeta}(\lambda)+f^{\vec \theta}(\lambda))^{-1}}
\right]}^{\top}e^{i(j-k)\lambda} d\lambda,
\]
\[
D^\zeta_{mn}(k,j)=\frac{1}{2\pi}\int_{-\pi}^{\pi}{\left[
{ f^{\vec \zeta}(\lambda)(f^{\vec \zeta}(\lambda)+f^{\vec
\theta}(\lambda))^{-1}}
\right]}^{\top}e^{i(j-k)\lambda} d\lambda,
\]
\[
R^\zeta_{mn}(k,j)=\frac{1}{
2\pi}\int_{-\pi}^{\pi}{\left[ { f^{\vec
\zeta}(\lambda)(f^{\vec \zeta}(\lambda)+f^{\vec
\theta}(\lambda))^{-1}f^{\vec \theta}(\lambda)}
\right]}^{\top}e^{i(j-k)\lambda} d\lambda,
\]
\[m,n=0,\dots,s-1,\]
\[k={M_{m}^T}
,\dots,M_{m}^T+N_{m+1}^T-1,\]
\[j={M_{n}^T}
,\dots,M_{n}^T+N_{n+1}^T-1.
\]
\end{theorem}

In the case of observations  without noise we have the following corollary.

\begin{corollary}
\label{cor3.5}
Let $\zeta(j)$ be a
T-PC stochastic sequence with the spectral density matrix $f^{\vec
\zeta}(\lambda)$ of T-variate stationary sequence $\vec \zeta(j)$.
Assume that $f^{\vec \zeta}(\lambda)$ satisfies the minimality
condition (\ref{3.9}). Then the optimal linear estimate of $A_s{\vec \zeta}$
based on observations of $\vec \zeta(j)$ at points   $j\in {\mathbb Z}
\setminus \tilde{S}$, is given by
$$
\widehat{{A}_{s}\vec\zeta}=\int_{-\pi}^{\pi}\vec h^{\top}(f^{\vec
\zeta})Z^{\vec\zeta}(d\lambda)=\int_{-\pi}^{\pi}\sum_{\nu=1}^{T}h_{\nu}(f^{\vec
\zeta})Z_{\nu}^{\vec\zeta}(d\lambda).
$$
The spectral characteristic $\vec h(f^{\vec \zeta})$ and the
mean square error $\Delta(f^{\vec \zeta})$ of $ \widehat{{A}_s \vec\zeta}$ are
calculated by formulas
\begin{equation}\label{3.15}
\vec h^{\top}(f^{\vec \zeta})= \tilde{A}^{\top}_s (e^{i\lambda}) -
\tilde{C}^{\top}_s (e^{i\lambda }) \left[ f^{\vec
\zeta}(\lambda) \right]^{-1},
\end{equation}
\begin{equation}\label{3.16}
\Delta(f^{\vec
\zeta})=\langle{\vec{c}^\zeta_{s},\vec{a}^{\zeta}_{s}}\rangle,
\end{equation}
unknown coefficients $\vec c(\tilde{j}),\, \tilde{j}\in \tilde{S}$ are determined from the relation  $$\vec{c}^\zeta_{s}=(\textbf{B}^{\zeta}_{s})^{-1}\vec{a}_{s},$$ operator $\textbf{B}^\zeta_{s}$
is a matrix composed with $\rho\times \rho$ matrix,
constructed from $T\cdot N_{m+1}^T\times T\cdot N_{n+1}^T$ block-matrices $B^\zeta_{mn}=\left\{B^\zeta_{mn}(k,j)\right\}_{k=M_{m}^T}^{M_{m}^T+N_{m+1}^T-1} {} _{j=M_{n}}^{M_{n}+N_{n+1}-1}$ with elements:
$$
B^\zeta_{mn}(j,k)=\frac{1}{2\pi}\int_{-\pi}^{\pi}{\left[
{(f^{\vec \zeta}(\lambda))^{-1}}
\right]}^{\top}e^{i(k-j)\lambda} d\lambda,
$$
\[m,n=0,\dots,s-1,\]
\[k={M_{m}^T}
,\dots,M_{m}^T+N_{m+1}^T-1,\]
\[j={M_{n}^T}
,\dots,M_{n}^T+N_{n+1}^T-1.
\]
\end{corollary}

\begin{example}
Let $\zeta(n),\,n\in \mathbb{Z}$ be a 2-PC
stochastic sequence such that  $\zeta(2n+1)=\eta(n)$ is a univariate
stationary Ornstein-Uhlenbeck sequence with the spectral density
 $f(\lambda)=\frac{1}{ |2+e^{i\lambda}|^{2}}$ and
$\zeta(2n)=\gamma(n)$ is an uncorrelated with $\eta(n)$ univariate
stationary Ornstein-Uhlenbeck sequence with the spectral density
 $g(\lambda)=\frac{1}{|3-e^{i\lambda}|^{2}}$.

Consider the problem of estimation of the functional $$A_{1}\zeta=\zeta(1)+\zeta(2)-\zeta(3)+\zeta(4).$$

Here $S=\{1,2,3,4\}$ and $N_1=4$ is a multiple of $T=2$. Rewrite $A_{1}\zeta$ in the form (\ref{veczeta})
$$
A_{1}\zeta=e^{2\pi i1\cdot0/2}\zeta(1+0\cdot 2)+e^{2\pi  i2\cdot 0/2}\zeta(2+0\cdot2)+e^{2\pi i1\cdot1/2}\zeta(1+1\cdot2)+e^{2\pi i2\cdot2/2}\zeta(2+1\cdot2)=
$$
$$
=\vec a^{\top}(0)\vec\zeta(0)+\vec a^{\top}(1)\vec \zeta(1)=A_{1}\vec\zeta,
$$
where $\vec a(\tilde{j})=(a(\tilde{j})e^{2\pi i1\cdot\tilde{j}/2},a(\tilde{j})e^{2\pi i2\cdot\tilde{j}/2})^{\top},\,$$a(0)=1,\,a(1)=1,\,$ $\vec \zeta(\tilde{j})=(\zeta(1+\tilde{j}\cdot2),\zeta(2+\tilde{j}\cdot2))^{\top},$ $\tilde{j }\in\tilde{S}=\{0,1\}.$
In this case the spectral density matrix of $\vec{\zeta}(n)$ is of the form
$$
f^{\vec
\zeta}(\lambda)=\begin{pmatrix} f(\lambda) & 0\\
0 & g(\lambda)
\end{pmatrix}
$$
and ${ [{f^{\vec \zeta}(\lambda)}]}^{-1}$ satisfies the minimality
condition (\ref{3.9}). The matrix $\textbf{B}_{1}$ and its inverse $(\textbf{B}_{1}^{\zeta})^{-1}$,
the vector of unknown coefficients $\vec{c}_{2}$ are of the form
$$
\textbf{B}^{\zeta}_{1}=\begin{pmatrix}
5 & 0 & 2 & 0\\
0 & 10 & 0 & -3\\
2 & 0 & 5 & 0\\
0 & -3 & 0 & 10
\end{pmatrix},\,(\textbf{B}_{1}^{\zeta})^{-1}=\frac{1}{273\pi}\begin{pmatrix}
65 & 0 & -26 & 0\\
0 & 30 & 0 & 9\\
-26 & 0 & 65 & 0\\
0 & 9 & 0 & 30
\end{pmatrix},$$
$$\vec{c}_{1}^{\zeta}=\frac{1}{273}\left(
91,
39,
-91,
39
\right)^{\top}.
$$
The spectral characteristic of the optimal estimate of $A_{1}\zeta$ is of the form
$$
\vec h^{\top}(f^{\vec \zeta})=\left(
-\frac{2}{3}e^{-i\lambda}+\frac{10}{3}e^{2i\lambda} ,
\frac{3}{7}e^{-i\lambda}+\frac{3}{7}e^{2i\lambda}\right),
$$
and the optimal linear estimate of $A_{1}\vec\zeta$ is of the form
$$
\widehat{{A}_{1}\vec\zeta}=-\frac{2}{3}\zeta_1(-1)+\frac{10}{3}\zeta_1(2)+\frac{3}{7}\zeta_2(-1)+\frac{3}{7}\zeta_2(2)=
$$
$$
=-\frac{2}{3}\zeta(-1)+\frac{10}{3}\zeta(5)+\frac{3}{7}\zeta(0)+\frac{3}{7}\zeta(6).
$$
The mean square error of this estimate
$$ \Delta(f^{\vec
\zeta})=\frac{20}{21}.
$$
\end{example}

\section{Minimax (robust) method of linear interpolation}

Let $f(\lambda)$ and $g(\lambda)$ be the spectral density
matrices of  $T$-variate stationary
sequences $\vec \zeta(j)$ and $\vec \theta(j)$, obtained by $T$-blocking (\ref{block}) of $T$-PC sequences  $\zeta(j)$ and $\theta(j)$, respectively.

