=\langle\mathbb{R}^{0}\mathbb{a},(\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a}\rangle+\langle\mathbb{Q}^{0}\mathbb{a},\mathbb{a}\rangle.\end{split}$
(131)

The minimax spectral characteristic $h^{0}=h(f_{0},g_{0})$ can be calculated
by the formula (129) if $h(f_{0},g_{0})\in H_{D}.$

The least favourable spectral densities $f_{0}(\lambda)$, $g_{0}(\lambda)$ and
the minimax spectral characteristic $h^{0}=h(f_{0},g_{0})$ form a saddle point
of the function $\Delta\left(h;f,g\right)$ on the set $H_{D}\times D$. The
saddle point inequalities

$\Delta\left(h;f_{0},g_{0}\right)\geq\Delta\left(h^{0};f_{0},g_{0}\right)\geq\Delta\left(h^{0};f,g\right)$
$\forall h\in H_{D},\forall f\in D_{f},\forall g\in D_{g}$

hold true if $h^{0}=h(f_{0},g_{0})$ and $h(f_{0},g_{0})\in H_{D},$ where
$(f_{0},g_{0})$ is a solution to the constrained optimization problem

$\sup\limits_{(f,g)\in D_{f}\times
D_{g}}\Delta\left(h(f_{0},g_{0});f,g\right)=\Delta\left(h(f_{0},g_{0});f_{0},g_{0}\right),$
(132)

where

$\begin{split}\Delta\left(h(f_{0},g_{0});f,g\right)&=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\frac{\left|A(e^{i\lambda})g_{0}(\lambda)+\sum\limits_{j=0}^{\infty}((\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a})_{j}e^{ij\lambda}\right|^{2}}{(f_{0}(\lambda)+g_{0}(\lambda))^{2}}f(\lambda)d\lambda\\\
&+\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\frac{\left|A(e^{i\lambda})f_{0}(\lambda)-\sum\limits_{j=0}^{\infty}((\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a})_{j}e^{ij\lambda}\right|^{2}}{(f_{0}(\lambda)+g_{0}(\lambda))^{2}}g(\lambda)d\lambda,\end{split}$
(133)

The constrained optimization problem (132) is equivalent to the unconstrained
optimization problem

$\Delta_{D}(f,g)=-\Delta(h(f_{0},g_{0});f,g)+\delta(f,g\left|D_{f}\times
D_{g}\right.)\rightarrow\inf,$ (134)

where $\delta(f,g\left|D_{f}\times D_{g}\right.)$ is the indicator function of
the set $D=D_{f}\times D_{g}$. Solution $(f_{0},g_{0})$ to the problem (134)
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 $(f_{0},g_{0})$. This condition makes it
possible to find the least favourable spectral densities in some special
classes of spectral densities $D$ (see books [20], [70], [72] for additional
details).

Note, that the form of the functional $\Delta(h(f_{0},g_{0});f,g)$ (133) is
convenient for application the Lagrange method of indefinite multipliers for
finding solution to the problem (132). Making use the method of Lagrange
multipliers and the form of subdifferentials of the indicator functions we
describe relations that determine least favourable spectral densities in some
special classes of spectral densities (see books [13, 58, 64] for additional
details).

######  Lemma 5.2.

Let $(f_{0},g_{0})$ be a solution to the optimization problem (134). The
spectral densities $f_{0}(\lambda)$, $g_{0}(\lambda)$ are the least favourable
in the class $D=D_{f}\times D_{g}$, and the spectral characteristic
$h^{0}=h(f_{0},g_{0})$ is minimax for the optimal estimate of the functional
$A\xi$, in the case where $h(f_{0},g_{0})\in H_{D}$.

###  5.3 Least favourable spectral densities in the class $D_{f}^{0}\times
D_{g}^{0}$

Consider the problem of the optimal estimation of the functional
$A\xi=\sum_{j=0}^{\infty}a(j)\xi(j)$ which depends on the unknown values of a
stationary stochastic sequence $\xi(j)$ based on observations of the sequence
$\xi(j)+\eta(j)$ at points of time $j=-1,-2,\dots$ in the case where the
spectral densities $f(\lambda)$, $g(\lambda)$ are from the class
$D=D_{f}^{0}\times D_{g}^{0}$, where

$D_{f}^{0}=\left\\{f(\lambda)\left|\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f(\lambda)d\lambda\leq
P_{1}\right.\right\\},$
$D_{g}^{0}=\left\\{g(\lambda)\left|\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}g(\lambda)d\lambda\leq
P_{2}\right.\right\\}.$

Let the densities $f_{0}(\lambda)\in D_{f}^{0}$, $g_{0}(\lambda)\in D_{g}^{0}$
and the functions $h_{f}(f_{0},g_{0})$, $h_{g}(f_{0},g_{0})$, determined by
the relations

$h_{f}(f_{0},g_{0})=\frac{\left|A(e^{i\lambda})g_{0}(\lambda)+\sum\limits_{j=0}^{\infty}((\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a})_{j}e^{ij\lambda}\right|^{2}}{(f_{0}(\lambda)+g_{0}(\lambda))^{2}},$
(135)
$h_{g}(g_{0},g_{0})=\frac{\left|A(e^{i\lambda})f_{0}(\lambda)-\sum\limits_{j=0}^{\infty}((\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a})_{j}e^{ij\lambda}\right|^{2}}{(f_{0}(\lambda)+g_{0}(\lambda))^{2}},$
(136)

be bounded. In this case the functional

$\Delta(h(f_{0},g_{0});f,g)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}h_{f}(f_{0},g_{0})f(\lambda)d\lambda+\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}h_{g}(f_{0},g_{0})g(\lambda)d\lambda$

is linear and continuous on the space $L_{1}\times L_{1}$ and we can apply the
method of Lagrange multipliers to find solution to the optimization problem
(134). We get the following relations that determine least favourable spectral
densities $f^{0}\in D^{0}_{f}$, $g^{0}\in D^{0}_{g}$

$\begin{split}&-\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}h_{f}(f_{0},g_{0})\rho(f(\lambda))d\lambda-\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}h_{g}(f_{0},g_{0})\rho(g(\lambda))d\lambda\\\
&+\alpha_{1}\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\rho(f(\lambda))d\lambda+\alpha_{2}\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\rho(g(\lambda))d\lambda=0,\end{split}$

where $\rho(f(\lambda))$ and $\rho(g(\lambda))$ are variations of the
functions $f(\lambda)$ and $g(\lambda)$, the constants $\alpha_{1}\geq 0$ and
$\alpha_{2}\geq 0$. From this relation we get that the least favourable
spectral densities $f_{0}(\lambda)\in D_{f}^{0}$, $g_{0}(\lambda)\in
D_{g}^{0}$ satisfy equations

$\left|A(e^{i\lambda})g_{0}(\lambda)+\sum\limits_{j=0}^{\infty}((\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a})_{j}e^{ij\lambda}\right|=\alpha_{1}(f_{0}(\lambda)+g_{0}(\lambda)),$
(137)
$\left|A(e^{i\lambda})f_{0}(\lambda)-\sum\limits_{j=0}^{\infty}((\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a})_{j}e^{ij\lambda}\right|=\alpha_{2}(f_{0}(\lambda)+g_{0}(\lambda)).$
(138)

Note, that $\alpha_{1}\neq 0$ in the case, where

$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f_{0}(\lambda)d\lambda=P_{1},$

and $\alpha_{2}\neq 0$ in the case, where

$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}g_{0}(\lambda)d\lambda=P_{2}.$

Summing up our reasoning we come to conclusion that the following theorem
holds true.

######  Theorem 5.2.

Let the spectral densities $f_{0}(\lambda)\in D_{f}^{0}$ and
$g_{0}(\lambda)\in D_{g}^{0}$ satisfy the minimality condition (121) and let
the functions $h_{f}(f_{0},g_{0})$ and $h_{g}(f_{0},g_{0})$, determined by the
formulas (135), (136), be bounded. The functions $f_{0}(\lambda)$ and
$g_{0}(\lambda)$, which give solution to the system of equations (137), (138)
are the least favourable spectral densities in the class $D=D_{f}^{0}\times
D_{g}^{0}$ for the optimal estimation of the functional
$A\xi=\sum_{j=0}^{\infty}a(j)\xi(j)$ which depends on the unknown values of a
stationary stochastic sequence $\xi(j)$ based on observations of the sequence
$\xi(j)+\eta(j)$ at points of time $j=-1,-2,\dots$, if they determine a
solution to the optimization problem (131). The function
$h^{0}(e^{i\lambda})$, determined by the formula (129), is minimax-robust
spectral characteristic of the optimal linear estimate of the functional
$A\xi$.

######  Theorem 5.3.

Let the spectral density $f(\lambda)$ be known and fixed and the spectral
density $g_{0}(\lambda)\in D_{g}^{0}$. Let the functions $f(\lambda)$ and
$g_{0}(\lambda)$ be such that the function $(f(\lambda)+g_{0}(\lambda))^{-1}$
is integrable and let the function $h_{g}(f,g_{0})$, determined by the formula
(136), be bounded. The spectral density $g_{0}(\lambda)$ is the least
favourable spectral densities in the class $D_{g}^{0}$ for the optimal
estimation of the functional $A\xi=\sum_{j=0}^{\infty}a(j)\xi(j)$ which
depends on the unknown values of a stationary stochastic sequence $\xi(j)$
based on observations of the sequence $\xi(j)+\eta(j)$ at points of time
$j=-1,-2,\dots$, if it is of the form

$g_{0}(\lambda)=\max\left\\{0,\alpha_{2}^{-1}\left|A(e^{i\lambda})f(\lambda)-\sum\limits_{j=0}^{\infty}((\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a})_{j}e^{ij\lambda}\right|-f(\lambda)\right\\}$

and the functions $f(\lambda)$ and $g_{0}(\lambda)$ determine a solution to
the optimization problem (131). The function $h^{0}(e^{i\lambda})$, determined
by the formula (129), is minimax-robust spectral characteristic of the optimal
linear estimate of the functional $A\xi$.

###  5.4 Least favourable spectral densities in the class $D_{v}^{u}\times
D_{\varepsilon}$

Consider the problem for the optimal estimation of the functional
$A\xi=\sum_{j=0}^{\infty}a(j)\xi(j)$ which depends on the unknown values of a
stationary stochastic sequence $\xi(j)$ based on observations of the sequence
$\xi(j)+\eta(j)$ at points of time $j=-1,-2,\dots$ in the case where the
spectral densities $f(\lambda)$, $g(\lambda)$ are from the class
$D=D_{v}^{u}\times D_{\varepsilon}$, where

$D_{v}^{u}=\left\\{f(\lambda)\left|v(\lambda)\leq f(\lambda)\leq
u(\lambda),\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f(\lambda)d\lambda\leq
P_{1}\right.\right\\},$
$D_{\varepsilon}=\left\\{g(\lambda)\left|g(\lambda)=(1-\varepsilon)g_{1}(\lambda)+\varepsilon
w(\lambda),\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}g(\lambda)d\lambda\leq
P_{2}\right.\right\\}.$

Here the spectral densities $v(\lambda),u(\lambda),g_{1}(\lambda)$ are known
and fixed and the densities $v(\lambda)$ and $u(\lambda)$ are bounded.

Let the densities $f^{0}(\lambda)\in D_{v}^{u}$, $g^{0}(\lambda)\in
D_{\varepsilon}$ determine bounded functions $h_{f}(f_{0},g_{0})$,
$h_{g}(f_{0},g_{0})$ with the help of formulas (135), (136). Then from the
condition $0\in\partial\Delta_{D_{f,g}}(f_{0},g_{0})$ we derive the following
equations that determine the least favourable spectral densities

$\left|A(e^{i\lambda})g_{0}(\lambda)+\sum\limits_{j=0}^{\infty}((\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a})_{j}e^{ij\lambda}\right|=(f_{0}(\lambda)+g_{0}(\lambda))(\gamma_{1}(\lambda)+\gamma_{2}(\lambda)+\alpha_{1}),$
(139)
$\left|A(e^{i\lambda})f_{0}(\lambda)-\sum\limits_{j=0}^{\infty}((\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a})_{j}e^{ij\lambda}\right|=(f_{0}(\lambda)+g_{0}(\lambda))(\varphi(\lambda)+\alpha_{2}),$
(140)

where $\gamma_{1}\leq 0$ and $\gamma_{1}=0$ in the case $f_{0}(\lambda)\geq
v(\lambda)$; $\gamma_{2}\geq 0$ and $\gamma_{2}=0$ in the case
$f_{0}(\lambda)\leq u(\lambda)$; $\varphi(\lambda)\leq 0$ and
$\varphi(\lambda)=0$ in the case
$g_{0}(\lambda)\geq(1-\varepsilon)g_{1}(\lambda)$.

The following theorems hold true.

######  Theorem 5.4.

Let the spectral densities $f_{0}(\lambda)\in D_{v}^{u}$ and
$g_{0}(\lambda)\in D_{\varepsilon}$ satisfy the minimality condition (121) and
let the functions $h_{f}(f_{0},g_{0})$ and $h_{g}(f_{0},g_{0})$, determined by
the formulas (135), (136), be bounded. The functions
$f_{0}(\lambda),g_{0}(\lambda)$, which give solution to the system of
equations (139), (140) are the least favourable spectral densities in the
class $D_{v}^{u}\times D_{\varepsilon}$, if they determine a solution to the
optimization problem (131). The function $h^{0}(e^{i\lambda})$, determined by
the formula (129), is minimax-robust spectral characteristic of the optimal
linear estimate of the functional $A\xi$.