Formulas (\ref{3.13})--(\ref{3.16}) may be applied for finding the spectral characteristic and
the mean square error of the optimal linear estimate of the functional $A_s \vec \zeta$
only under the condition that the spectral density
matrices $f(\lambda)$ and $g(\lambda)$ are exactly known.
If the density matrices are not known exactly while a set
$D=D_{f}\times D_{g}$ of possible spectral densities is given, the
minimax (robust) approach to estimation of functionals from unknown
values of stationary sequences is reasonable. In this case we find the
estimate which minimizes the mean square error for all spectral
densities from the given set simultaneously.

\begin{definition} For a given class of
pairs of spectral densities $D=D_{f}\times D_{g}$ the spectral density
matrices $f^{0}(\lambda)\in D_{f}$, $g^{0}(\lambda)\in D_{g}$ are
called  {the least favorable} in $D$ for the optimal linear
estimation of the functional $A_{s}\vec\zeta$ if
$$
\Delta(f^0,g^0)=\Delta(\vec h(f^0,g^0);f^0,g^0)={\max_{\substack
{(f,g)\in D}}} \Delta (\vec h(f,g);f,g).
$$
\end{definition}

\begin{definition}
For a given class of
pairs of spectral densities $D=D_{f}\times D_{g}$ the spectral
characteristic $\vec h^0(\lambda)$ of the optimal linear estimate of
the functional $A_{s}\vec \zeta$ is called minimax (robust) if
$$
\vec h^0(\lambda)\in H_{D}=\mathop {\bigcap }\limits_{(f,g)\in D}
L_{2}^{s-}(f+g),$$
$$\mathop {\min_{\substack {\vec h\in H_{D}}}} {\max_
{\substack {(f,g)\in D}}} \Delta(\vec h;f,g)=\mathop {\max_{\substack
{(f,g)\in D}}} \Delta(\vec h^0;f,g).
$$
\end{definition}

Taking into consideration these definitions and the obtained relations we can
verify that the following lemmas hold true.

\begin{lemma}
\label{lem4.1}
The spectral density matrices
$f^{0}(\lambda)\in D_{f}$, $g^{0}(\lambda)\in D_{g}$, that satisfy
 condition (\ref{3.5}), are the least favorable in D for the optimal
linear  estimation of $A_{s}\vec\zeta$, if the Fourier coefficients of
the matrix functions
$$
(f^0(\lambda)+g^0(\lambda))^{-1},\quad
f^0(\lambda)(f^0(\lambda)+g^0(\lambda))^{-1},
$$
$$
f^0(\lambda)(f^0(\lambda)+g^0(\lambda))^{-1}g^0(\lambda)
$$
define matrices $\textbf{B}^{\zeta}_{s}{}^0, \textbf{D}^{\zeta}_{s}{}^0, \textbf{R}^{\zeta}_{s}{}^0$, that determine a solution of the constrained optimization problem
$$
{\max_{\substack {(f,g)\in
D}}}(\langle{\vec{a}^{\zeta}_{s},\textbf{R}^{\zeta}_{s}\vec{a}^{\zeta}_{s}}\rangle+\langle{(\textbf{B}^{\zeta}_{s})^{-1}\textbf{D}^{\zeta}_{s}\vec{a}^{\zeta}_{s},\textbf{D}^{\zeta}_{s}\vec{a}^{\zeta}_{s}}\rangle)=\langle{\vec{a}^{\zeta}_{s},\textbf{R}^{\zeta}_{s}{}^0\vec{a}^{\zeta}_{s}}\rangle+\langle{(\textbf{B}^{\zeta}_{s}{}^0)^{-1}\textbf{D}^{\zeta}_{s}{}^0\vec{a}^{\zeta}_{s},\textbf{D}^{\zeta}_{s}{}^0\vec{a}^{\zeta}_{s}}\rangle.
$$
The minimax spectral characteristic
$\vec h^0=\vec h(f^0,g^0)$ is given by (\ref{3.7}), if $\vec h(f^0,g^0)\in H_{D}$.
\end{lemma}

\begin{lemma}
\label{lem4.2}
The spectral density matrix
$f^{0}(\lambda)\in D_{f}$, that satisfies  condition (\ref{3.9}), is
the least favorable in $D_{f}$ for the optimal linear estimation
of $A_{s}\vec\zeta$ based on observations of the sequence $\vec \zeta(j)$ at points
$j\in {\mathbb Z} \setminus \tilde{S}$, if the Fourier
coefficients of the matrix function $(f^0(\lambda))^{-1}$ define the matrix
$\textbf{B}^{\zeta}_{s}{}^0$, that determine a solution of the constrained optimization problem
$$
\max_{\substack {f\in D_f}}
\langle{(\textbf{B}^{\zeta}_{s})^{-1}\vec{a}^{\zeta}_{s},\vec{a}^{\zeta}_{s}}\rangle=\langle{(\textbf{B}^{\zeta}_{s}{}^0)^{-1}\vec{a}^{\zeta}_{s},\vec{a}^{\zeta}_{s}}\rangle.
$$
The minimax spectral characteristic $\vec h^0=\vec h(f^0)$
is given by (\ref{3.10}), if $\vec h(f^0)\in H_{D}$.
\end{lemma}

The least favorable spectral densities $f^{0}(\lambda)\in D_{f}$,
$g^{0}(\lambda)\in D_{g}$ and the minimax spectral characteristic
$\vec h^0=\vec  h(f^0,g^0)$ form a saddle point of the function $\Delta(\vec h;f,g)$
on the set $H_{D}\times D$. The saddle point inequalities
$$
\Delta(\vec h^0;f,g)\leq\Delta(\vec h^0;f^0,g^0)\leq\Delta(\vec h;f^0,g^0), \quad \forall
\vec h\in H_{D}, \forall f\in D_{f}, \forall g\in D_{g}
$$
\noindent hold when $\vec h^0=\vec h(f^0,g^0)$, $\vec h(f^0,g^0)\in H_{D}$ and
$(f^0,g^0)$ is a solution of the constrained optimization problem
\begin{equation} \label{copt1}
\sup\limits_{(f,g)\in D_f\times D_g}\Delta\left(\vec h(f^0,g^0);f,g\right)=\Delta\left(\vec h(f^0,g^0);f^0,g^0\right).
\end{equation}