######  Theorem 5.5.

Let the spectral density $f(\lambda)$ be known and fixed and let the spectral
density $g_{0}(\lambda)\in D_{\varepsilon}$. Let the function
$f(\lambda)+g_{0}(\lambda)$ satisfy the minimality condition (121), and let
the function $h_{g}(f,g_{0})$, determined by the formula (136), be bounded.
The spectral density $g_{0}(\lambda)$ is the least favourable spectral
densities in the class $D_{\varepsilon}$ for the optimal linear estimate of
the functional $A\xi$, based on observations of the sequence $\xi(j)+\eta(j)$
at points of time $j=-1,-2,\dots$, if it is of the form

$g_{0}(\lambda)=\max\left\\{(1-\varepsilon)g_{1}(\lambda),\alpha_{2}\left|A(e^{i\lambda})f(\lambda)-\sum\limits_{j=0}^{\infty}((\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a})_{j}e^{ij\lambda}\right|-f(\lambda)\right\\}$

and the functions $f(\lambda),g_{0}(\lambda)$ determine a solution to the
optimization problem (131). The function $h^{0}(e^{i\lambda})$, determined by
the formula (129), is minimax-robust spectral characteristic of the optimal
linear estimate of the functional $A\xi$.

###  5.5 Least favourable spectral densities in the class
$D_{2\varepsilon_{1}}\times D_{1\varepsilon_{2}}$

Consider the problem for the optimal estimation of the functional
$A\xi=\sum_{j=0}^{\infty}a(j)\xi(j)$ which depends on the unknown values of a
stationary stochastic sequence $\xi(j)$ based on observations of the sequence
$\xi(j)+\eta(j)$ at points of time $j=-1,-2,\dots$ in the case where the
spectral densities $f(\lambda)$, $g(\lambda)$ are from the class
$D_{2\varepsilon_{1}}\times D_{1\varepsilon_{2}}$, which describe the model of
”$\varepsilon$-neighbourhood” of spectral densities in the space $L_{2}\times
L_{1}$. Let

$D_{2\varepsilon_{1}}=\left\\{f(\lambda)\left|\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left|f(\lambda)-f_{1}(\lambda)\right|^{2}d\lambda\leq\varepsilon_{1}\right.\right\\}$

be the “$\varepsilon$-neighbourhood” in the space $L_{2}$ of a given bounded
spectral density $f_{1}(\lambda)$ and let

$D_{1\varepsilon_{2}}=\left\\{g(\lambda)\left|\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left|g(\lambda)-g_{1}(\lambda)\right|d\lambda\leq\varepsilon_{2}\right.\right\\}$

be the “$\varepsilon$-neighbourhood” in the space $L_{1}$ of a given bounded
spectral density $g_{1}(\lambda)$.

Let the spectral densities $f_{0}(\lambda)\in D_{2\varepsilon_{1}}$,
$g_{0}(\lambda)\in D_{1\varepsilon_{2}}$ determine the bounded functions
$h_{f}(f_{0},g_{0})$, $h_{g}(f_{0},g_{0})$ with the help of formulas (135),
(136). Then from the condition $0\in\partial\Delta_{D_{f,g}}(f_{0},g_{0})$ for
$D=D_{2\varepsilon_{1}}\times D_{1\varepsilon_{2}}$ we derive the following
equations that determine the least favourable spectral densities

$\left|A(e^{i\lambda})g_{0}(\lambda)+\sum\limits_{j=0}^{\infty}((\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a})_{j}e^{ij\lambda}\right|^{2}=(f_{0}(\lambda)+g_{0}(\lambda))^{2}(f_{0}(\lambda)-f_{1}(\lambda))\alpha_{1},$
(141)
$\left|A(e^{i\lambda})f_{0}(\lambda)-\sum\limits_{j=0}^{\infty}((\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a})_{j}e^{ij\lambda}\right|^{2}=(f_{0}(\lambda)+g_{0}(\lambda))^{2}\Psi(\lambda)\alpha_{2},$
(142)

where $\left|\Psi(\lambda)\right|\leq 1$ and
$\Psi(\lambda)=sign(g_{0}(\lambda)-g_{1}(\lambda))$, in the case
$g_{0}(\lambda)\neq g_{1}(\lambda)$, $\alpha_{1},\alpha_{2}$ are constants.

Equations (141), (142) with the optimization problem (131) and the normalizing
conditions

$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left|f(\lambda)-f_{1}(\lambda)\right|^{2}d\lambda=\varepsilon_{1}$
(143)
$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left|g(\lambda)-g_{1}(\lambda)\right|d\lambda=\varepsilon_{2}$
(144)

determine the least favourable spectral densities in the class
$D=D_{2\varepsilon_{1}}\times D_{1\varepsilon_{2}}$.

The following theorems hold true.

######  Theorem 5.6.

Let the spectral densities $f_{0}(\lambda)\in D_{2\varepsilon_{1}}$,
$g_{0}(\lambda)\in D_{1\varepsilon_{2}}$ satisfy the minimality condition
(121) and let the functions $h_{f}(f_{0},g_{0})$ and $h_{g}(f_{0},g_{0})$,
determined by the formulas (135), (136), be bounded. The spectral densities
$f_{0}(\lambda),g_{0}(\lambda)$, which give solution to the system of
equations (141)–(144) are the least favourable spectral densities in the class
$D_{2\varepsilon_{1}}\times D_{1\varepsilon_{2}}$, if they determine a
solution to the optimization problem (131). The function
$h^{0}(e^{i\lambda})$, determined by the formula (129), is minimax-robust
spectral characteristic of the optimal linear estimate of the functional
$A\xi$.

######  Theorem 5.7.

Let the spectral density $f(\lambda)$ be known, and let the spectral density
$g_{0}(\lambda)\in D_{1\varepsilon_{2}}$. Let the function
$f(\lambda)+g_{0}(\lambda)$ satisfy the minimality condition (121), and let
the function $h_{g}(f,g_{0})$, determined by the formula (136), be bounded.
The spectral density $g_{0}(\lambda)$ is the least favourable spectral
densities in the class $D_{1\varepsilon_{2}}$ for the optimal linear estimate
of the functional $A\xi$, based on observations of the sequence $\xi(j)$ at
points of time $j=-1,-2,\dots$, if it is of the form

$g_{0}(\lambda)=\max\left\\{g_{1}(\lambda),\alpha_{2}\left|A(e^{i\lambda})f(\lambda)-\sum\limits_{j=0}^{\infty}((\mathbb{B}^{0})^{-1}\mathbb{R}^{0}\mathbb{a})_{j}e^{ij\lambda}\right|-f(\lambda)\right\\}$

and the functions $f(\lambda),g_{0}(\lambda)$ determine a solution to the
optimization problem (131). The function $h^{0}(e^{i\lambda})$, determined by
the formula (129), is minimax-robust spectral characteristic of the optimal
linear estimate of the functional $A\xi$.

###  5.6 Conclusions

In this section we propose methods of solution of the problem of the mean-
square optimal linear estimation of the functional
$A\xi=\sum\limits_{j=0}^{\infty}a(j)\xi(j)$ which depends on the unknown
values of the stationary stochastic sequence $\xi(j)$. Estimates are based on
observations of the sequence $\xi(j)+\eta(j)$ at points of time
$j=-1,-2,\dots$. Here $\eta(j)$ is an uncorrelated with $\xi(j)$ stationary
sequence. We provide formulas for calculating the values of the mean square
error and the spectral characteristic of the optimal linear estimate of the
functional in the case of spectral certainty, where the spectral densities
$f(\lambda)$ and $g(\lambda)$ of the sequences $\xi(j)$ and $\eta(j)$ are
exactly known. In the case of spectral uncertainty, where the spectral
densities $f(\lambda)$ and $g(\lambda)$ are not known, but a set of admissible
spectral densities is given, the minimax approach is applied to estimation of
the functionals. We obtain formulas that determine the least favourable
spectral densities and the minimax spectral characteristics of the optimal
linear estimates of the functional $A\xi$ for concrete classes of admissible
spectral densities.

For the relative results on the mean-square optimal linear extrapolation of
linear functionals for stationary stochastic sequences and processes based on
observations with noise see papers by Moklyachuk [46] – [48], [52] – [57],
book by Moklyachuk and Masyutka [64].

##  6 Interpolation problem for stationary sequences

In this section we deal with the problem of the mean-square optimal estimation
of the linear functional $A_{N}\xi=\sum\limits_{j=0}^{N}a(j)\xi(j)$ which
depends on the unknown values of a stationary stochastic sequence
$\xi(j),j\in\mathbb{Z},$ based on observations of the sequence at points of
time $j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$. The problem is investigated
in the case of spectral certainty, where the spectral density of the
stationary stochastic sequence $\xi(j)$ is exactly known. In this case the
classical Hilbert space projection method of linear estimation of the
functional is applied. Formulas are derived for calculation the value of the
mean square error and the spectral characteristic of the mean-square optimal
estimate of the linear functional. In the case of spectral uncertainty, where
the spectral density of the stationary stochastic sequence is not exactly
known, but a class of admissible spectral densities is given the minimax-
robust procedure to linear estimation of the functional is applied. Relations
which determine the least favourable spectral densities and the minimax
spectral characteristics are proposed for some special sets of admissible
spectral densities.

###  6.1 The classical Hilbert space projection method of linear interpolation

Let $\xi(j),\,j\in\mathbb{Z}$, be a (wide sense) stationary stochastic
sequence. We will consider values of $\xi(j),\,j\in\mathbb{Z}$, as elements of
the Hilbert space $H=L_{2}(\Omega,\mathcal{F},P)$ of complex valued random
variables with zero first moment, $E\xi=0$, finite second moment,
$E|\xi|^{2}<\infty$, and the inner product
$\left(\xi,{\eta}\right)=E\xi\overline{\eta}$. The correlation function
$R(k)=\left(\xi(j+k),\xi(j)\right)=E\xi(j+k)\overline{\xi(j)}$ of the
stationary stochastic sequence $\xi(j),\,j\in\mathbb{Z}$, admits the spectral
representation [12]

$R(k)=\int\limits_{-\pi}^{\pi}e^{ik\lambda}F(d\lambda),$

where $F(d\lambda)$ is the spectral measure of the sequence. We will consider
stationary stochastic sequences with absolutely continuous spectral measures
and the correlation functions of the form

$R(k)=\,\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}e^{ik\lambda}f(\lambda)d\lambda,$

where $f(\lambda)$ is the spectral density function of the sequence $\xi(j)$
that satisfies the minimality condition

$\int\limits_{-\pi}^{\pi}f^{-1}(\lambda)d\lambda<\infty.$ (145)

Under this condition the error-free interpolation of the unknown values of the
sequence is impossible [73].

The stationary stochastic sequence $\xi(j),\,j\in\mathbb{Z}$, admits the
spectral representation [12, 22]

$\xi(j)=\int\limits_{-\pi}^{\pi}e^{ij\lambda}{Z}(d\lambda),$ (146)

where ${Z}(\Delta)$ is the orthogonal stochastic measure of the sequence such
that

$E{Z}(\Delta_{1})\overline{{Z}(\Delta_{2})}=F(\Delta_{1}\cap\Delta_{2})=\,\frac{1}{2\pi}\int_{\Delta_{1}\cap\Delta_{2}}f(\lambda)d\lambda.$

Consider the problem of the mean-square optimal estimation of the linear
functional

$A_{N}\xi=\sum\limits_{j=0}^{N}a(j)\xi(j)$

which depends on the unknown values of a stationary stochastic sequence
$\xi(j),j\in\mathbb{Z},$ from observations of the sequence at points of time
$j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$.

It follows from the spectral representation (146) of the sequence $\xi(j)$
that we can represent the functional $A_{N}\xi$ in the form

$A_{N}\xi=\int\limits_{-\pi}^{\pi}A_{N}(e^{i\lambda}){Z}(d\lambda),$ (147)

where

$A_{N}(e^{i\lambda})=\sum\limits_{j=0}^{N}a(j)e^{ij\lambda}.$

Denote by $H^{N}(\xi)$ the subspace of the Hilbert space
$H=L_{2}(\Omega,\mathcal{F},P)$ generated by elements
$\\{\xi(j):j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$. Let $L_{2}(f)$ be the
Hilbert space of complex-valued functions that are square-integrable with
respect to the measure whose density is $f(\lambda).$ Denote by $L_{2}^{N}(f)$
the subspace of $L_{2}(f)$ generated by functions
$\\{e^{ij\lambda},\,j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}\\}.$ The mean
square optimal linear estimate $\hat{A}_{N}\xi$ of the functional $A_{N}\xi$
from observations of the sequence $\xi(j)$ at points of time
$j\in\mathbb{Z}\\{0,1,\dots,N\\}$ is an element of the $H^{N}(\xi)$. It can be
represented in the form

$\hat{A}_{N}\xi=\,\int\limits_{-\pi}^{\pi}h(e^{i\lambda}){Z}(d\lambda),$ (148)

where $h(e^{i\lambda})\in L_{2}^{N}(f)$ is the spectral characteristic of the
estimate $\hat{A}_{N}\xi.$

The mean square error $\Delta(h;f)$ of the estimate $\hat{A}_{N}\xi$ is given
by the formula

$\Delta(h;f)=E\left|A_{N}\xi-\hat{A}_{N}\xi\right|^{2}=\,\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left|A_{N}(e^{i\lambda})-h(e^{i\lambda})\right|^{2}f(\lambda)d\lambda.$

The Hilbert space projection method proposed by A. N. Kolmogorov [24] makes it
possible to find the spectral characteristic $h(e^{i\lambda})$ and the mean
square error $\Delta(h;f)$ of the optimal linear estimate of the functional
$A_{N}\xi$ in the case where the spectral density $f(\lambda)$ of the sequence
$\xi(j),j\in\mathbb{Z},$ is exactly known and the minimality condition (145)
is satisfied. The spectral characteristic can be found from the following
conditions:

$\begin{split}1)&h(e^{i\lambda})\in L_{2}^{N}(f),\\\
2)&A_{N}(e^{i\lambda})-h(e^{i\lambda})\bot\,L_{2}^{N}(f).\end{split}$

It follows from the second condition that for any $\eta\in H^{N}(\xi)$ the
following equation should be satisfied

$\left(A_{N}\xi-\hat{A}_{N}\xi,\eta\right)=E(A_{N}\xi-\hat{A}_{N}\xi)\overline{\eta}=0.$

The last relation is equivalent to equations

$E(A_{N}\xi-\hat{A}_{N}\xi)\overline{\xi}_{k}=0,\,\,\,k\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}.$

By using representations (147), (148) and definition of the inner product in
the Hilbert space $H=L_{2}(\Omega,\mathcal{F},P)$ we get

$\begin{split}E\left(\int\limits_{-\pi}^{\pi}\left(A_{N}(e^{i\lambda})-h(e^{i\lambda})\right)Z(d\lambda)\cdot\int\limits_{-\pi}^{\pi}e^{-ik\lambda}\overline{Z(d\lambda)}\right)=\\\
=\,\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left(A_{N}(e^{i\lambda})-h(e^{i\lambda})\right)f(\lambda)e^{-ik\lambda}d\lambda=0,\,\,k\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}.\end{split}$

It follows from these equations that the function
$(A_{N}(e^{i\lambda})-h(e^{i\lambda}))f(\lambda)$ is of the form

$(A_{N}(e^{i\lambda})-h(e^{i\lambda}))f(\lambda)=C_{N}(e^{i\lambda}),$ (149)
$C_{N}(e^{i\lambda})=\sum\limits_{j=0}^{N}c(j)e^{ij\lambda},$

where $c(j),j=0,1,\dots,N,$ are unknown coefficients that we have to find.

From the relation (149) we deduce that the spectral characteristic
$h(e^{i\lambda})$ of the optimal linear estimate of the functional $A_{N}\xi$
is of the form

$h(e^{i\lambda})=\,A_{N}(e^{i\lambda})-C_{N}(e^{i\lambda})f^{-1}(\lambda).$
(150)

To find equations for determination the unknown coefficients
$c(j),j=0,1,\dots,N,$ we use the decomposition of the function
$f^{-1}(\lambda)$ into the Fourier series

$f^{-1}(\lambda)=\sum\limits_{m=-\infty}^{\infty}r_{m}e^{im\lambda},$ (151)

where $r_{m}$ are the Fourier coefficients of the function $f^{-1}(\lambda)$

$r_{m}=\,\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f^{-1}(\lambda)e^{-im\lambda}d\lambda.$

Inserting (151) into (150) we obtain the following representation of the
spectral characteristic

$\begin{split}h(e^{i\lambda})=\left(\sum\limits_{j=0}^{N}a(j)e^{ij\lambda}\right)-\left(\sum\limits_{j=0}^{N}c(j)e^{ij\lambda}\right)\left(\sum\limits_{m=-\infty}^{\infty}r_{m}e^{im\lambda}\right).\end{split}$
(152)

It follows from the first condition, $h(e^{i\lambda})\in L_{2}^{N}(f)$, which
determines the spectral characteristic, that the Fourier coefficients of the
function $h(e^{i\lambda})$ are equal to zero for $j=0,1,\dots,N$, namely

$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}h(e^{i\lambda})e^{-ij\lambda}d\lambda=0,\,\,j=0,1,\dots,N.$

Using the last relations and (152) we get the following system of equations
that determine the unknown coefficients $c(j),j=0,1,\dots,N,$

$\begin{split}a(0)-&\sum\limits_{j=0}^{N}c(j)r_{-j}=0;\\\
a(1)-&\sum\limits_{j=0}^{N}c(j)r_{1-j}=0;\\\ &\ldots\\\
a(N)-&\sum\limits_{j=0}^{N}c(j)r_{N-j}=0.\\\ \end{split}$ (153)

Denote by $\vec{\mathbb{a}}_{N}=(a(0),a(1),\ldots,a(N))$ and let
$\vec{\mathbb{c}}_{N}=(c(0),c(1),\ldots,c(N))$ be a vector constructed from
the unknown coefficients $c(j),j=0,1,\dots,N.$ Let $B_{N}$ be an
$(N+1)\times(N+1)$ matrix

$B_{N}=\left(\begin{array}[]{cccc}B_{00}&B_{01}&\ldots&B_{0s}\\\
B_{10}&B_{11}&\ldots&B_{1s}\\\ \vdots&\vdots&\ddots&\vdots\\\
B_{N0}&B_{N1}&\ldots&B_{NN}\end{array}\right),$

with elements that are Fourier coefficients of the function $f^{-1}(\lambda):$

$B_{kj}=\,\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f^{-1}(\lambda)e^{-i(k-j)\lambda}d\lambda=\,r_{k-j},\quad
k=0,1,\ldots,N,\,\,j=0,1,\ldots,N.$

Making use the introduced notations we can write equations (153) in the form
of equation

$\vec{\mathbb{a}}_{N}=B_{N}\vec{\mathbb{c}}_{N},$ (154)

Since the matrix $B_{N}$ is reversible [74], we get the formula

$\vec{\mathbb{c}}_{N}=B_{N}^{-1}\vec{\mathbb{a}}_{N}.$ (155)

Hence, the unknown coefficients $c(j),j=0,1,\dots,N,$ are calculated by the
formula

$c(j)=\left(B_{N}^{-1}\vec{\mathbb{a}}_{N}\right)_{j},$

where $\left(B_{N}^{-1}\vec{\mathbb{a}}_{N}\right)_{j}$ is the $j$ component
of the vector $B_{N}^{-1}\vec{\mathbb{a}}_{N},$ and the formula for
calculating the spectral characteristic of the estimate $\hat{A}_{N}\xi$ is of
the form

$\begin{split}h(e^{i\lambda})=\left(\sum\limits_{j=0}^{N}a(j)e^{ij\lambda}\right)-\left(\sum\limits_{j=0}^{N}\left(B_{N}^{-1}\vec{\mathbb{a}}_{N}\right)_{j}e^{ij\lambda}\right)\left(\sum\limits_{m=-\infty}^{\infty}r_{m}e^{im\lambda}\right).\end{split}$
(156)

The mean square error of the estimate of the function can be calculated by the
formula

$\begin{split}\Delta(h;f)&=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left|C_{N}(e^{i\lambda})\right|^{2}f^{-1}(\lambda)d\lambda=\\\
&=\int\limits_{-\pi}^{\pi}\left(\sum\limits_{k=0}^{N}c(k)e^{ik\lambda}\right)\left(\sum\limits_{j=0}^{N}\overline{c(j)}e^{-ij\lambda}\right)\left(\sum\limits_{m=-\infty}^{\infty}r_{m}e^{im\lambda}\right)d\lambda=\\\
&=\left<\vec{\mathbb{c}}_{N},\vec{\mathbb{c}}_{N}B_{N}\right>=\left<B_{N}^{-1}\vec{\mathbb{a}}_{N},\vec{\mathbb{a}}_{N}\right>,\end{split}$
(157)

where $\left<\cdot,\cdot\right>$ is the inner product in $C^{(N+1)}.$

Let us summarize our results and present them in the form of a theorem.

######  Theorem 6.1.

Let $\xi(j)$ be a stationary stochastic sequence with the spectral density
$f(\lambda)$ that satisfies the minimality condition (145). The spectral
characteristic $h(e^{i\lambda})$ and the mean square error $\Delta(h,f)$ of
the optimal linear estimate $\hat{A}_{N}\xi$ of the functional $A_{N}\xi$
based on observations of the sequence $\xi(j)$ at points of time
$j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$ can be calculated by formulas
(156), (157).

######  Example 1.

Consider the problem of linear estimation of the functional
$A_{1}\xi=a\xi(0)+b\xi(1)$ which depends on the unknown values $\xi(0)$,
$\xi(1)$ of the stationary stochastic sequence $\xi(j)$ from observations of
the sequence $\xi(j)$ at points $j\in\mathbb{Z}\backslash\\{0,1\\}.$ In this
case the spectral characteristic (152) of the estimate $\hat{A}_{1}\xi$ can be
calculated by the formula

$h(f)=a+be^{i\lambda}-\left(c(0)+c(1)e^{i\lambda}\right)f^{-1}(\lambda),$
(158)

where $f(\lambda)$ is a known spectral density, coefficients $c(0),c(1)$ are
determined by the system of equations

$\displaystyle a=c(0)\alpha+c(1)\beta,$ $\displaystyle
b=c(0)\bar{\beta}+c(1)\alpha,$

where

$\alpha={\frac{1}{2\pi}}\int_{\pi}^{\pi}f^{-1}(\lambda)\,d\lambda,$
$\beta={\frac{1}{2\pi}}\int_{\pi}^{\pi}e^{i\lambda}f^{-1}(\lambda)\,d\lambda,$

The matrix $B_{1}$ is of the form

$B_{1}=\left(\begin{array}[]{cc}\alpha&\beta\\\
\bar{\beta}&\alpha\end{array}\right),$

The determinant $D=det(B_{1})$ of the matrix $B_{1}$ is as follows
$D=det(B_{1})=\alpha^{2}-|\beta|^{2}$

We get the following formulas for calculating the coefficients $c(0),c(1)$

$c(0)=(a\alpha-b\beta)/(\alpha^{2}-|\beta|^{2}),$
$c(1)=(b\alpha-a\bar{\beta})/(\alpha^{2}-|\beta|^{2}).$

Thus, the unknown coefficients $c(0),c(1)$ in (158) are determined.

The mean square error of the estimate is calculated by the formula

$\Delta(f)=\left[\left(|a|^{2}+|b|^{2}\right)\alpha-(\bar{a}b+a\bar{b})\beta\right]/(\alpha^{2}-|\beta|^{2}).$
(159)

###  6.2 Minimax-robust method of interpolation

The traditional methods of estimation of the functional $A_{N}\xi$ which
depends on the unknown values of a stationary stochastic sequence $\xi(j)$ can
be applied in the case where the spectral density $f(\lambda)$ of the
considered stochastic sequence $\xi(j)$ is exactly known. In practice,
however, we do not have complete information on spectral density of the
sequence. For this reason we apply the minimax(robust) method of estimation of
the functional $A_{N}\xi$, that is we find an estimate that minimizes the
maximum of the mean square errors for all spectral densities from a given
class of admissible spectral densities $D$.

######  Definition 6.1.

For a given class of spectral densities $D$ a spectral density
$f_{0}(\lambda)\in D$ is called the least favourable in $D$ for the optimal
linear estimation of the functional $A_{N}\xi$ if the following relation holds
true

$\Delta\left(f_{0}\right)=\Delta\left(h\left(f_{0}\right);f_{0}\right)=\max\limits_{f\in
D}\Delta\left(h\left(f\right);f\right).$

######  Definition 6.2.

For a given class of spectral densities $D$ the spectral characteristic
$h^{0}(e^{i\lambda})$ of the optimal linear estimate of the functional
$A_{N}\xi$ is called minimax-robust if

$h^{0}(e^{i\lambda})\in H_{D}=\bigcap\limits_{f\in D}L_{2}^{N}(f),$
$\min\limits_{h\in H_{D}}\max\limits_{f\in
D}\Delta\left(h;f\right)=\sup\limits_{f\in D}\Delta\left(h^{0};f\right).$

It follows from the introduced definitions and the obtained formulas that the
following statement holds true.

######  Lemma 6.1.

The spectral density $f_{0}(\lambda)\in D$ is the least favourable in a class
of admissible spectral densities $D$ for the optimal linear estimation of the
functional $A_{N}\xi$ if the Fourier coefficients of the function
$f_{0}^{-1}(\lambda)$ define a matrix $B_{N}^{0}$ that determines a solution
to the optimization problem

$\max\limits_{f\in
D}\left<B_{N}^{-1}\vec{\mathbb{a}}_{N},\,\vec{\mathbb{a}}_{N}\right>=\left<(B_{N}^{0})^{-1}\vec{\mathbb{a}}_{N},\,\vec{\mathbb{a}}_{N}\right>.$
(160)

The minimax spectral characteristic $h^{0}=h(f_{0})$ can be calculated by the
formula (156) if $h(f_{0})\in H_{D}.$

The least favourable spectral density $f_{0}$ and the minimax spectral
characteristic $h^{0}$ form a saddle point of the function
$\Delta\left(h;f\right)$ on the set $H_{D}\times D$. The saddle point
inequalities

$\Delta\left(h;f_{0}\right)\geq\Delta\left(h^{0};f_{0}\right)\geq\Delta\left(h^{0};f\right)\hskip
28.45274pt\forall f\in D,\forall h\in H_{D}$

hold true if $h^{0}=h(f_{0})$ and $h(f_{0})\in H_{D},$ where $f_{0}$ is a
solution to the constrained optimization problem

$\tilde{\Delta}(f)=-\Delta\left(h^{0};f\right)=-\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\frac{\left|C_{N}^{0}(e^{i\lambda})\right|^{2}}{f_{0}^{2}(\lambda)}f(\lambda)d\lambda\rightarrow\inf,\hskip
28.45274ptf(\lambda)\in D,$ (161)

where

$C_{N}^{0}(e^{i\lambda})=\,\sum\limits_{j=0}^{N}\left((B_{N}^{0})^{-1}\vec{\mathbb{a}}_{N}\right)_{j}e^{ij\lambda}.$

The constrained optimization problem (161) is equivalent to the unconstrained
optimization problem

$\Delta_{D}(f)=\tilde{\Delta}(f)+\delta(f\left|D\right.)\rightarrow\inf,$

where $\delta(f\left|D\right.)$ is the indicator function of the set $D.$
Solution $f_{0}$ to this problem is characterized by the condition
$0\in\partial\Delta_{D}(f_{0}),$ where $\partial\Delta_{D}(f_{0})$ is the
subdifferential of the convex functional $\Delta_{D}(f)$ at point $f_{0}$.
This condition makes it possible to find the least favourable spectral
densities in some special classes of spectral densities $D$ [20], [70], [72].