\noindent
The linear functional $\Delta(\vec h(f^0,g^0);f,g)$
is calculated by the formula
$$\Delta(\vec h(f^0,g^0);f,g)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\left(\tilde{A}_{s}(e^{i\lambda})g^0(\lambda)+
\tilde{C}^0_{s}(e^{i\lambda})\right)^{\top}
(f^0(\lambda)+g^0(\lambda))^{-1}f(\lambda)\times$$
$$
(f^0(\lambda)+g^0(\lambda))^{-1}\overline{\left(\tilde{{A}}_{s}(e^{i\lambda})g^0(\lambda)+
\tilde{{C}}^0_{s}(e^{i\lambda})\right)}d\lambda+ \frac{1}{2\pi}\int_{-\pi}^{\pi}\left(\tilde{A}_{s}(e^{i\lambda})f^0(\lambda)-
\tilde{C}^0_{s}(e^{i\lambda})\right)^{\top}\times$$
$$(f^0(\lambda)+g^0(\lambda))^{-1}g(\lambda)(f^0(\lambda)+g^0(\lambda))^{-1}
\overline{\left(\tilde{A}_{s}(e^{i\lambda})f^0(\lambda)-
\tilde{C}^0_{s}(e^{i\lambda})\right)}d\lambda.
$$

The constrained optimization problem (\ref{copt1}) is equivalent to the unconstrained optimization problem \cite{Pshenichnyj}:
\begin{equation} \label{copt2}
\Delta_D(f,g)=-\Delta(\vec h(f^0,g^0);f,g)+\delta((f,g)\left|D_f\times D_g\right.)\rightarrow \inf,
\end{equation}
where $\delta((f,g)|D_f\times D_g)$ is the indicator function of the set $D=D_f\times D_g$.
A solution of the problem (\ref{copt2}) is characterized by the condition $0 \in \partial\Delta_D(f^0,g^0),$ where $\partial\Delta_D(f^0,g^0)$ is the subdifferential of the convex functional $\Delta_D(f,g)$ at point $(f^0,g^0)$ \cite{Rockafellar}.

The form of the functional $\Delta(\vec h(f^0,g^0);f,g)$  admits finding the derivatives and differentials of the functional in the space $L_1\times L_1$. Therefore the complexity  of the optimization problem (\ref{copt2}) is determined by the complexity of calculating of subdifferentials of the indicator functions  $\delta((f,g)|D_f\times D_g)$  of the sets $D_f\times D_g$ \cite{Ioffe}.

Taking into consideration the introduced definitions and the derived relations we can verify that the following lemma holds true.

\begin{lemma}
\label{lem4.3}
Let $(f^0,g^0)$ be a solution to the optimization problem (\ref{copt2}). The spectral densities  $f^0(\lambda)$, $g^0(\lambda)$ are the least favorable in the class $D=D_f\times D_g$ and the spectral characteristic  $\vec h^0= \vec h(f^0,g^0)$ is the minimax of the optimal linear estimate of the functional  $A_s\vec{\zeta}$  if  $\vec h(f^0,g^0) \in H_D$.
\end{lemma}

In the case of estimation of the functional based on observations without noise we have the following statement.
\begin{lemma}
\label{lem4.4}
Let $f^0(\lambda)$ satisfies the
condition (\ref{3.9}) and be a solution of the constrained optimization problem
\begin{equation}\label{4.17}
\Delta(\vec h(f^0);f)\rightarrow{sup},
f(\lambda)\in D_{f},\end{equation}
\[
\Delta(\vec h(f^0);f)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\left(\tilde{C}^0_{s}(e^{i\lambda})\right)^{\top}(f^0(\lambda))^{-1}f(\lambda)(f^0(\lambda))^{-1}
\overline{\left(\tilde{C}^0_{s}(e^{i\lambda})\right)}d\lambda.
\]
Then $f^0(\lambda)$ is the least favorable
spectral density matrix for the optimal linear estimation of
$A_{s}\vec\zeta$ based on observations of the sequence $\vec \zeta(j)$ at points  $j\in
{\mathbb Z} \setminus\tilde{ S}$. The minimax spectral
characteristic $\vec h^0=\vec h(f^0)$ is given by (\ref{3.10}), if $\vec h(f^0)\in
H_{D}$.
\end{lemma}

\section{The least favorable spectral densities in $D_{0}^{-}$ }

Let $\zeta(j),\, j\in \mathbb{Z}$ be $T$-PC sequence and let $\vec\zeta(j)$ be $T$-variate stationary sequence, obtained by $T$-blocking (\ref{block}) of $T$-PC sequence  $\zeta(j)$.
Assume that the number of missed observations of the functional
$A_{s}\vec\zeta$ at each of the intervals is a multiple of the period $T$  $$K_1=T \cdot K_1^T, K_2=T\cdot K_2^T,\dots, K_{s-1}=T \cdot K_{s-1}^T$$
and the number of observations at each of the intervals is a multiple of $T$ $$N_1=T \cdot N_1^T, N_2=T \cdot N_2^T,\dots, N_{s}=T \cdot N_s^T,$$
and coefficients $a(j), j\in S$ are of the form
(\ref{aj}).

Consider the problem of minimax estimation of the functional
$A_{s}\vec\zeta$ from observations of the sequence $\vec\zeta(j)$ at points $j\in {\mathbb Z}
\setminus \tilde{S}$ without noise, under the condition that the spectral
density matrix $f(\lambda)$ of $T$-variate stationary sequence $\vec \zeta(j)$ belongs to the set
$$
D_0^{-}={\left\{ {f(\lambda)|\,\frac{1}{2\pi}\int_{-\pi}^{\pi}f^{-1}(\lambda)d\lambda=P}
\right\}},
$$

\noindent where $P=\left\{ p_{\nu\mu} \right\}_{\nu,\mu=1}^{T}$ is a given
positive definite matrix and $det P\neq0$. With the help of
Lemma \ref{lem4.4} and the method of Lagrange multipliers  we can find that a solution
$f^{0}(\lambda)$ of the constrained optimization problem (\ref{4.17}) satisfy
the following relation:
\begin{equation}\label{5.18}
{\left[
(f^{0}(\lambda))^{-1}\right]}^{\top}
\tilde{C}_{s}^0(e^{i\lambda})={\left[
(f^{0}(\lambda))^{-1}\right]}^{\top} {\vec \alpha},
\end{equation}
\noindent where $\vec \alpha=(\alpha_1,\dots,\alpha_{T})^{\top}$ is a
vector of  Lagrange multipliers,
$$\tilde{C}_{s}^0(e^{i\lambda})=\sum_{l=0}^{s-1}\sum_{\tilde{j}={M_l^T}}^{M_l^T+N_{l+1}^T-1} \vec c^{}{}^0(\tilde{j})e^{i\tilde{j}\lambda},
$$
$$\vec c^{ \zeta}_s{}^0=\left(\vec c^0(0),\dots,\vec c^0\left({N_1^T-1}\right), \dots, \vec c^0(M_{s-1}^T),\dots, \vec c^0\left(M_{s-1}^T+N_s^T-1\right)\right)^{\top},$$
unknown coefficients $\vec c^0(\tilde{j}),\, \tilde{j}\in \tilde{S}$ are determined from relation $\vec c^{\zeta}_{s}{}^0=(\textbf{B}_{s}^{0})^{-1}\vec{a}^{\zeta}_{s},$ the matrix $\textbf{B}_{s}^0$ is
constructed from the Fourier coefficients
$$
B_{s}^0(k,j)=R^{\top}(j-k)=\frac{1}{2\pi}\int_{-\pi}^{\pi}{\left[
(f^{0}(\lambda))^{-1}\right]}^{\top} e^{i(j-k)\lambda}
d\lambda,\,\,k,j\in \tilde{S}
$$
of the matrix function
${\left[ (f^{0}(\lambda))^{-1}\right]}^{\top}.$

The Fourier coefficients $R(k)=R^*(-k),k\in \tilde{S}$, found from
the equation
$$\textbf{B}_s^{0} {\vec \alpha}_s={\vec a^{\zeta}}_s,
$$
\noindent  for ${\vec \alpha}_s=(\vec \alpha,\vec 0,\dots,\vec
0)^{\top}$, satisfy relation (\ref{5.18}) and $\textbf{B}^{0}_{s}{\vec c^{\zeta}_s{}^0}=\vec{a}_{s}$. From equations above we obtain that
$$
R(k)=\begin{cases}
P(\vec a(0))^{-1}{\vec a}^{\top}(k),\,\,k\in \tilde{S},\\
0,\,\, k\in \{0,\dots,M_{s-1}^T+N_s^T-1\}\backslash \tilde{S},
\end{cases}\\
$$
\noindent  where $\left[ (\vec a(0))^{-1}\right]^{\top }\cdot \vec a
(0)=1$. The equality $R(0)=P$ follows as a consequence of the restriction on the spectral densities from the class $D_0^-$.