Note, that the form of the functional $\Delta\left(h^{0};f\right)$ is
convenient for application the Lagrange method of indefinite multipliers for
finding solution to the problem (161). Making use the method of Lagrange
multipliers and the form of subdifferentials of the indicator functions we
describe relations that determine least favourable spectral densities in some
special classes of spectral densities (see books [13, 58, 64] for additional
details).

###  6.3 Least favourable spectral densities in the class $D_{0}^{-}$

Consider the problem of the optimal estimation of the functional
$A_{N}\xi=\sum_{j=1}^{N}a(j)\xi(j)$ which depends on the unknown values of a
stationary stochastic sequence $\xi(j)$ from observations of the sequence
$\xi(j)$ at points of time $j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$ in the
case where the spectral density is from the class

$D_{0}^{-}=\left\\{f(\lambda)\left|\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f^{-1}(\lambda)d\lambda\geq
P\right.\right\\}.$

Let the sequence $a(k),k=0,1,\dots,N,$ that determines the functional
$A_{N}\xi$, be strictly positive. To find solutions to the constrained
optimization problem (161) we use the Lagrange multipliers method. With the
help of this method we get the equation

$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left[\frac{\left|C_{N}^{0}(e^{i\lambda})\right|^{2}}{f_{0}^{2}(\lambda)}-p_{0}^{2}\frac{1}{f_{0}^{2}(\lambda)}\right]\rho(f(\lambda))d\lambda=0,$

where $p_{0}^{2}$ is a constant (the Lagrange multiplier), $\rho(f(\lambda))$
is a variation of the function $f(\lambda)$. From a generalization of the
Lagrange lemma we get that the Fourier coefficients of the function
$f_{0}^{-1}$ satisfy the equation

$\left|\sum\limits_{k=0}^{N}c(k)e^{ik\lambda}\right|^{2}=p_{0}^{2},$ (162)

where $c(k),k=0,1,\dots,N,$ are components of the vector
$\vec{\mathbb{c}}_{N}$ that satisfies the equation

$B_{N}^{0}\vec{\mathbb{c}}_{N}=\vec{\mathbb{a}}_{N},$ (163)

the matrix $B_{N}^{0}$ is determined by the Fourier coefficients of the
function $f_{0}^{-1}(\lambda)$

$B_{N}^{0}(k,j)=\,\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f_{0}^{-1}(\lambda)e^{-i(k-j)\lambda}d\lambda=r_{k-j}^{0},$
$\begin{split}&k=0,1,\ldots,N,\\\ &j=0,1,\ldots,N.\\\ \end{split}$

The Fourier coefficients $r_{k}=r_{-k},k=0,1,\dots,N,$ satisfy both equation
(162) and equation (163). These coefficients can be found from the equation

$B_{N}^{0}\vec{\mathbb{p}}_{N}^{0}=\vec{\mathbb{a}}_{N},\quad\vec{\mathbb{p}}_{N}^{0}=(p_{0},0,\ldots,0).$

The last relation can be presented in the form of the system of equations

$r_{k}p_{0}=a(k),\,k=0,1,\dots,N.$

From the first equation of the system (for $k=0$) we find the unknown value

$p_{0}=a(0)(r_{0})^{-1}.$

It follows from the extremum condition (160) and the restriction on the
spectral densities from the class $D_{0}^{-}$ that the Fourier coefficient

$r_{0}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f_{0}^{-1}(\lambda)d\lambda=P.$

Thus,

$r_{k}=Pa(k)a^{-1}(0),\,\,k=0,1,\dots,N.$

We can represent the function $f_{0}^{-1}(\lambda)$ in the form

$f_{0}^{-1}(\lambda)=\sum_{k=-N}^{N}r_{|k|}e^{ik\lambda}.$

Since the sequence $a(k),k=0,1,\dots,N,$ is strictly positive, the sequence
$r_{k},k=0,1,\ldots,N$, is also strictly positive and the function
$f_{0}^{-1}(\lambda)$ is positive, so it can be represented in the form [25]

$f_{0}^{-1}(\lambda)=\left|\sum\limits_{k=0}^{N}\gamma_{k}e^{-ik\lambda}\right|^{2},\,\,\lambda\in\left[-\pi,\pi\right],$

Hence, $f_{0}(\lambda)$ is the spectral density of the autoregressive
stochastic sequence of order $N$ generated by the equation

$\sum\limits_{k=0}^{N}\gamma_{k}\xi(n-k)=\varepsilon_{n},$ (164)

where $\varepsilon_{n}$ is a “white noise” sequence.

The minimax spectral characteristic $h(f_{0})$ of the optimal linear estimate
of the functional $A_{N}\xi$ can be calculated by the formula (150), where

$C_{N}(e^{i\lambda})=\sum\limits_{k=0}^{N}c(k)e^{ik\lambda}=p_{0}=P^{-1}a(0),$

namely

$\begin{gathered}h(f_{0})=\sum\limits_{k=0}^{N}a(k)e^{ik\lambda}-P^{-1}a(0)\sum\limits_{k=-N}^{N}r_{k}e^{ik\lambda}=\sum\limits_{k=1}^{N}a(k)e^{-ik\lambda}.\end{gathered}$
(165)

Let the sequence $a(N),a(N-1),\dots,a(0)$ that determines the functional
$A_{N}\xi$ be strictly positive. In this case the Fourier coefficients
$r_{k}=r_{-k},k=0,1,\dots,N,$ can be found from the equation

$B_{N}^{0}\vec{\mathbb{p}}_{N}^{0}=\vec{\mathbb{a}}_{N},\quad\vec{\mathbb{p}}_{N}^{0}=(0,\ldots,0,p_{0}).$

From this equation we find the unknown values of

$r_{k}=Pa(N-k)a^{-1}(N),\,\,k=0,1,\dots,N.$

The minimax spectral characteristic $h(f_{0})$ of the optimal linear estimate
of the functional $A_{N}\xi$ can be calculated by the formula

$h(f_{0})=\sum\limits_{k=0}^{N}a(k)e^{ik\lambda}-P^{-1}a(N)e^{iN\lambda}\sum\limits_{k=-N}^{N}r_{|k|}e^{ik\lambda}=\sum\limits_{k=1}^{N}a(N-k)e^{-i(N+k)\lambda}.$
(166)

Summing up our reasoning we come to conclusion that the following theorem
holds true.

######  Theorem 6.2.

The least favourable in the class $D_{0}^{-}$ spectral density for the optimal
linear estimation of the functional $A_{N}\xi$ determined by strictly positive
sequence $a(0),a(1),\dots,a(N)$ is the spectral density of the autoregressive
sequence (164) whose Fourier coefficients are
$r_{k}=r_{-k}=Pa(k)a^{-1}(0),k=0,1,\dots,N$. The minimax spectral
characteristics $h(f_{0})$ is given by formula (165). The least favourable in
the class $D_{0}^{-}$ spectral density for the optimal linear estimation of
the functional $A_{N}\xi$ determined by strictly positive sequence
$a(N),a(N-1),\dots,a(0)$ is the spectral density of the autoregressive
sequence whose Fourier coefficients are
$r_{k}=r_{-k}=Pa(N-k)a^{-1}(N),k=0,1,\dots,N$. The minimax spectral
characteristics $h(f_{0})$ is given by formula (166).

######  Example 2.

Consider the problem of the optimal linear estimation of the functional
$A_{1}\xi=a\xi(0)+b\xi(1)$ which depends on the unknown values $\xi(0)$,
$\xi(1)$ of the stationary stochastic sequence
$\left\\{\xi(j):j\in\mathbb{Z}\right\\}$ from observations of the sequence at
points of time $\mathbb{Z}\backslash\left\\{0,1\right\\}.$

The least favourable spectral density in the class $D_{0}^{-}$ is of the form

$f_{0}(\lambda)=1\big{/}{|x+ye^{i\lambda}|^{2}},$

where

$x=\pm\left(P\left(1\pm\left(1-4(b/a)^{2}\right)^{1/2}\right)\Big{/}2\right)^{1/2},$
$y=\pm\left(P\left(1\mp\left(1-4(b/a)^{2}\right)^{1/2}\right)\Big{/}2\right)^{1/2}$

under the condition $|b/a|<1/2$. For example, in the case of $a=4$,
$b=\sqrt{3}$ the least favourable spectral density $f_{0}(\lambda)$ and the
minimax spectral characteristic are calculated by the formulas

$f_{0}(\lambda)=4\Big{/}P|\sqrt{3}+e^{i\lambda}|^{2},$
$h(f_{0})=-\sqrt{3}e^{-i\lambda}.$

Under the condition $|b/a|>2$ the least favourable spectral density
$f_{0}(\lambda)$ is as follows

$f_{0}(\lambda)=1\Big{/}|x+ye^{i\lambda}|^{2},$

where

$x=\pm\left(P\left(1\pm\left(1-4(a/b)^{2}\right)^{1/2}\right)\Big{/}2\right)^{1/2},$
$y=\pm\left(P\left(1\mp\left(1-4(a/b)^{2}\right)^{1/2}\right)\Big{/}2\right)^{1/2}.$

For example, in the case of $a=\sqrt{3}$, $b=4$ the least favourable spectral
density $f_{0}(\lambda)$ and the minimax spectral characteristic are
calculated by the formulas

$f_{0}(\lambda)=4\Big{/}P|1+\sqrt{3}e^{i\lambda}|^{2},$
$h(f_{0})=-\sqrt{3}e^{2i\lambda}.$

###  6.4 Least favourable spectral densities in the class $D_{M}$

Consider the problem of the optimal estimation of the functional
$A_{N}\xi=\sum_{j=1}^{N}a(j)\xi(j)$ which depends on the unknown values of a
stationary stochastic sequence $\xi(j)$ from observations of the sequence
$\xi(j)$ at points of time $j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$ in the
case where the spectral density is from the set of spectral densities with
restrictions on the moments of the function $f^{-1}(\lambda).$ Let

$D_{M}=\left\\{f(\lambda)\left|\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f^{-1}(\lambda)\cos(m\lambda)d\lambda=r_{m}\right.,m=0,1,\ldots,M\right\\},$

where $r_{m},m=0,1,\ldots,M$ is a strictly positive sequence. There is an
infinite number of functions in the class $D_{M}$ [25] and the function

$f^{-1}(\lambda)=\sum\limits_{m=-M}^{M}r_{\left|m\right|}e^{im\lambda}>0,\,\,\,\lambda\in\left[-\pi,\pi\right].$

To find solutions to the constrained optimization problem (161) for the set
$D_{M}$ of admissible spectral densities we use the Lagrange multipliers
method and get the equation

$\left|\sum\limits_{k=0}^{N}c(k)e^{ik\lambda}\right|^{2}=\sum\limits_{m=0}^{M}\alpha_{m}\cos(m\lambda)=\left|\sum\limits_{m=0}^{M}p(m)e^{im\lambda}\right|^{2},$
(167)

where $\alpha_{m},m=0,1,\ldots,M$ are the Lagrange multipliers and
$c(k),k=0,\ldots,N$ are solutions to the equation
$B_{N}^{0}\vec{{c}}_{N}=\vec{{a}}_{N}$.

Consider two cases: $M\geq N$ and $M<N.$ Let $M\geq N.$ In this case the given
Fourier coefficients $r_{m}$ define the matrix $B_{N}^{0}$ and the
optimization problem (160) is degenerate. If we take $p(N+1)=\cdots=p(M)=0$
and components $(p(0),\dots,p(N))$ of the vector $\vec{{p}}_{N}$ find from the
equation $B_{N}^{0}\vec{{p}}_{N}=\vec{{a}}_{N}$ then the relation (167) holds
true. Thus the least favorable is every density $f(\lambda)\in D_{M}$ and the
density of the autoregression stochastic sequence

$f_{0}(\lambda)=1/{\sum\limits_{m=-M}^{M}r_{\left|m\right|}e^{im\lambda}}=1/{\left|\sum\limits_{k=0}^{M}\gamma_{k}e^{ik\lambda}\right|}$
(168)

is least favourable, too.

Let $M<N.$ Then the matrix $B_{N}$ is determined by the known $r_{m}$,
$m=0,1,\dots,M$ and the unknown $r_{m}$, $m=M+1,\dots,N$, Fourier coefficients
of the function $f^{-1}(\lambda).$

The unknown coefficients $p(k)$, $k=0,1,\dots,M$, and $r_{m}$,
$m=M+1,\dots,N$, can be found from the equation
$B_{N}\vec{{p}}_{N}^{0}=\vec{{a}}_{N}$ with
$p_{N}^{0}=(p(0),\dots,p(M),0,\dots,0)$, or from the equation
$B_{N}\vec{{p}}^{N}_{0}=\vec{{a}}_{N}$ with
$p_{0}^{N}=(0,\dots,0,p(M),p(M-1),\dots,p(0))$.

If the sequence $r_{m}$, $m=0,1,\dots,N$, that is constructed from the
strictly positive sequence $r_{m}$, $m=0,1,\dots,M$, and the calculated
coefficients $r_{m}$, $m=M+1,\dots,N$, is also strictly positive, then the
least favourable spectral density $f_{0}(\lambda)$ is determined by the
Fourier coefficients $r_{m}$, $m=0,1,\dots,N$, of the function
$f_{0}^{-1}(\lambda)$

$f_{0}(\lambda)=1/{\sum\limits_{m=-N}^{N}r_{\left|m\right|}e^{im\lambda}}=1/{\left|\sum\limits_{k=0}^{N}\gamma_{k}e^{ik\lambda}\right|}$
(169)

Let us summarize our results and present them in the form of a theorem.