Let the vector-valued sequence $\vec a (k),k\in\tilde{ S}$, be such
that the matrix function $$(f^0(\lambda))^{-1}=\sum_{k=-\left(M_{s-1}^T+N_s^T-1\right)}^{M_{s-1}^T+N_s^T-1}
R(k)e^{ik\lambda}$$ is positive definite and has nonzero
determinant. Then $(f^0(\lambda))^{-1}$ can be represented
in the form \cite{Hannan1974}
$$
(f^0(\lambda))^{-1}=\left(
\sum_{k=0}^{M_{s-1}^T+N_s^T-1} Q(k) e^{-ik\lambda} \right)\cdot \left( \sum_{k=0}^{M_{s-1}^T+N_s^T-1}
Q(k) e^{-ik\lambda} \right)^*,
$$
\noindent  where $Q(k)=0_{T\times T},\, k\in \{0,\dots,M_{s-1}^T+N_s^T-1\}\setminus\tilde{ S}$. Thus $f^0(\lambda)$  is the spectral
density of the vector autoregression stochastic sequence of
order $M_{s-1}^T+N_s^T-1$ generated by the equation
\begin{equation}\label{5.19}
\sum_{k=0}^{M_{s-1}^T+N_s^T-1} Q(k) \vec \zeta (n-k)=\vec \varepsilon(n),
\end{equation}
\noindent where  $\vec \varepsilon(n)$ is a vector "white noise"
sequence. The minimax spectral characteristic $\vec h(f^0)$ is given by
\begin{equation}\label{5.20}
\vec h(f^0)=- \sum_{k=1}^{M_{s-1}^T+N_s^T-1}\overline{R(k)} (P^T)^{-1}\vec
a(0)e^{-ik\lambda}.
\end{equation}

Hence the following theorem holds true.

\begin{theorem}
\label{theorem5.1}
Let the sequence $\vec a(k)=(a_1(k),a_2(k),\dots ,a_{T}(k))^T,$ $a_\nu(k)=a(k)e^{2\pi
i\nu k/T},$ $\nu=1,\dots ,T$, which determine the linear functional
$A_s\vec\zeta$ from  observations of sequence $\vec \zeta(j)$ at points $ j\in {\mathbb Z}
\setminus\tilde{ S}$, be such that the matrix function
$$\sum_{k=-\left(M_{s-1}^T+N_s^T-1\right)}^{M_{s-1}^T+N_s^T-1} R(k)e^{ik\lambda},$$ where
$$ R(k)=R^*(-k)=\begin{cases}
P(\vec a(0))^{-1}{\vec a}^{\top}(k),\,\,k\in \tilde{S},\\
0_{T\times T},\,\, k\in \{0,\dots,M_{s-1}^T+N_s^T-1\}\backslash \tilde{S},
\end{cases}\\
$$
 \noindent is positive definite and has nonzero determinant. Then
 the least favorable in the class $D_0^-$ spectral density for the
 optimal linear estimate of $A_s\vec\zeta$ is given by the formula
\begin{equation}\label{5.21}f^{0}(\lambda)=\left( \sum_{k=-\left(M_{s-1}^T+N_s^T-1\right)}^{M_{s-1}^T+N_s^T-1}
R(k) e^{ik\lambda} \right)^{-1}.\end{equation}
\noindent The minimax spectral characteristic $\vec h(f^{0})$ is given by
(\ref{5.20}).The greatest value of the mean square error of
$\widehat{{A}_s\vec\zeta}$ is calculated by the formula
\begin{equation}\label{5.22}\Delta(f^0)=<\vec c^{\zeta}_s{}^0,\vec a^{\zeta}_s>.
\end{equation}
\end{theorem}

\begin{example}
\label{5.1}
Let $\zeta(n)$ be a 2-PC
stochastic sequence. Consider the problem of minimax estimation of the functional $$A_2\zeta=5\zeta(1)+5\zeta(2)+2\zeta(5)+2\zeta(6)$$ from observation of the sequence $\zeta(j)$ at points $j\in\mathbb{Z}\setminus \{1,2,5,6\}$ on the set $D_0^{-}$ with $P=\begin{pmatrix}
23 & 22 \\
22 & 23
\end{pmatrix}$.

Rewrite $A_2\zeta$ in the form (\ref{veczeta})
$$A_2\zeta=5\zeta(1)+5\zeta(2)+2\zeta(5)+2\zeta(6)=$$
$$
=\vec a^{\top}(0)\vec \zeta(0)+\vec a^{\top}(2)\vec \zeta(2)=A_2\vec \zeta,
$$
where $\vec a(0)=(5,5)^{\top},\, \vec a(2)=(2,2)^{\top}.$
The matrix function $$\sum_{k={-2,0,2}}
R(k)e^{ik\lambda}$$ and the representation  $$\left(
\sum_{k=0,2} Q(k) e^{-ik\lambda} \right)\cdot \left( \sum_{k=0,2}
Q(k) e^{-ik\lambda} \right)^*$$ are of the form
$$
\begin{pmatrix}
9e^{-2i\lambda}+23+9e^{2i\lambda} & 9e^{-2i\lambda}+22+9e^{2i\lambda} \\
9e^{-2i\lambda}+22+9e^{2i\lambda} & 9e^{-2i\lambda}+23+9e^{2i\lambda}
\end{pmatrix}=$$
$$=\begin{pmatrix}
2+3e^{-2i\lambda} & 1+3e^{-2i\lambda} \\
1+3e^{-2i\lambda} & 2+3e^{-2i\lambda}
\end{pmatrix}\cdot \begin{pmatrix}
2+3e^{2i\lambda} & 1+3e^{2i\lambda} \\
1+3e^{2i\lambda} & 2+3e^{2i\lambda}
\end{pmatrix}.$$

The   least favorable spectral density in the class $D_0^-$  for the
 optimal linear estimate of $A_2\vec\zeta$ by (\ref{5.21}) is of the form
$$
f^0(\lambda)=\frac{1}{45-18e^{-2i\lambda}-18e^{2i\lambda}}\begin{pmatrix}
9e^{-2i\lambda}+23+9e^{2i\lambda} & -9e^{-2i\lambda}-22-9e^{2i\lambda} \\
-9e^{-2i\lambda}-22-9e^{2i\lambda} & 9e^{-2i\lambda}+23+9e^{2i\lambda}
\end{pmatrix}.$$

The minimax spectral characteristic, calculated by (\ref{5.20}), is given by the formula
$$
\vec h(f^0)=-\begin{pmatrix}
2  \\
2
\end{pmatrix}e^{-2i\lambda}.
$$
The greatest value of the mean square error of
$\widehat{{A}_2\vec \zeta}$ takes value
$$
\Delta(f^0)=\frac{10}{9}.
$$
\end{example}

\section{The least favorable spectral densities in $D_{G}^{-}$ }

Let $\zeta(j),\, j\in \mathbb{Z}$ be $T$-PC sequence and $\vec\zeta(j)$ be $T$-variate stationary sequence, obtained by $T$-blocking (\ref{block}) of $T$-PC sequence  $\zeta(j)$.
Assume that the number of missed observations of the functional
$A_{s}\vec\zeta$ at each of the intervals is a multiple of the period $T$  $$K_1=T \cdot K_1^T, K_2=T\cdot K_2^T,\dots, K_{s-1}=T \cdot K_{s-1}^T$$
and the number of observations at each of the intervals is a multiple of $T$ $$N_1=T \cdot N_1^T, N_2=T \cdot N_2^T,\dots, N_{s}=T \cdot N_s^T,$$
and coefficients $a(j), j\in S$ are of the form
(\ref{aj}).