######  Theorem 6.3.

The least favourable spectral density in the class $D_{M}$ for the optimal
linear estimation of the functional $A_{N}\xi$ in the case where $M\geq N$ is
the spectral density (168) of the autoregression stochastic sequence of order
$M$ determined by coefficients $r_{m},m=0,1,\ldots,M.$ If $M<N$ and solutions
$r_{m}$, $m=M+1,\dots,N$, to the equation
$B_{N}\vec{{p}}_{N}^{0}=\vec{{a}}_{N}$ with
$p_{N}^{0}=(p(0),\dots,p(M),0,\dots,0)$, or to the equation
$B_{N}\vec{{p}}^{N}_{0}=\vec{{a}}_{N}$ with
$p_{0}^{N}=(0,\dots,0,p(M),p(M-1),\dots,p(0))$ together with coefficients
$r_{m}$, $m=0,1,\dots,M$, form a strictly positive sequence, the least
favourable spectral density in $D_{M}$ is the density (169) of the
autoregression stochastic sequence of the order $N.$ The minimax
characteristic of the estimate is calculated by formula (156).

Similar statement holds true for the set of spectral densities

$D_{M,\cal
R}=\left\\{f(\lambda)\left|\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f^{-1}(\lambda)\cos(w\lambda)d\lambda=r_{m}\right.,\,m=0,1,\dots,M;\vec{r}_{{M}}\in\cal
R\right\\},$

where $\cal R$ is a convex compact which have a strictly positive sequence as
an interior point.

###  6.5 Least favourable spectral densities in the class $D_{v}^{u-}$

Consider the problem of the optimal estimation of the functional
$A_{N}\xi=\sum_{j=1}^{N}a(j)\xi(j)$ which depends on the unknown values of a
stationary stochastic sequence $\xi(j)$ from observations of the sequence
$\xi(j)$ at points of time $j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$ in the
case where the spectral density is from the set of spectral densities

$D_{v}^{u-}=\left\\{f(\lambda)\left|0\leq v(\lambda)\leq f(\lambda)\leq
u(\lambda),\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f^{-1}(\lambda)d\lambda=P\right.\right\\},$

where $v(\lambda),u(\lambda)$ are given bounded spectral densities. Let the
sequence $\\{a(0),a(1),\dots,a(N)\\}$ (or the sequence
$\\{a(N),a(N-1),\dots,a(0)\\}$) be strictly positive. To find solutions to the
constrained optimization problem (161) for the set $D_{v}^{u-}$ of admissible
spectral densities we use the condition $0\in\partial\Delta_{D}(f_{0})$. It
follows from the condition $0\in\partial\Delta_{D}(f_{0})$ for $D=D_{v}^{u-}$
that the Fourier coefficients of the function $f_{0}^{-1}$ satisfy both
equation

$B_{N}^{0}\vec{{c}}_{N}=\vec{{a}}_{N}$

and the equation

$\left|\sum\limits_{k=0}^{N}\left(\left(B_{N}^{0}\right)^{-1}\vec{{a}}_{N}\right)_{k}e^{ik\lambda}\right|^{2}=\psi_{1}(\lambda)+\psi_{2}(\lambda)+p_{0}^{-2},$

where $\psi_{1}(\lambda)\geq 0$ and $\psi_{1}(\lambda)=0$ if
$f_{0}(\lambda)\geq v(\lambda);$ $\psi_{2}(\lambda)\leq 0$ and
$\psi_{2}(\lambda)=0$ if $f_{0}(\lambda)\leq u(\lambda).$

Therefore, in the case where $v(\lambda)\leq f_{0}(\lambda)\leq u(\lambda)$,
the function $f_{0}^{-1}(\lambda)$ is of the form

$f_{0}^{-1}(\lambda)=\sum\limits_{k=-N}^{N}r_{{|k|}}e^{ik\lambda}=\left|\sum\limits_{k=0}^{N}\gamma_{k}e^{ik\lambda}\right|^{2},$

with $r_{{k}}=Pa(k)a^{-1}(0)$, in the case where the sequence
$\\{a(0),a(1),\dots,a(N)\\}$ is strictly positive, and with
$r_{{k}}=r_{{-k}}=Pa(N-k)a^{-1}(N)$, in the case where the sequence
$\\{a(N),a(N-1),\dots,a(0)\\}$ is strictly positive.

The least favourable in the class $D=D_{v}^{u-}$ is the density of the
autoregression stochastic sequence of the order $N$ if the following
inequality holds true

$v^{-1}(\lambda)\geq\sum\limits_{k=0}^{N}(r_{k}e^{ik\lambda}+r_{-k}e^{-ik\lambda})=\left|\sum\limits_{k=0}^{N}\gamma_{k}e^{ik\lambda}\right|^{2}\geq
u^{-1}(\lambda),\,\,\lambda\in\left[-\pi,\pi\right].$ (170)

In general case the least favourable density is of the form

$f_{0}(\lambda)=\max\left\\{v(\lambda),\min\left\\{u(\lambda),\left|p_{0}\sum\limits_{k=0}^{N}\left(\left(B_{N}^{0}\right)^{-1}\vec{{a}}_{N}\right)_{k}e^{ik\lambda}\right|^{-2}\right\\}\right\\}.$
(171)

The following theorem holds true.

######  Theorem 6.4.

If the sequence $\\{a(0),a(1),\dots,a(N)\\}$ is strictly positive and
coefficients $r_{{k}}=r_{{-k}}=Pa(k)a^{-1}(0)\,$, $k=0,\dots,\,N$, satisfy the
inequality (170) (or the sequence $\\{a(N),a(N-1),\dots,a(0)\\}$ is strictly
positive and coefficients $r_{{k}}=r_{{-k}}=Pa(N-k)\\*a^{-1}(N)$ satisfy the
inequality (170)), then the least favourable in the class $D_{v}^{u-}$
spectral density for the optimal linear estimate of the functional $A_{N}\xi$
is density of the autoregression stochastic sequence (164) of order $N$. The
minimax characteristic $h(f_{0})$ of the estimate can be calculated by the
formula (165). If the inequality (170) is not satisfied, then the least
favourable spectral density in $D_{v}^{u-}$ is determined by relation (171)
and the extremum condition (160). The minimax characteristic of the estimate
is calculated by formula (156).

###  6.6 Conclusions

In this section we propose methods of solution of the problem of the mean-
square optimal linear estimation of the functional
$A_{N}\xi=\sum\limits_{j=0}^{N}a(j)\xi(j)$ which depends on the unknown values
of the stationary stochastic sequence $\xi(j)$. Estimates are based on
observations of the sequence $\xi(j)$ at points
$j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$. We provide formulas for
calculating the values of the mean square error and the spectral
characteristic of the optimal linear estimate of the functional in the case of
spectral certainty, where the spectral density of the sequence $\xi(j)$ is
exactly known. In the case of spectral uncertainty, where the spectral density
is unknown, but a set of admissible spectral densities is given, the minimax
approach is applied to estimation of the functional. We obtain formulas that
determine the least favourable spectral densities and the minimax spectral
characteristics of the optimal linear estimates of the functional $A_{N}\xi$
for concrete classes of admissible spectral densities. It is shown that
spectral densities the autoregressive stochastic sequences are the least
favourable in some classes of spectral densities.

The minimax-robust approach to the problem of estimation of one missed value
of the stationary stochastic sequences based on convex optimization methods
was initiated in papers by Franke [9, 10]. See also papers by Hosoya [19],
Taniguchi [75], and survey by Kassam and Poor [23].

For the relative results on the mean-square optimal linear interpolation of
linear functionals for stationary stochastic sequences and processes see
papers by Moklyachuk [50] – [57], book by Moklyachuk and Masyutka [64], papers
by Moklyachuk and Sidei[67], Moklyachuk and Ostapenko[65].

##  7 Interpolation problem for stationary sequences from observations with
noise

In this section we consider the problem of the mean-square optimal estimation
of the linear functional $A_{N}\xi=\sum\limits_{j=0}^{N}a(j)\xi(j)$ which
depends on the unknown values of a stationary stochastic sequence
$\xi(j),j\in\mathbb{Z},$ from observations of the sequence $\xi(j)+\eta(j)$ at
points of time $j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$. The problem is
investigated in the case of spectral certainty, where the spectral densities
of the stationary stochastic sequences $\xi(j)$ and $\eta(j)$ are exactly
known. In this case the classical Hilbert space projection method of linear
estimation of the functional is applied. Formulas are derived for calculation
the value of the mean square error and the spectral characteristic of the
mean-square optimal estimate of the linear functional. In in the case of
spectral uncertainty, where the spectral densities of the stationary
stochastic sequences $\xi(j)$ and $\eta(j)$ are not exactly known, but classes
of admissible spectral densities are given, the minimax-robust procedure to
linear estimation of the functional is applied. Relations which determine the
least favourable spectral densities and the minimax spectral characteristics
are proposed for some special sets of admissible spectral densities.

###  7.1 The classical Hilbert space projection method of linear interpolation

Let $\xi(j),j\in\mathbb{Z}$, and $\eta(j),j\in\mathbb{Z}$, be (wide sense)
stationary stochastic sequences with zero mathematical expectations
$E\xi(j)=0$, $E\eta(j)=0$. The correlation functions
$R_{\xi}(k)=E\xi(j+k)\overline{\xi(j)}$ and
$R_{\eta}(k)=E\eta(j+k)\overline{\eta(j)}$ of the stationary stochastic
sequences $\xi(j),j\in\mathbb{Z}$, and $\eta(j),j\in\mathbb{Z}$, admit the
spectral representations [12]

$R_{\xi}(k)=\int\limits_{-\pi}^{\pi}e^{ik\lambda}F(d\lambda),\quad
R_{\eta}(k)=\int\limits_{-\pi}^{\pi}e^{ik\lambda}G(d\lambda),$

where $F(d\lambda)$ and $G(d\lambda)$ are the spectral measures of the
sequences. We will consider stationary stochastic sequences with absolutely
continuous spectral measures $F(d\lambda)$ and $G(d\lambda)$ and the
correlation functions of the form

$R_{\xi}(k)=\,\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}e^{ik\lambda}f(\lambda)d\lambda,\quad
R_{\eta}(k)=\,\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}e^{ik\lambda}g(\lambda)d\lambda,$

where $f(\lambda)$ and $g(\lambda)$ are the spectral density functions of the
sequences $\xi(j),j\in\mathbb{Z}$, and $\eta(j),j\in\mathbb{Z}$,
correspondingly.

We will suppose that the spectral density functions $f(\lambda)$ and
$g(\lambda)$ satisfy the minimality condition

$\int\limits_{-\pi}^{\pi}\frac{1}{f(\lambda)+g(\lambda)}d\lambda<\infty.$
(172)

Under this condition the error-free interpolation of the unknown values of the
sequence $\xi(j)+\eta(j)$ is impossible [73].

The stationary stochastic sequences $\xi(j)$ and $\eta(j)$ admit the spectral
representations [12, 22]

$\xi(j)=\int\limits_{-\pi}^{\pi}e^{ij\lambda}dZ_{\xi}(\lambda),\hskip
28.45274pt\eta(j)=\int\limits_{-\pi}^{\pi}e^{ij\lambda}dZ_{\eta}(\lambda),$

where $Z_{\xi}(d\lambda)$ and $Z_{\eta}(d\lambda)$ are orthogonal stochastic
measure of the sequences $\xi(j)$ and $\eta(j)$ such that

$E{Z_{\xi}}(\Delta_{1})\overline{{Z_{\xi}}(\Delta_{2})}=F(\Delta_{1}\cap\Delta_{2})=\,\frac{1}{2\pi}\int_{\Delta_{1}\cap\Delta_{2}}f(\lambda)d\lambda,$
$E{Z_{\eta}}(\Delta_{1})\overline{{Z_{\eta}}(\Delta_{2})}=G(\Delta_{1}\cap\Delta_{2})=\,\frac{1}{2\pi}\int_{\Delta_{1}\cap\Delta_{2}}g(\lambda)d\lambda.$

Consider the problem of the mean-square optimal estimation of the linear
functional

$A_{N}\xi=\sum\limits_{j=0}^{N}a(j)\xi(j)$

which depends on the unknown values of a stationary stochastic sequence
$\xi(j),j\in\mathbb{Z},$ from observations of the sequence $\xi(j)+\eta(j)$ at
points of time $j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$, where
$\eta(j),j\in\mathbb{Z},$ is uncorrelated with $\xi(j),j\in\mathbb{Z},$
stationary stochastic sequence.

It follows from the spectral decomposition of the sequence $\xi(j)$ that we
can represent the functional $A_{N}\xi$ in the form

$A_{N}\xi=\int\limits_{-\pi}^{\pi}A_{N}(e^{i\lambda}){Z}_{\xi}(d\lambda),$
(173)

where

$A_{N}(e^{i\lambda})=\sum\limits_{j=0}^{N}a(j)e^{ij\lambda}.$

Denote by $\hat{A}_{N}\xi$ the mean square optimal linear estimate of the
functional $A_{N}\xi$ from observations of the sequence $\xi(j)+\eta(j)$ at
points of time $j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$. Denote by
$\Delta(f,g)=E\left|A_{N}\xi-\hat{A}_{N}\xi\right|^{2}$ the mean square error
of the estimate $\hat{A}_{N}\xi$. To find the estimate $\hat{A}_{N}\xi$ we
will use the Hilbert space projection method proposed by A. N. Kolmogorov
[24]. We will consider random values $\xi(j),j\in\mathbb{Z}$, and
$\eta(j),j\in\mathbb{Z}$, as elements of the Hilbert space
$H=L_{2}(\Omega,\mathcal{F},P)$ of complex valued random variables with zero
first moment, $E\xi=0$, finite second moment, $E|\xi|^{2}<\infty$, and the
inner product $\left(\xi,{\eta}\right)=E\xi\overline{\eta}$.