Consider the problem of minimax estimation of the functional
$A_{s}\vec\zeta$ from observations $\vec\zeta(j)$ at points $j\in {\mathbb Z}
\setminus\tilde{ S}$ without noise, under the condition that the spectral
density matrix $f(\lambda)$ of the  vector stationary sequence $\vec \zeta(j)$  belongs to the set
$$
D_G^{-}={\left\{ {f(\lambda)|\,\frac{1}{2\pi}\int_{-\pi}^{\pi}f^{-1}(\lambda)\cos(g\lambda)d\lambda=P(g),
\, g=0,1,\dots,G}
\right\}},
$$

\noindent where the sequence of matrices $P(g)=\left\{ p_{\nu\mu}(g) \right\}_{\nu,\mu=1}^{T}, \, P(g)=P^*(g), g=0,1,\dots,G,$ is such that the  matrix function $\sum_{g=-G}^{G}
P(g)e^{ig\lambda}$ is positive definite  and has the determinant that does not equal zero.
With the help of Lemma \ref{lem4.4} and the method of Lagrange multipliers we can find that solution
$f^{0}(\lambda)$ of the constrained optimization problem (\ref{4.17}) satisfy
the following relation:
\begin{equation}\label{5.23}
{\left[
\left(f^{0}(\lambda)\right)^{-1}\right]}^{\top}
\tilde{C}_{s}^0(e^{i\lambda})={\left[
\left(f^{0}(\lambda)\right)^{-1}\right]}^{\top} \left(\sum_{g=0}^G {\vec \alpha_g}e^{ig\lambda}\right),
\end{equation}
\noindent where $\vec \alpha_g,\, g=0,1,\dots,G$ are
Lagrange multipliers. Relation (\ref{5.23}) holds true if
$$
\sum_{\tilde{j}\in \tilde{S}} \vec{c}^0(\tilde{j}) e^{i\tilde{j}\lambda}=\sum_{g=0}^G \vec \alpha_g e^{ig\lambda}.
$$

Consider two cases: $G\geq M_{s-1}^T+N_s^T-1$ and $G<  M_{s-1}^T+N_s^T-1$.

Let $G\geq  M_{s-1}^T+N_s^T-1$. Then the Fourier coefficients of the function  $\left(f^0({\lambda})^{-1}\right)^{\top}$ determine the matrix $\mathbf{B}_s^0$ and extremum problem (\ref{4.17}) is degenerate. Let
$$
\vec \alpha_{ M_{s-1}^T+N_s^T}=\dots=\vec \alpha_G=\vec 0\,\, \mbox{and}\,\,\vec \alpha_g=0,\,g\notin \tilde{S},
$$
and $\vec \alpha_0,\dots,\vec \alpha_{ M_{s-1}^T+N_s^T-1} $ find from the equation $$\mathbf{B}_s^0 \vec \alpha_s^0=\vec a^{\vec\zeta}_s,$$ where $\vec \alpha_s^0=\left(\vec \alpha_0,\dots,\vec \alpha_{ M_{s-1}^T+N_s^T-1}\right)^{\top}$. Then the least favorable is every density $f(\lambda)\in D_G^-$ and the density
\begin{equation}\label{1g}
 f^0(\lambda) =\left(\sum_{g=-G}^{G}
P(g)e^{ig\lambda}\right)^{-1}=
\end{equation}
$$
=\left(\left(\sum_{g=0}^{G}
Q(g)e^{-ig\lambda}\right)\left(\sum_{g=0}^{G}
Q(g)e^{-ig\lambda}\right)^{*}\right)^{-1}
$$
of the vector stochastic autoregression sequence of the order $G$
\begin{equation} \label{1gg}
\sum_{g=0}^{G} Q(g) \vec \zeta(l-g)=\vec \varepsilon_l.
\end{equation}

Let $G<  M_{s-1}^T+N_s^T-1$.  Then the matrix  $\mathbf{B}_s$ is defined by the Fourier coefficients of the  function $\left(f({\lambda})^{-1}\right)^{\top}$. Among them  $P(g),
g\in \{0,\dots,G\}\cap \tilde{S},$ are known and  $P(g), g\in \tilde{S}\setminus \{0,\dots,G\},$ are unknown. The unknown $\vec
\alpha_g,\, g\in \{0,\dots,G\}\cap S$ and $P(g), g\in\tilde{ S}\setminus \{0,\dots,G\}$ we find from the equation
\begin{equation}\label{g}
\mathbf{B}_s \vec \alpha_G^0=\vec a^{\vec\zeta}_s,
\end{equation}
where ${\mathbf{\vec \alpha}}_G^0=(\vec \alpha_0,\dots,\vec \alpha_{G'},\vec
0,\dots,\vec 0)^{\top}$, $G'$ is defined from the relation $\{0,\dots,G\}\cap \tilde{S}=\{0,\dots,G'\}$. The equation (\ref{g}) can be represented  as a system of the following equations
$$
\sum_{g\in\{0,\dots,G\}\cap \tilde{S}}B_s(0,g)\vec \alpha(g)=\vec a(0),
$$
$$
\vdots
$$
$$
\sum_{g\in\{0,\dots,G\}\cap \tilde{S}}B_s(G',g)\vec \alpha(g)=\vec a(G'),
$$
$$
\vdots
$$
$$
\sum_{g\in\{0,\dots,G\}\cap \tilde{S}}B_s\left( M_{s-1}^T+N_s^T-1,g\right)\vec \alpha(g)=\vec a\left( M_{s-1}^T+N_s^T-1\right).
$$
From the first $G'$ equations we can find coefficients $\vec \alpha_0,\dots,\vec \alpha_{G'}$ and from the next equations we can find matrices $P(g), g\in \tilde{S}\setminus \{0,\dots,G\}$.

If the sequence of matrices $P(g), g\in \tilde{S},$ is such that $P(g)=P^*(g), g\in S$, the matrix function $$\sum_{g=-\left( M_{s-1}^T+N_s^T-1\right)}^{ M_{s-1}^T+N_s^T-1}
P(g)e^{ig\lambda}$$ is positive-definite and has the determinant which does not equal zero identically, then the least favorable spectral density
$ f^0(\lambda)$ is defined by the formula
\begin{equation} \label{2g}
f^0(\lambda) =\left(\sum_{g=-\left( M_{s-1}^T+N_s^T-1\right)}^{ M_{s-1}^T+N_s^T-1}
P(g)e^{ig\lambda}\right)^{-1}=
\end{equation}
$$
=\left(\left(\sum_{g=0}^{ M_{s-1}^T+N_s^T-1}
Q(g)e^{-ig\lambda}\right)\left(\sum_{g=0}^{ M_{s-1}^T+N_s^T-1}
Q(g)e^{-ig\lambda}\right)^{*}\right)^{-1}
$$
and is the density of the vector stochastic autoregression sequence of order $ M_{s-1}^T+N_s^T-1$
\begin{equation} \label{2gg}
\sum_{g=0}^{ M_{s-1}^T+N_s^T-1} Q(g) \vec\zeta(l-g)=\vec \varepsilon_l.
\end{equation}

Thus, the following theorem holds true.

\begin{theorem}
\label{theorem6.1}
The spectral density (\ref{1g}) of the vector stochastic autoregression sequence (\ref{1gg}) of order $G$, that is determined by matrices $P(g), g\in \{0,1,\dots,G\}$, is the least favorable in the class $D^{-}_G$  for the optimal estimation of the functional  $A_s
\vec\zeta$ in the case  where $G\geq  M_{s-1}^T+N_s^T-1$. If $G<  M_{s-1}^T+N_s^T-1$ and solutions $P(g),
g\in \tilde{S}\cap \{0,1,\dots,G\},$ of the equation $\mathbf{B}_s \vec \alpha_G^0=\vec a^{\vec\zeta}_s$ with coefficients $P(g),
g\in \tilde{S}\backslash \{0,1,\dots,G\},$ form a positive-definite matrix function  $\sum_{g=-\left(  M_{s-1}^T+N_s^T-1\right)}^{  M_{s-1}^T+N_s^T-1}P(g)e^{i g\lambda}$, with the determinant which does not equal zero identically, then the spectral density (\ref{2g}) of the  vector stochastic autoregression sequence  (\ref{2gg})  of order
$ M_{s-1}^T+N_s^T-1$ is the least favorable in the class $D^{-}_G$. \noindent The minimax spectral characteristic $h(f^{0})$ is calculated by the formula
(\ref{3.10}).
\end{theorem}

\section{Conclusion}

We propose formulas for calculating the mean square error and the
spectral characteristic of the optimal linear estimate of the
functional $$A_s{\zeta}=\sum_{l=0}^{s-1}\sum_{j=M_l+1}^{M_l+N_{l+1}}{a}(j){\zeta}(j),\,\,M_l=\sum_{k=0}^l(N_k+K_k),\,\,N_0=K_0=0,$$ which depends
on the unobserved values of a periodically correlated stochastic
sequence ${\zeta}(j)$. Estimates are based on observations of the sequence
${\zeta}(j)+{\theta}(j)$ at points $j\in{\mathbb Z}\setminus S$, where $S=\bigcup _{l=0}^{s-1}\{M_l+1,\dots,M_l+N_{l+1}\}$.
The sequence ${\theta}(j)$ is an uncorrelated with ${\zeta}(j)$
periodically correlated stochastic sequence. This problem is
investigated in two cases. In the first case the spectral density matrices
$f(\lambda)$ and $g(\lambda)$ of the
$T$-variate stationary sequences $\vec \zeta(n)$ and $\vec \theta(n)$, obtained by $T$-blocking of $T$-PC sequences  $\zeta(j)$ and $\theta(j)$, respectively, are suppose to be known exactly.
In this case we derived formulas for calculating the spectral characteristic and the mean-square error of the optimal estimate of the functional.
In the second case where the spectral density matrices are
not exactly known while a class $D=D_{f} \times D_{g}$ of admissible
spectral densities is given.  Formulas that determine the least favorable spectral
densities and the minimax spectral characteristic of the optimal estimate of the functional
$A_{s}\zeta$ are proposed. The problem is investigated in details for two special classes of admissible spectral densities.
Some examples of application of the obtained  results for finding optimal estimates of linear functionals and determining the least favorable spectral
densities of the optimal estimates are presented.

redpajama_set_name: RedPajamaArXiv


text:
Q: Visual Studio Team Services: How to ensure build server (AzureVM) is running before queuing a build We have an AzureVM being used for a VSTS build server.  In order to save costs, we also have that build server turn off at 6pm EST every day.
We'd like VSTS to automatically be able to restart the VM (if not already running) upon queuing a build, but there doesnt seem to be any support for this scenario.
Thought about doing a powershell task, but thats after the build starts, so you still get the "cant schedule a build" error because the VM that all the build tasks will run from is still off.
How has anybody else solved this problem?

redpajama_set_name: RedPajamaStackExchange


text:
HMS Redbreast was an Archer-class brig of the British Royal Navy. She captured some small merchant vessels and privateers. She also participated in two actions that would in 1847 earn her surviving crew members clasps to the Naval General service Medal (NGSM). The Navy transferred in 1816 to His Majesty's Customs. She was finally sold in 1850.

Napoleonic wars
Redbreast was commissioned by Lieutenant John Maxwell in May 1805, for the Channel. He was promoted to Commander on 22 January 1806. His replacement was Lieutenant John Bayby Harrison.Redbreast was on the Portuguese coast in July 1807.

On 19 March 1808, Sir George Mouat Keith was appointed to Redbreast on the North Sea station.

Redbreast operated in the Channel until 1808, when she took part in the Baltic expedition. Detachments from the frigate , the sloop , and Redbreast manned two gunboats on 9 and 11 August 1808 on the Jade and Weser rivers. There they first captured a Danish privateer Mosin; four days later they captured Dutch gunboat No.106. Mosin was armed with one 4-pounder gun and had a crew of 11 men. The gunboat was armed with one 18-pounder and two 4-pounder guns. It had a crew of 23 men under the command of Lieutenant Henry Meyer. Neither side suffered any casualties.

On 16 May 1809 Redbreat captured Anna Sophia, Griffstadt, and Wannerne;  on 24 July she captured Twee Gesisters.

In 1811 Redbreast was involved in a skirmish in the Jahde. On 1 August, a squadron, consisting of the 32-gun frigate , 16-gun brig , the gun-brigs  and Redbreast, and hired cutters  and , were cruising off Texel. Quebec captured a Vaisseau de Guerre of the Imperial Customs Service, later named as a privateer Christine Charlotte. She had a crew of one officer and twelve men, one of whom was killed before she struck. She was leaving Nordeney (East Frisian Islands) with a merchant vessel in tow.

The sight of four enemy gun-brigs lying at anchor at Nordeney induced Captain Hawtayne of Quebec to despatch ten boats from the squadron, with 117 seamen and marines, to cut them out. The boats attacked, and after a hard fight captured French gunboats Nos. 22, 28, 31, 71. British casualties were four men killed (one from Redbreast), and 14 wounded; enemy casualties were two men killed and 12 wounded. In 1847 the Admiralty awarded the NGSM to all surviving claimants from the action.

On 28 November 1811 , Captain Arthur Farquarh, arrived at Cuxhaven from Bremerlehe to join the squadron that Captain Greene, of , had gathered. The British vessels had come to support the Russian troops under Colonel Alexander Radlinger, who were besieging the town. The squadron comprised , , , , Redbreast, and the gun-boats Nos. 1, 2, 3, 4, 5, 6, and 10. The two key positions were the French forts Napoleon and Phare. While the gunboats fired on Phare, the British landed a number of guns and established a battery of six 18-pounders, two 32-pounders, and two 6-pounders. Before the battery could start firing on the 30th, the French surrendered both forts. The French surrendered 26 heavy guns, two 13" mortars, and a blockhouse with a garrison of three hundred men and officers, all of whom became prisoners of war. Redbreast carried the news back to England, together with the officers captured at Phare.

On 8 July 1812 Exertion hit a shifting sandbank unknown to her pilots. The next day, as enemy gunboats approached, the decision was taken to abandon her. Redbreast removed Exertions crew and then set fire to her to prevent Exertion falling into enemy hands.

Redbreast and  were in company on 1 November 1812 at the capture of Mercurius, Pusche, master. Mercurius arrived at Yarmouth on 11 November.

Redbreast captured a Danish vessel Manly and her cargo in March 1813. This may have been the former , which the Danes had captured in 1811, though the prize money notice does not make the connection.

On 3 August 1813 Redbreast and Princess Augusta captured five small vessels called "ewers".

In August 1813 Redbreast was in company with  when they captured the Danish droits Haabet and Evers, No. 73 and 123, on the 13th and 14th.

From late December 1813 the squadron, comprising Desiree, Hearty, Blazer, Piercer, Shamrock, Redbreast, and eight gun-boats (the seven from Cuxhaven plus No. 12), supported Swedish forces under the command of General Baron de Boye against the Danes in the attack on the town and fortress of Glückstadt at the Elbe river during the War of the Sixth Coalition. The British arrived on 23 December and by 25 December had erected a battery of two 32-pounder guns. (They had landed six but the poor quality of the roads meant that only two were in place.) To support the bombardment on the next day, Farquhar sent in the gunboats and the brigs, whose armament he had reinforced with two 18-pounders each from Desiree. The bombardment continued to the 28th. Glückstadt having failed to surrender, Farquhar established three more batteries, one of two 18-pounders, one of four 32-pounders, and one of the two 13" mortars captured at Cuxhaven. An English rocket brigade under Lieutenant Amherst Wright contributed to the bombardment. These batteries commenced bombarding the town on 1 January 1814, keeping it up for the next two days. On 4 January Farquhar sent in a flag of truce, and after negotiations, the governor surrendered on 5 January. British casualties overall were light. In 1847 the Admiralty awarded the NGSM to all surviving claimants from the campaign.