Denote by $H^{N}(\xi+\eta)$ the subspace of the Hilbert space
$H=L_{2}(\Omega,\mathcal{F},P)$ generated by elements
$\\{\xi(j)+\eta(j):j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$. Denote by
$L_{2}(f+g)$ be the Hilbert space of complex-valued functions that are square-
integrable with respect to the measure whose density is
$f(\lambda)+g(\lambda).$ Denote by $L_{2}^{N}(f+g)$ the subspace of
$L_{2}(f+g)$ generated by functions
$\\{e^{ij\lambda},\,j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}\\}.$

The mean square optimal linear estimate $\hat{A}_{N}\xi$ of the functional
$A_{N}\xi$ from observations of the sequence $\xi(j)+\eta(j)$ at points of
time $j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$ is an element of the
$H^{N}(\xi+\eta)$. It can be represented in the form

$\hat{A}_{N}\xi=\,\int\limits_{-\pi}^{\pi}h(e^{i\lambda})(Z_{\xi}(d\lambda)+Z_{\eta}(d\lambda)),$
(174)

where $h(e^{i\lambda})\in L_{2}^{N}(f+g)$ is the spectral characteristic of
the estimate $\hat{A}_{N}\xi.$

The mean square error $\Delta(h;f,g)$ of the estimate $\hat{A}_{N}\xi$ is
given by the formula

$\Delta(h;f,g)=E\left|A_{N}\xi-\hat{A}_{N}\xi\right|^{2}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left|A_{N}(e^{i\lambda})-h(e^{i\lambda})\right|^{2}f(\lambda)d\lambda+\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left|h(e^{i\lambda})\right|^{2}g(\lambda)d\lambda.$

The Hilbert space projection method proposed by A. N. Kolmogorov [24] makes it
possible to find the spectral characteristic $h(e^{i\lambda})$ and the mean
square error $\Delta(h;f,g)$ of the optimal linear estimate of the functional
$A_{N}\xi$ in the case where the spectral densities $f(\lambda)$ and
$g(\lambda)$ of the sequences $\xi(j),j\in\mathbb{Z},$ and
$\eta(j),j\in\mathbb{Z},$ are exactly known and the minimality condition (172)
is satisfied. The spectral characteristic can be found from the following
conditions:

$\begin{split}1)&\hat{A}_{N}\xi\in H^{N}(\xi+\eta),\\\
2)&A_{N}\xi-\hat{A}_{N}\xi\bot H^{N}(\xi+\eta).\end{split}$

It follows from the second condition that for any
$j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$ the following equations should be
satisfied

$\begin{split}&E\left[\left(A_{N}\xi-\hat{A}_{N}\xi\right)\left(\overline{\xi(j)}+\overline{\eta(j)}\right)\right]=\\\
&=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left(A_{N}(e^{i\lambda})-h(e^{i\lambda})\right)e^{-ij\lambda}f(\lambda)d\lambda-\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}h(e^{i\lambda})e^{-ij\lambda}g(\lambda)d\lambda=0.\end{split}$

The last equations are equivalent to equations

$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left[A_{N}(e^{i\lambda})f(\lambda)-h(e^{i\lambda})(f(\lambda)+g(\lambda))\right]e^{-ij\lambda}d\lambda=0,\quad
j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}.$

It follows from these equations that the function
$\left[A_{N}(e^{i\lambda})f(\lambda)-h(e^{i\lambda})(f(\lambda)+g(\lambda))\right]$
is of the form

$A_{N}(e^{i\lambda})f(\lambda)-h(e^{i\lambda})(f(\lambda)+g(\lambda))=C_{N}(e^{i\lambda}),$
(175) $C_{N}(e^{i\lambda})=\sum\limits_{j=0}^{N}c(j)e^{ij\lambda},$

where $c(j),j=0,1,\dots,N,$ are unknown coefficients that we have to find.

From the relation (175) we deduce that the spectral characteristic
$h(e^{i\lambda})$ of the optimal linear estimate of the functional $A_{N}\xi$
is of the form

$\begin{split}h(e^{i\lambda})=&\frac{A_{N}(e^{i\lambda})f(\lambda)-C_{N}(e^{i\lambda})}{f(\lambda)+g(\lambda)}=\\\
=A_{N}(e^{i\lambda})-&\frac{A_{N}(e^{i\lambda})g(\lambda)+C_{N}(e^{i\lambda})}{f(\lambda)+g(\lambda)}.\end{split}$
(176)

It follows from the first condition, which determines the spectral
characteristic $h(e^{i\lambda})\in L_{2}^{N}(f+g)$ of the optimal linear
estimate of the functional $A_{N}\xi$, that the Fourier coefficients of the
function $h(e^{i\lambda})$ are equal to zero for $j=0,1,\dots,N$, namely

$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}h(e^{i\lambda})e^{-ij\lambda}d\lambda=0,\,\,j=0,1,\dots,N.$

Using the last relations and (176) we get the following system of equations

$\begin{split}\int\limits_{-\pi}^{\pi}\left(A_{N}(e^{i\lambda})\frac{f(\lambda)}{f(\lambda)+g(\lambda)}-\frac{C_{N}(e^{i\lambda})}{f(\lambda)+g(\lambda)}\right)e^{-ij\lambda}d\lambda=0,\hskip
28.45274ptj=0,1,\dots,N.\end{split}$

The last equations can be written in the form

$\begin{split}\sum_{k=0}^{N}a(k)\int\limits_{-\pi}^{\pi}\frac{e^{i(k-j)\lambda}f(\lambda)}{f(\lambda)+g(\lambda)}d\lambda-\sum\limits_{k=0}^{N}c(k)\int\limits_{-\pi}^{\pi}\frac{e^{i(k-j)\lambda}}{f(\lambda)+g(\lambda)}d\lambda=0,\,\,j=0,1,\dots,N.\end{split}$
(177)

Let us introduce the following notations

$R_{j,k}^{N}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}e^{-i(j-k)\lambda}\frac{f(\lambda)}{f(\lambda)+g(\lambda)}d\lambda,$
$B_{j,k}^{N}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}e^{-i(j-k)\lambda}\frac{1}{f(\lambda)+g(\lambda)}d\lambda,$
$Q_{j,k}^{N}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}e^{-i(j-k)\lambda}\frac{f(\lambda)g(\lambda)}{f(\lambda)+g(\lambda)}d\lambda.$

Making use the introduced notations we can write equations (177) in the form

$\begin{split}\sum\limits_{k=0}^{N}R^{N}_{j,k}a(k)=\sum\limits_{k=0}^{N}B^{N}_{j,k}c(k),\,\,j=0,1,\dots,N.\end{split}$

The derived equations can be written in the matrix form

$\begin{split}\mathbb{R}_{N}\mathbb{a}_{N}=\mathbb{B}_{N}\mathbb{c}_{N},\end{split}$

where $\mathbb{a}_{N}=(a(0),a(1),\dots,a(N))$ is a vector constructed from the
coefficients that determine the functional $A_{N}\xi$,
$\mathbb{c}_{N}=(c(0),c(1),\dots,c(N))$ is a vector constructed from the
unknown coefficients $c(k),k=0,1,\dots,N$, $\mathbb{B}_{N}$ and
$\mathbb{R}_{N}$ are linear operators in $\mathbb{C}^{N+1}$, which are
determined by matrices with elements $(\mathbb{B}_{N})_{j,k}=B_{j,k}^{N}$,
$(\mathbb{R}_{N})_{j,k}=R_{j,k}^{N}$, $j,k=0,1,\dots,N.$

Since the matrix $\mathbb{B}_{N}$ is reversible [74], we get the formula

$\mathbb{c}_{N}=\mathbb{B}_{N}^{-1}\mathbb{R}_{N}\mathbb{a}_{N},$ (178)

Hence, the unknown coefficients $c(j),j=0,1,\dots,N,$ are calculated by the
formula

$c(j)=\left(\mathbb{B}_{N}^{-1}\mathbb{R}_{N}\mathbb{a}_{N}\right)_{j},$

where $\left(\mathbb{B}_{N}^{-1}\mathbb{R}_{N}\mathbb{a}_{N}\right)_{j}$ is
the $j$-th component of the vector
$\mathbb{B}_{N}^{-1}\mathbb{R}_{N}\mathbb{a}_{N},$ and the formula for
calculating the spectral characteristic of the estimate $\hat{A}_{N}\xi$ is of
the form

$\begin{split}h(e^{i\lambda})=A_{N}(e^{i\lambda})\frac{f(\lambda)}{f(\lambda)+g(\lambda)}-\frac{\sum\limits_{k=0}^{N}(\mathbb{B}_{N}^{-1}\mathbb{R}_{N}\mathbb{a}_{N})_{k}e^{ik\lambda}}{f(\lambda)+g(\lambda)}.\end{split}$
(179)

The mean square error of the estimate of the function can be calculated by the
formula

$\begin{split}\Delta(h;f,g)=E\left|A_{N}\xi-\hat{A}_{N}\xi\right|^{2}&=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\frac{\left|A_{N}(e^{i\lambda})g(\lambda)+\sum\limits_{k=0}^{N}(\mathbb{B}_{N}^{-1}\mathbb{R}_{N}\mathbb{a}_{N})_{k}e^{ik\lambda}\right|^{2}}{(f(\lambda)+g(\lambda))^{2}}f(\lambda)d\lambda\\\
&+\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\frac{\left|A_{N}(e^{i\lambda})f(\lambda)-\sum\limits_{k=0}^{N}(\mathbb{B}_{N}^{-1}\mathbb{R}_{N}\mathbb{a}_{N})_{k}e^{ik\lambda}\right|^{2}}{(f(\lambda)+g(\lambda))^{2}}g(\lambda)d\lambda\\\
&=\langle\mathbb{R}_{N}\mathbb{a}_{N},\mathbb{B}_{N}^{-1}\mathbb{R}_{N}\mathbb{a}_{N}\rangle+\langle\mathbb{Q}_{N}\mathbb{a}_{N},\mathbb{a}_{N}\rangle,\end{split}$
(180)

where $\mathbb{Q_{N}}$ is a linear operator in $\mathbb{C}^{N+1}$, which is
determined by matrix with elements $(\mathbb{Q}_{N})_{j,k}=Q^{N}_{j,k}$,
$j,k=0,1,\dots,N.$

Let us summarize our results and present them in the form of a theorem.

######  Theorem 7.1.

Let $\xi(j)$ and $\eta(j)$ be uncorrelated stationary stochastic sequences
with the spectral densities $f(\lambda)$ and $g(\lambda)$ that satisfy the
minimality condition (172). The spectral characteristic $h(e^{i\lambda})$ and
the mean square error $\Delta(h;f,g)$ of the optimal linear estimate
$\hat{A}_{N}\xi$ of the functional $A_{N}\xi$ based on observations of the
sequence $\xi(j)+\eta(j)$ at points of time
$j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$ can be calculated by formulas
(179), (180).

###  7.2 Minimax-robust method of interpolation

The traditional methods of estimation of the functional $A_{N}\xi$ which
depends on the unknown values of a stationary stochastic sequence $\xi(j)$ can
be applied in the case where the spectral densities $f(\lambda)$ and
$g(\lambda)$ of the considered stochastic sequences $\xi(j)$ and $\eta(j)$ are
exactly known. In practise, however, we do not have complete information on
spectral densities of the sequences. For this reason we apply the
minimax(robust) method of estimation of the functional $A_{N}\xi$, that is we
find an estimate that minimizes the maximum of the mean square errors for all
spectral densities from the given class of admissible spectral densities $D$.

######  Definition 7.1.

For a given class of spectral densities $D=D_{f}\times D_{g}$ the spectral
densities $f_{0}(\lambda)\in D_{f}$, $g_{0}(\lambda)\in D_{g}$ are called the
least favourable in $D$ for the optimal linear estimation of the functional
$A_{N}\xi$ if the following relation holds true

$\Delta\left(f_{0},g_{0}\right)=\Delta\left(h\left(f_{0},g_{0}\right);f_{0},g_{0}\right)=\max\limits_{(f,g)\in
D_{f}\times D_{g}}\Delta\left(h\left(f,g\right);f,g\right).$

######  Definition 7.2.

For a given class of spectral densities $D=D_{f}\times D_{g}$ the spectral
characteristic $h^{0}(e^{i\lambda})$ of the optimal linear estimate of the
functional $A_{N}\xi$ is called minimax-robust if

$h^{0}(e^{i\lambda})\in H_{D}=\bigcap\limits_{(f,g)\in D_{f}\times
D_{g}}L_{2}^{N}(f+g),$ $\min\limits_{h\in H_{D}}\max\limits_{(f,g)\in
D}\Delta\left(h;f,g\right)=\sup\limits_{(f,g)\in
D}\Delta\left(h^{0};f,g\right).$

It follows from the introduced definitions and the obtained formulas that the
following statement holds true.