In 1814 the Navy re-rated Redbreast as a sloop, so that Keith could continue to command her on his promotion on 16 March 1814 from lieutenant to commander.

Post-war
Redbreast was in Ordinary at Portsmouth in 1815. She then was converted to a customs hulk, and in 1830 a lazaretto for the Quarantine Service at Liverpool.

Fate
Redbreast was sold in Liverpool on 14 June 1850.

Notes, citations, and references
Notes

Citations

References

1805 ships
Brig-sloops of the Royal Navy

redpajama_set_name: RedPajamaWikipedia


text:
The 1944 United States presidential election in Delaware took place on November 7, 1944, as part of the 1944 United States presidential election. State voters chose three representatives, or electors, to the Electoral College, who voted for president and vice president.

Delaware was won by incumbent President Franklin D. Roosevelt (D–New York), running with Senator Harry S. Truman, with 54.38% of the popular vote, against Governor Thomas E. Dewey (R–New York), running with Governor John W. Bricker, with 45.27% of the popular vote.

Results

See also
 United States presidential elections in Delaware

References

Delaware
1944
1944 Delaware elections


The retention of radioactivity after doses of 14C- and 3H-labelled folic acid is described. Radioactivity was retained in liver, kidney and gut of rats for some time after administration of the dose. The retained radioactivity could not be displaced by large doses of unlabelled folic acid or unlabelled 5-methyltetrahydrofolate. 14C- and 3H-labbelled folates showed similar chromatographic behaviour onion-exchange chromatography to 5-methyltetrahydrofolate, and on ion-exchange and gel-permeation chromatography to synthetic pteroylhepta-gamma-glutamate.


Look Outdoor Advertising Co. is a Texas partnership that began operating in 1994 with the purchase of four neglected billboard structures near Amarillo, Texas. Since that time, we have grown into a full-time outdoor advertising company that specializes in highway directional signs. We now own and operate over 450 faces in Texas, Oklahoma, New Mexico and southwest Kansas.
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From concept to final installation, Look Outdoor will do whatever it takes to deliver the perfect display to you target audience.
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Not every child is lucky enough to come into the world at the right time and in the best of health. For children who require help sucking after their birth, NUK has developed special products that are used in most German clinics with neonatal wards. For it is our aim to help every child develop healthily right from the start. MedicPro products have been developed for many years in close collaboration with medical professionals and meet the highest quality standards. Most notably, the teats from the NUK Clinic range are used in 95% of all German clinics, and many hospitals around Australia.


text:
Equality, Religion And Gender In Israel
• Comments Off on Equality, Religion And Gender In Israel
Women In Israel
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Crimes Against Women
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In 1998, the Knesset passed a law for "Prevention of Sexual Harassment". Earlier this year, the Israeli public celebrated the success of the blockbuster film "Wonder Woman" and its homegrown lead actress, Gal Gadot.
They generally apply in instances the place the circumstances do not meet the criteria established by regulation. The impact of the payments submitted by liberal members of the Knesset relating to removal of restrictions to abortions on reigniting the abortion debate in Israel is unclear. In an article published in October 2009, an Israeli scholar warned against rising tensions regarding abortions in Israel in connection with the 2009 bill. The writer, Dr. Yael Hashiloni-Dolev, advised that as compared with the United States, Germany, and Ireland, abortions have by no means been the topic of a serious public debate in Israel.
Budget Travel Idea Wins Israel21c Student Project Contest
Indeed, during her term as Minister of Justice, Shaked has not only appointed tons of of conservative judges, including six supreme courtroom justices, however has additionally played a serious position in promoting Israel's controversial Nation-State Law. As a result, in recent years, some have actually come to speculate that she might ultimately turn out to be Israel's second female Prime Minister. In the 1994 Bavli case (Bavli v. Rabbinical Court of Appeals, 48 P.D. , 221), the High Court of Justice imposed on the rabbinical courts the duty to abide by the principle of equality in the division of matrimonial property. Justice Barak held that the Jewish law principle of separation of matrimonial property could not fulfill this requirement because it resulted in women receiving a negligible share of the property on divorce. The Court's choice was in conformity with the Women's Equal Rights Law, which excluded solely the matter of license or prohibition in marriage and divorce from the purview of the equality precept and not associated issues, such as property, maintenance and custody.
The first time researchers identified a clear fashionable gender voting gap in Israel was after the elections in 2009 during which Livni ran against Netanyahu (Gedalya, Herzog, and Shamir, 2010). Then once more in 2013 when three different parties were led by women (Shamir and Gedalya-Lavy, 2015). Similar to voting patterns around the world, we observe men growing their assist for excessive proper parties – focusing on nationalist rhetoric – within the Israeli case the emphasis being on Jewish nationalism. In March 2015, after noticing the scarcity of female speakers at business conferences in Israel, serial entrepreneur Hana Rado launched Supersonas. Designed to increase the ratio of ladies speakers—which Rado estimates is just 10 p.c—Supersonas is a digital Rolodex of women on the high of their fields.
Men on common out-earn them, usually tend to have extensive skilled networks, and do not face as a lot social pressure to prioritize their roles as family caretakers. Furthermore, despite Tzipi Livni's tenure as head of the Ministry of Foreign Affairs, no women has ever been appointed to go the other three most prestigious ministries, namely the Defense, Finance, and Interior Ministries.
There are several NGOs that provide information to Israelis wishing to bear children. For instance, the Hen Association for Fertility and Life web site states that the affiliation's goal is to help couples who suffer from infertility.The affiliation provides common data on relevant laws and reproductive providers. Some NGOs, nonetheless, are energetic in providing info regarding reproductive rights and in attempting to impression Israeli public opinion either directly or by way of lobbing the Knesset (Parliament).
Lusky's NGO operates in many international locations that don't maintain relations with Israel, similar to Syria. Not solely does her team challenge stereotypes by helping out these often thought of an enemy, but also helps break down limitations on the opposite aspect of the border. IFA provides individuals across the world with meals, medical supplies, health providers and infrastructure no matter their nationality, religion, politics or gender, underlying the fact that regardless of who we are, we're all human. The model and media personality is not only one of many few Israeli-Arab celebrities within the Holy Land, but additionally the proud winner of Israel's first transgender beauty pageant. She shot to fame upon profitable the 2016 competitors, and has since participated in numerous TV shows, bringing to the fore her lovely self in addition to consciousness and recognition of the native LGBTQ community.
When it comes to inclusion, says Henig Shaked, "I think Silicon Valley is extra superior. They have heads of diversity, which few companies in Israel have." Almost all of these nonprofit organizations and groups of ladies are supported by major American tech corporations that give them free house to carry events. For example, Microsoft hosted the Yazamiyot accelerator; Google hosts their campus for mothers; and LeadWith's babies events are hosted by Facebook.
Israel's reproductive care coverage appears to reflect Jewish religious, cultural, and social norms relating to fertility. Parenthood is considered a primary human right primarily based on biblical and different Jewish religious sources. The personal want for parenthood, and particularly motherhood, has been engrained in Jewish culture and has been strengthened by the historical persecution of Jews in the Diaspora and particularly by the genocide perpetrated against Jews in the Holocaust.
In the Nineties, the Knesset created a particular committee for ladies's points. Today there is an official Authority for the Advancement on the Status of Women as a part of the Prime Minister's workplace. In current years, the Knesset has passed legal guidelines that guarantee non-discrimination primarily based on gender. Controversial points relating to gender equality in Israel at present relate to tensions between non secular and secular Jews. In the present Israeli parliament of one hundred twenty, 23% are women, as in comparison with 20% within the US Congress, 18% in European parliaments, and fewer than 7% in Arab countries.
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redpajama_set_name: RedPajamaCommonCrawl