######  Lemma 7.1.

The spectral densities $f_{0}(\lambda)\in D_{f},$ $g_{0}(\lambda)\in D_{g}$
are the least favourable in the class of admissible spectral densities
$D=D_{f}\times D_{g}$ for the optimal linear estimate of the functional
$A_{N}\xi$ if the Fourier coefficients of the functions

$(f_{0}(\lambda)+g_{0}(\lambda))^{-1},\quad
f_{0}(\lambda)(f_{0}(\lambda)+g_{0}(\lambda))^{-1},\quad
f_{0}(\lambda)g_{0}(\lambda)(f_{0}(\lambda)+g_{0}(\lambda))^{-1}$

define operators $\mathbb{B}_{N}^{0},\mathbb{R}_{N}^{0},\mathbb{Q}_{N}^{0}$
that determine a solution to the optimization problem

$\begin{split}\max\limits_{(f,g)\in D_{f}\times
D_{g}}\langle\mathbb{R}_{N}\mathbb{a}_{N},\mathbb{B}_{N}^{-1}\mathbb{R}_{N}\mathbb{a}_{N}\rangle+\langle\mathbb{Q}_{N}\mathbb{a}_{N},\mathbb{a}_{N}\rangle\\\
=\langle\mathbb{R}_{N}^{0}\mathbb{a}_{N},(\mathbb{B}_{N}^{0})^{-1}\mathbb{R}_{N}^{0}\mathbb{a}_{N}\rangle+\langle\mathbb{Q}_{N}^{0}\mathbb{a}_{N},\mathbb{a}_{N}\rangle.\end{split}$
(181)

The minimax spectral characteristic $h^{0}=h(f_{0},g_{0})$ can be calculated
by the formula (179) if $h(f_{0},g_{0})\in H_{D}.$

The least favourable spectral densities $f_{0}(\lambda)$, $g_{0}(\lambda)$ and
the minimax spectral characteristic $h^{0}=h(f_{0},g_{0})$ form a saddle point
of the function $\Delta\left(h;f,g\right)$ on the set $H_{D}\times D$. The
saddle point inequalities

$\Delta\left(h;f_{0},g_{0}\right)\geq\Delta\left(h^{0};f_{0},g_{0}\right)\geq\Delta\left(h^{0};f,g\right)$
$\forall h\in H_{D},\forall f\in D_{f},\forall g\in D_{g}$

hold true if $h^{0}=h(f_{0},g_{0})$ and $h(f_{0},g_{0})\in H_{D},$ where
$(f_{0},g_{0})$ is a solution to the constrained optimization problem

$\sup\limits_{(f,g)\in D_{f}\times
D_{g}}\Delta\left(h(f_{0},g_{0});f,g\right)=\Delta\left(h(f_{0},g_{0});f_{0},g_{0}\right),$
(182)

where

$\begin{split}\Delta\left(h(f_{0},g_{0});f,g\right)&=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\frac{\left|A_{N}(e^{i\lambda})g_{0}(\lambda)+C_{N}^{0}(e^{i\lambda})\right|^{2}}{(f_{0}(\lambda)+g_{0}(\lambda))^{2}}f(\lambda)d\lambda\\\
&+\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\frac{\left|A_{N}(e^{i\lambda})f_{0}(\lambda)-C_{N}^{0}(e^{i\lambda})\right|^{2}}{(f_{0}(\lambda)+g_{0}(\lambda))^{2}}g(\lambda)d\lambda,\end{split}$
$C_{N}^{0}(e^{i\lambda})=\sum\limits_{j=0}^{N}((\mathbb{B}_{N}^{0})^{-1}\mathbb{R}_{N}^{0}\mathbb{a}_{N})_{j}e^{ij\lambda},$

The constrained optimization problem (182) is equivalent to the unconstrained
optimization problem

$\Delta_{D}(f,g)=-\Delta(h(f_{0},g_{0});f,g)+\delta(f,g\left|D_{f}\times
D_{g}\right.)\rightarrow\inf,$ (183)

where $\delta(f,g\left|D_{f}\times D_{g}\right.)$ is the indicator function of
the set $D=D_{f}\times D_{g}$. Solution $(f_{0},g_{0})$ to the problem (183)
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 $(f_{0},g_{0})$. This condition makes it
possible to find the least favourable spectral densities in some special
classes of spectral densities $D$ [20], [70], [72].

Note, that the form of the functional $\Delta(h(f_{0},g_{0});f,g)$ is
convenient for application the Lagrange method of indefinite multipliers for
finding solution to the problem (182). Making use the method of Lagrange
multipliers and the form of subdifferentials of the indicator functions we
describe relations that determine least favourable spectral densities in some
special classes of spectral densities (see books [13, 58, 64] for additional
details).

######  Lemma 7.2.

Let $(f_{0},g_{0})$ be a solution to the optimization problem (183). The
spectral densities $f_{0}(\lambda)$, $g_{0}(\lambda)$ are the least favourable
in the class $D=D_{f}\times D_{g}$, and the spectral characteristic
$h^{0}=h(f_{0},g_{0})$ is minimax for the optimal estimate of the functional
$A_{N}\xi$ if $h(f_{0},g_{0})\in H_{D}$.

###  7.3 Least favourable spectral densities in the class $D_{f}^{0}\times
D_{g}^{0}$

Consider the problem of the optimal estimation of the functional
$A_{N}\xi=\sum_{j=1}^{N}a(j)\xi(j)$ which depends on the unknown values of a
stationary stochastic sequence $\xi(j)$ from observations of the sequence
$\xi(j)+\eta(j)$ at points of time $j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$
in the case where the spectral densities $f(\lambda)$, $g(\lambda)$ are from
the class $D=D_{f}^{0}\times D_{g}^{0}$, where

$D_{f}^{0}=\left\\{f(\lambda)\left|\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f(\lambda)d\lambda\leq
P_{1}\right.\right\\},$
$D_{g}^{0}=\left\\{g(\lambda)\left|\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}g(\lambda)d\lambda\leq
P_{2}\right.\right\\}.$

Let the densities $f_{0}(\lambda)\in D_{f}^{0}$, $g_{0}(\lambda)\in D_{g}^{0}$
and the functions $h_{f}(f_{0},g_{0})$, $h_{g}(f_{0},g_{0})$, which are
determined by the relations

$h_{f}(f_{0},g_{0})=\frac{\left|A_{N}(e^{i\lambda})g_{0}(\lambda)+C_{N}^{0}(e^{i\lambda})\right|^{2}}{(f_{0}(\lambda)+g_{0}(\lambda))^{2}},$
(184)
$h_{g}(g_{0},g_{0})=\frac{\left|A_{N}(e^{i\lambda})f_{0}(\lambda)-C_{N}^{0}(e^{i\lambda})\right|^{2}}{(f_{0}(\lambda)+g_{0}(\lambda))^{2}},$
(185)

be bounded. In this case the functional

$\Delta(h(f_{0},g_{0});f,g)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}h_{f}(f_{0},g_{0})f(\lambda)d\lambda+\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}h_{g}(f_{0},g_{0})g(\lambda)d\lambda$

is linear and continuous on the space $L_{1}\times L_{1}$ and we can apply the
method of Lagrange multipliers to find solution to the optimization problem
(183). We get the following relations that determine the least favourable
spectral densities $f^{0}\in D^{0}_{f}$, $g^{0}\in D^{0}_{g}$

$\begin{split}&-\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}h_{f}(f_{0},g_{0})\rho(f(\lambda))d\lambda-\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}h_{g}(f_{0},g_{0})\rho(g(\lambda))d\lambda\\\
&+\alpha_{1}\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\rho(f(\lambda))d\lambda+\alpha_{2}\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\rho(g(\lambda))d\lambda=0,\end{split}$

where $\rho(f(\lambda))$ and $\rho(g(\lambda))$ are variations of the
functions $f(\lambda)$ and $g(\lambda)$, the constants $\alpha_{1}\geq 0$ and
$\alpha_{2}\geq 0$. From this relation we get that the least favourable
spectral densities $f_{0}(\lambda)\in D_{f}^{0}$, $g_{0}(\lambda)\in
D_{g}^{0}$ satisfy equations

$\left|A_{N}(e^{i\lambda})g_{0}(\lambda)+C_{N}^{0}(e^{i\lambda})\right|=\alpha_{1}(f_{0}(\lambda)+g_{0}(\lambda)),$
(186)
$\left|A_{N}(e^{i\lambda})f_{0}(\lambda)-C_{N}^{0}(e^{i\lambda})\right|=\alpha_{2}(f_{0}(\lambda)+g_{0}(\lambda)).$
(187)

Note, that $\alpha_{1}\neq 0$ in the case where

$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f_{0}(\lambda)d\lambda=P_{1},$

and $\alpha_{2}\neq 0$ in the case where

$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}g_{0}(\lambda)d\lambda=P_{2}.$

Summing up our reasoning we come to conclusion that the following theorem
holds true.

######  Theorem 7.2.

Let the spectral densities $f_{0}(\lambda)\in D_{f}^{0}$ and
$g_{0}(\lambda)\in D_{g}^{0}$ satisfy the minimality condition (172) and let
the functions $h_{f}(f_{0},g_{0})$ and $h_{g}(f_{0},g_{0})$, determined by the
formulas (184), (185), be bounded. The functions
$f_{0}(\lambda),g_{0}(\lambda)$, which give solution to the system of
equations (186), (187) are the least favourable spectral densities in the
class $D=D_{f}^{0}\times D_{g}^{0}$, if they determine a solution to the
optimization problem (181). The function $h^{0}(e^{i\lambda})$, determined by
the formula (176), is minimax-robust spectral characteristic of the optimal
linear estimate of the functional $A_{N}\xi$.

######  Theorem 7.3.

Let the spectral density $f(\lambda)$ be known and fixed, and let the spectral
density $g_{0}(\lambda)\in D_{g}^{0}$. Let the functions $f(\lambda)$ and
$g_{0}(\lambda)$ be such that the function $(f(\lambda)+g_{0}(\lambda))^{-1}$
is integrable and let the function $h_{g}(f,g_{0})$, determined by the formula
(185), be bounded. The spectral density $g_{0}(\lambda)$ is the least
favourable spectral densities in the class $D_{g}^{0}$ for the optimal linear
estimate of the functional $A_{N}\xi$, if it is of the form

$g_{0}(\lambda)=\max\left\\{0,\alpha_{2}^{-1}\left|A_{N}(e^{i\lambda})f(\lambda)-C_{N}^{0}(e^{i\lambda})\right|-f(\lambda)\right\\}$

and the functions $f(\lambda),g_{0}(\lambda)$ determine a solution to the
optimization problem (181). The function $h^{0}(e^{i\lambda})$, determined by
the formula (176), is minimax-robust spectral characteristic of the optimal
linear estimate of the functional $A_{N}\xi$.

######  Theorem 7.4.

Let the spectral density $f_{0}(\lambda)\in D_{f}^{0}$, let the function
$f_{0}^{-1}(\lambda)$ be integrable, and let the function $h(f_{0})$,
determined by the formula (150), be bounded. The spectral density
$f_{0}(\lambda)$ is the least favourable spectral densities in the class
$D_{f}^{0}$ for the optimal linear estimate of the functional $A_{N}\xi$,
based on observations of the sequence $\xi(j)$ at points of time
$j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$, if it satisfies the relation

$f_{0}(\lambda)=\alpha_{1}\left|C_{N}^{0}(e^{i\lambda})\right|$

and $f_{0}(\lambda)$ determine a solution to the optimization problem (160).
The function $h^{0}(e^{i\lambda})$, determined by the formula (150), is
minimax-robust spectral characteristic of the optimal linear estimate of the
functional $A_{N}\xi$.

###  7.4 Least favourable spectral densities in the class $D_{v}^{u}\times
D_{\varepsilon}$

Consider the problem of the optimal estimation of the functional
$A_{N}\xi=\sum_{j=1}^{N}a(j)\xi(j)$ which depends on the unknown values of a
stationary stochastic sequence $\xi(j)$ from observations of the sequence
$\xi(j)+\eta(j)$ at points of time $j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$
in the case where the spectral densities $f(\lambda)$, $g(\lambda)$ are from
the class $D=D_{v}^{u}\times D_{\varepsilon}$, where

$D_{v}^{u}=\left\\{f(\lambda)\left|v(\lambda)\leq f(\lambda)\leq
u(\lambda),\,\,\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f(\lambda)d\lambda\leq
P_{1}\right.\right\\},$
$D_{\varepsilon}=\left\\{g(\lambda)\left|g(\lambda)=(1-\varepsilon)g_{1}(\lambda)+\varepsilon
w(\lambda),\,\,\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}g(\lambda)d\lambda\leq
P_{2}\right.\right\\}.$

Here the spectral densities $v(\lambda),u(\lambda),g_{1}(\lambda)$ are known
and fixed and the densities $v(\lambda)$ and $u(\lambda)$ are bounded.

Let the densities $f^{0}(\lambda)\in D_{v}^{u}$, $g^{0}(\lambda)\in
D_{\varepsilon}$ determine the bounded functions $h_{f}(f_{0},g_{0})$,
$h_{g}(f_{0},g_{0})$ with the help of formulas (184), (185). Then from the
condition $0\in\partial\Delta_{D_{f,g}}(f_{0},g_{0})$ we derive the following
equations that determine the least favourable spectral densities

$\left|A_{N}(e^{i\lambda})g_{0}(\lambda)+C_{N}^{0}(e^{i\lambda})\right|=(f_{0}(\lambda)+g_{0}(\lambda))(\gamma_{1}(\lambda)+\gamma_{2}(\lambda)+\alpha_{1}^{-1}),$
(188)
$\left|A_{N}(e^{i\lambda})f_{0}(\lambda)-C_{N}^{0}(e^{i\lambda})\right|=(f_{0}(\lambda)+g_{0}(\lambda))(\varphi(\lambda)+\alpha_{2}^{-1}).$
(189)

Here $\gamma_{1}\leq 0$ and $\gamma_{1}=0$ in the case $f_{0}(\lambda)\geq
v(\lambda)$; $\gamma_{2}\geq 0$ and $\gamma_{2}=0$ in the case
$f_{0}(\lambda)\leq u(\lambda)$; $\varphi(\lambda)\leq 0$ and
$\varphi(\lambda)=0$ in the case
$g_{0}(\lambda)\geq(1-\varepsilon)g_{1}(\lambda)$.