text:
function Diff-CSV {
	 <#
    .SYNOPSIS
        PowerShell wrapper for diff-table.exe a tool to diff csv files
    .DESCRIPTION
        PowerShell wrapper for diff-table.exe tool.
	    Generates row/column wise diffs conmparing two .csv files
        This function requires diff-table.exe tool.
        https://github.com/chop-dbhi/diff-table
        Instructions:
        - Download https://github.com/chop-dbhi/diff-table/releases/download/0.4.0/diff-table-windows-amd64.zip
        - extract and update location to exePath in line 35
    .PARAMETER CSV1
		Path to the first csv file to compare.
	.PARAMETER CSV2
	    Path to the second csv file to compare
    .PARAMETER Key
	    Primary key (column name) that is common in both csv files (can be a combination of multiple columns provided as a comma separted list)
    .EXAMPLE
    	Diff-CSV test1.csv test2.csv name
        #Compares both files based on the "name" column as the primary key
    .LINK
        https://github.com/chop-dbhi/diff-table
    #>
    [CmdletBinding()]
    Param(
        [Parameter(Mandatory,Position=0)]
        $CSV1,
        [Parameter(Mandatory,Position=1)]
        $CSV2,
        [Parameter(Mandatory,Position=2)]
        $Key
    )

    $exePath = "C:\Scripts\tools\diffTable\diff-table.exe"
    if (!(Test-Path $exePath)){
        $message = @'
Generates row/column wise diffs conmparing two .csv files
This function requires diff-table.exe tool.
https://github.com/chop-dbhi/diff-table
Instructions:
- Download https://github.com/chop-dbhi/diff-table/releases/download/0.4.0/diff-table-windows-amd64.zip
- extract and update location to exePath in line 35
'@
        Write-Warning $message
        exit
    }
    $parameters = New-Object System.Collections.ArrayList
    $null = $parameters.Add("-csv1")
    $null = $parameters.Add($(Resolve-Path $CSV1).Path)
    $null = $parameters.Add("-csv2")
    $null = $parameters.Add($(Resolve-Path $CSV2).Path)
    $null = $parameters.Add("-csv1.sort")
    $null = $parameters.Add("-csv2.sort")
    $null = $parameters.Add("-key")
    $null = $parameters.Add($Key)
    $null = $parameters.Add("-events")
    & $exePath $parameters | ConvertFrom-Json
}

#$csvPath = "C:\Scripts\tools\diffTable"
#$res = Diff-CSV $csvPath\test.csv $csvPath\test2.csv name

redpajama_set_name: RedPajamaGithub


text:
Q: Transfer Zend App to new VPS show blank page I bought a new VPS and I try to transfer my working ZendApp (work in previous hosting with DirectAdmin). Now i install nginx with php5 and mysql i this is my php configuration - works well
http://178.216.200.85/info.php
I turn on display errors in php.ini and it shows me error in writable /_cache/ folder soo i set it to 777 permission. Error resolved but i still have error and don't see anything in my site http://178.216.200.85/index.php
I don't have any .htaccess file
Here is my /etc/nginx/sites-avaible/default content
server {
    listen 80 default_server;
    listen [::]:80 default_server;

    # SSL configuration
    #
    # listen 443 ssl default_server;
    # listen [::]:443 ssl default_server;
    #
    # Self signed certs generated by the ssl-cert package
    # Don't use them in a production server!
    #
    # include snippets/snakeoil.conf;

    root /var/www/html;

    # Add index.php to the list if you are using PHP
    index index.php index.html index.htm index.nginx-debian.html;

    server_name 178.216.200.85;

    location / {
        # First attempt to serve request as file, then
        # as directory, then fall back to displaying a 404.
        try_files $uri $uri/ =404;
    }

    location ~ \.php$ {
        include snippets/fastcgi-php.conf;
        fastcgi_pass unix:/var/run/php5-fpm.sock;
    }

    location ~ /\.ht {
        deny all;
    }
}

What i should to add or someting to run my app in new server?
Any advice will be very valuable

A: The problem (General error: 1449) occurs when you access views, triggers, or procedures, for example, when you move a database, when a connection to a new location occurs on behalf of another user.
You have created some data under a certain user ('bunga'@'localhost'), which is not currently on the new server in MySQL. The easiest way is to create this user.
Learn More — MySQL error 1449: The user specified as a definer does not exist


text:
Next Up Jennifer Lopez Reveals She's The New Face Of Versace & Wears Iconic Grammy Dress To Celebrate
Expert Says
January 11, 2020 11:53PM EST
Prince Harry & Meghan Markle Will Struggle With The Queen On Their Status — She's 'The Boss'
Cassie Gill and Allison Swan
Deals with The Queen? Not so fast says, says this royal expert! Prince Harry and Meghan Markle still have plenty to sort out as they move forward.
Prince Harry, 34, and Meghan Markle, 38, seem more determined than ever to forage a new path — but simply "stepping away" from royal life isn't as easy at it seems. "You can't sort of do deals with the Queen, even if she is your grandmother," Ingrid Seward, editor-in-chief of Majesty magazine, told HollywoodLife in an EXCLUSIVE new interview. "I mean the trouble is that they seem to fail to realize that the Monarchy's like a firm, and everybody works for the boss, who's the Queen. And you don't have your own agenda," she continued, referencing one of Princess Diana's best-known nicknames for the royal family ("the firm").
While Harry and Meghan are drawing comparisons to his great-uncle King Edward VIII, who abdicated from the throne in 1936 to marry Wallis Simpson, Harry and Meghan's scenario is unique in that they still wish to do official work on behalf of the Queen — and therefore, remain working royals in a part-time sense. "If they had said, 'Look, we're out. We don't want our titles, we want to live a completely private life, and we're off,' then that would be one thing. But wanting to have one foot in, and one foot out is very difficult," Ingrid explains, which adds to the confusion. "They want to stay working. They want to be working members of the royal family and therefore they want to be funded for when they work for the royal family, and they don't want to be funded what they don't."
The Duke and Duchess of Sussex shared a fair amount of information about their financial status on their new website, and explained that they were also seeking to gain "financial independence" — suggesting that in addition to retaining some royal duties, they still want to generate income from other work. "Nobody knows the answers, and that's why it's caused such a furor here. Because no one's ever done this," she continued. "You can't just step back from your heritage." While it's unclear what other work the couple may engage in, they did recently produce their own documentary Harry & Meghan: An African Journey, and Meghan, of course, comes from an acting background. "I think they would have been much better just to say, 'Look, we don't need anything.', Because Harry's got a lot of money if his mother got $17 million pounds in her divorce," Ingrid notes.
Though other royals, like Harry's younger cousins Princess Beatrice, 31, and Princess Eugenie, 29, have their titles along with their own independent jobs (Beatrice works in finance and Eugenie has a position at an art gallery), Harry and Meghan's situation still differs. "[Eugenie and Beatrice] are not in a direct line to the Royal throne [they are 9th and 10th in line, underneath their father Prince Andrew who is 8th] and they don't have royal protection," Ingrid notes. "But Harry and Meghan are saying they want to carry on [having protection], they want to keep doing their charity duties and they want to work for the Commonwealth, which is working for the Queen." While Meghan has returned to Canada, Queen Elizabeth has called a meeting on Monday, Jan. 13 with Prince Charles, Prince William and Prince Harry to discuss the situation further.
Meghan Markle	Prince Harry
Prince William Is 'Sad' That Prince Harry & Meghan Markle Will Step Back But 'Supports' Them -- Report
Prince Harry & Meghan Markle's Friends Say They Were 'Driven Out' Of The Royal Family
Queen Elizabeth 'Plans' To Work With Prince Harry & Meghan Markle On 'Step Back' From Royal Life