The following theorems hold true.

######  Theorem 7.5.

Let the spectral densities $f_{0}(\lambda)\in D_{v}^{u}$, $g_{0}(\lambda)\in
D_{\varepsilon}$ satisfy the minimality condition (172) and let the functions
$h_{f}(f_{0},g_{0})$ and $h_{g}(f_{0},g_{0})$, determined by the formulas
(184), (185), be bounded. The functions $f_{0}(\lambda),g_{0}(\lambda)$, which
give solution to the system of equations (188), (189) are the least favourable
spectral densities in the class $D_{v}^{u}\times D_{\varepsilon}$, if they
determine a solution to the optimization problem (181). The function
$h^{0}(e^{i\lambda})$, determined by the formula (176), is minimax-robust
spectral characteristic of the optimal linear estimate of the functional
$A_{N}\xi$.

######  Theorem 7.6.

Let the spectral density $f(\lambda)$ be known, and let the spectral density
$g_{0}(\lambda)\in D_{\varepsilon}$. Let the function
$f(\lambda)+g_{0}(\lambda)$ satisfy the minimality condition (172), and let
the function $h_{g}(f,g_{0})$, determined by the formula (185), be bounded.
The spectral density $g_{0}(\lambda)$ is the least favourable spectral
densities in the class $D_{\varepsilon}$ for the optimal linear estimate of
the functional $A_{N}\xi$, if it is of the form

$g_{0}(\lambda)=\max\left\\{(1-\varepsilon)g_{1}(\lambda),\alpha_{2}\left|A_{N}(e^{i\lambda})f(\lambda)-C_{N}^{0}(e^{i\lambda})\right|-f(\lambda)\right\\}$

and the functions $f(\lambda),g_{0}(\lambda)$ determine a solution to the
optimization problem (181). The function $h^{0}(e^{i\lambda})$, determined by
the formula (176), is minimax-robust spectral characteristic of the optimal
linear estimate of the functional $A_{N}\xi$.

######  Theorem 7.7.

Let the spectral density $f_{0}(\lambda)\in D_{v}^{u}$, let the function
$f_{0}^{-1}(\lambda)$ be integrable, and let the function $h(f_{0})$,
determined by the formula (150), be bounded. The spectral density
$f_{0}(\lambda)$ is the least favourable spectral densities in the class
$D_{v}^{u}$ for the optimal linear estimate of the functional $A_{N}\xi$,
based on observations of the sequence $\xi(j)$ at points of time
$j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$, if it satisfies the relation

$f_{0}(\lambda)=\max\left\\{v(\lambda),\min\left\\{u(\lambda),\alpha_{1}\left|C_{N}^{0}(e^{i\lambda})\right|\right\\}\right\\}$

and $f_{0}(\lambda)$ determine a solution to the optimization problem (160).
The function $h^{0}(e^{i\lambda})$, determined by the formula (150), is
minimax-robust spectral characteristic of the optimal linear estimate of the
functional $A_{N}\xi$.

###  7.5 Least favourable spectral densities in the class
$D_{2\varepsilon_{1}}\times D_{1\varepsilon_{2}}$

Consider the problem of the optimal estimation of the functional
$A_{N}\xi=\sum_{j=1}^{N}a(j)\xi(j)$ which depends on the unknown values of a
stationary stochastic sequence $\xi(j)$ from observations of the sequence
$\xi(j)+\eta(j)$ at points of time $j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$
in the case where the spectral densities $f(\lambda)$, $g(\lambda)$ are from
the class $D_{2\varepsilon_{1}}\times D_{1\varepsilon_{2}}$, which describe
the models of “$\varepsilon$-neighbourhood” of spectral densities in the space
$L_{2}\times L_{1}$. Let

$D_{2\varepsilon_{1}}=\left\\{f(\lambda)\left|\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left|f(\lambda)-f_{1}(\lambda)\right|^{2}d\lambda\leq\varepsilon_{1}\right.\right\\}$

be “$\varepsilon$-neighbourhood” in the space $L_{2}$ of a given bounded
spectral density $f_{1}(\lambda)$, and let

$D_{1\varepsilon_{2}}=\left\\{g(\lambda)\left|\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left|g(\lambda)-g_{1}(\lambda)\right|d\lambda\leq\varepsilon_{2}\right.\right\\}$

be ”$\varepsilon$-neighbourhood” in the space $L_{1}$ of a given bounded
spectral density $g_{1}(\lambda)$.

Let the spectral densities $f_{0}(\lambda)\in D_{2\varepsilon_{1}}$,
$g_{0}(\lambda)\in D_{1\varepsilon_{2}}$ determine the bounded functions
$h_{f}(f_{0},g_{0})$, $h_{g}(f_{0},g_{0})$ with the help of formulas (184),
(185). Then from the condition $0\in\partial\Delta_{D_{f,g}}(f_{0},g_{0})$ for
$D=D_{2\varepsilon_{1}}\times D_{1\varepsilon_{2}}$ we derive the following
equations that determine the least favourable spectral densities

$\left|A_{N}(e^{i\lambda})g_{0}(\lambda)+C_{N}^{0}(e^{i\lambda})\right|^{2}=(f_{0}(\lambda)+g_{0}(\lambda))^{2}(f_{0}(\lambda)-f_{1}(\lambda))\alpha_{1},$
(190)
$\left|A_{N}(e^{i\lambda})f_{0}(\lambda)-C_{N}^{0}(e^{i\lambda})\right|^{2}=(f_{0}(\lambda)+g_{0}(\lambda))^{2}\Psi(\lambda)\alpha_{2},$
(191)

where $\left|\Psi(\lambda)\right|\leq 1$ and
$\Psi(\lambda)=sign(g_{0}(\lambda)-g_{1}(\lambda))$, in the case
$g_{0}(\lambda)\neq g_{1}(\lambda)$, $\alpha_{1},\alpha_{2}$ are constants.

Equations (190), (191) with the optimization problem (160) and the normalising
conditions

$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left|f(\lambda)-f_{1}(\lambda)\right|^{2}d\lambda=\varepsilon_{1}$
(192)
$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left|g(\lambda)-g_{1}(\lambda)\right|d\lambda=\varepsilon_{2}$
(193)

determine the least favourable spectral densities in the class
$D=D_{2\varepsilon_{1}}\times D_{1\varepsilon_{2}}$.

The following theorems hold true.

######  Theorem 7.8.

Let the spectral densities $f_{0}(\lambda)\in D_{2\varepsilon_{1}}$,
$g_{0}(\lambda)\in D_{1\varepsilon_{2}}$ satisfy the minimality condition
(172) and let the functions $h_{f}(f_{0},g_{0})$ and $h_{g}(f_{0},g_{0})$,
determined by the formulas (184), (185), be bounded. The spectral densities
$f_{0}(\lambda),g_{0}(\lambda)$, which give solution to the system of
equations (190)–(193) are the least favourable spectral densities in the class
$D_{2\varepsilon_{1}}\times D_{1\varepsilon_{2}}$, if they determine a
solution to the optimization problem (181). The function
$h^{0}(e^{i\lambda})$, determined by the formula (176), is minimax-robust
spectral characteristic of the optimal linear estimate of the functional
$A_{N}\xi$.

######  Theorem 7.9.

Let the spectral density $f(\lambda)$ be known, and let the spectral density
$g_{0}(\lambda)\in D_{1\varepsilon_{2}}$. Let the function
$f(\lambda)+g_{0}(\lambda)$ satisfy the minimality condition (172), and let
the function $h_{g}(f,g_{0})$, determined by the formula (185), be bounded.
The spectral density $g_{0}(\lambda)$ is the least favourable spectral
densities in the class $D_{1\varepsilon_{2}}$ for the optimal linear estimate
of the functional $A_{N}\xi$, if it is of the form

$g_{0}(\lambda)=\max\left\\{g_{1}(\lambda),\alpha_{2}^{-1}\left|A_{N}(e^{i\lambda})f(\lambda)-C_{N}^{0}(e^{i\lambda})\right|-f(\lambda)\right\\}$

and the functions $f(\lambda),g_{0}(\lambda)$ determine a solution to the
optimization problem (181). The function $h^{0}(e^{i\lambda})$, determined by
the formula (176), is minimax-robust spectral characteristic of the optimal
linear estimate of the functional $A_{N}\xi$.

######  Theorem 7.10.

Let the spectral density $f_{0}(\lambda)\in D_{2\varepsilon_{1}}$, let the
function $f_{0}^{-1}(\lambda)$ be integrable, and let the function $h(f_{0})$,
determined by the formula (150), be bounded. The spectral density
$f_{0}(\lambda)$ is the least favourable spectral densities in the class
$D_{2\varepsilon_{1}}$ for the optimal linear estimate of the functional
$A_{N}\xi$, based on observations of the sequence $\xi(j)$ at points of time
$j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$, if it satisfies the relation

$\left|C_{N}^{0}(e^{i\lambda})\right|^{2}=(f_{0}(\lambda))^{2}(f_{0}(\lambda)-f_{1}(\lambda))\alpha_{1}$

and $f_{0}(\lambda)$ determine a solution to the optimization problem (160).
The function $h^{0}(e^{i\lambda})$, determined by the formula (150), is
minimax-robust spectral characteristic of the optimal linear estimate of the
functional $A_{N}\xi$.

###  7.6 Conclusions

In this section we propose methods of solution of the problem of the mean-
square optimal linear estimation of the functional
$A_{N}\xi=\sum\limits_{j=0}^{N}a(j)\xi(j)$ which depends on the unknown values
of a stationary stochastic sequence $\xi(j)$. Estimates are based on
observations of the sequence $\xi(j)+\eta(j)$ at points
$j\in\mathbb{Z}\backslash\\{0,1,\dots,N\\}$, where $\eta(j)$ is an
uncorrelated with $\xi(j)$ stationary sequence. We provide formulas for
calculating values of the mean square error and the spectral characteristic of
the optimal linear estimate of the functional in the case of spectral
certainty where the spectral densities $f(\lambda)$ and $g(\lambda)$ of the
sequences $\xi(j)$ and $\eta(j)$ are exactly known. In the case of spectral
uncertainty where the spectral densities $f(\lambda)$ and $g(\lambda)$ are not
known, but a set of admissible spectral densities is given, the minimax
approach is applied. We obtain formulas that determine the least favourable
spectral densities and the minimax spectral characteristics of the optimal
linear estimates of the functional $A_{N}\xi$ for concrete classes of
admissible spectral densities.

For the relative results on the mean-square optimal linear interpolation of
linear functionals for stationary stochastic sequences and processes based on
observations with noise see papers by Moklyachuk [55] – [57], book by
Moklyachuk and Masyutka [64].

##  8 Conclusion Remarks

In the proposed paper we describe methods of solution of the problems of the
mean-square optimal linear extrapolation and interpolation of linear
functionals which depend on the unknown values of a stationary stochastic
sequence $\xi(k)$ based on observations of the sequence $\xi(k)$ as well as
observations of the sequence $\xi(k)+\eta(k)$, where $\eta(k)$ is an
uncorrelated with the sequence $\xi(k)$ stationary stochastic sequence. The
corresponding methods of solution of the problem of the mean-square optimal
linear filtering of stationary stochastic sequences are described in the paper
by Luz and Moklyachuk [33].

Following the Ulf Grenander [15] approach to investigation the problem of
optimal linear estimation of the functional which depends on the unknown
values of the stationary stochastic continuous parameter process we consider
the problem as a two-person zero-sum game in which the first player chooses a
stationary stochastic sequence $\xi(j)$ from the class $\Xi$ of stationary
stochastic sequences with $E\xi(j)=0$ and $E|\xi(j)|^{2}=1$ which maximizes
the value of the mean square error of estimate. The second player is looking
for an estimate of the linear functional which minimizes the value of the mean
square error. It is show that this game has equilibrium point. The maximum
error gives a one-sided moving average stationary sequence which is least
favourable in the given class of stationary sequences. The greatest value of
the error and the least favourable sequence are determined by the largest
eigenvalue and the corresponding eigenvector of the operator determined by
coefficients which determine the functionals. Note, that this approach can be
applied to a specific class of estimation problems.

The second approach to the estimation problems we applied is based on the
Kolmogorov [24] Hilbert space projection method which we apply in the case of
spectral certainty, where the spectral densities of the sequences $\xi(n)$ and
$\eta(n)$ are exactly known, and the convex optimization method proposed by
Franke [9, 10] which is applied in the case of spectral uncertainty, where the
spectral densities of the sequences are not exactly known, but, instead, a set
of admissible spectral densities is given. Formulas for calculation the mean-
square errors and the spectral characteristics of the optimal estimates of
functionals are derived in the case of spectral certainty. In the case of
spectral uncertainty the minimax-robust estimation method is applied. Formulas
that determine the least favourable spectral densities and the minimax-robust
spectral characteristics of the optimal linear estimates of the functionals
are derived.

In the papers by Moklyachuk [35] – [58] problems of extrapolation,
interpolation and filtering for stationary processes and sequences were
studied. The corresponding problems for vector-valued stationary sequences and
processes were investigated by Moklyachuk and Masyutka [59] – [64]. In the
articles by Dubovets’ka and Moklyachuk [3] \- [7] and in the book by
Golichenko and Moklyachuk [13] the minimax estimation problems were
investigated for another generalization of stationary processes – periodically
correlated stochastic sequences and stochastic processes. Luz and Moklyachuk
[26] – [32], [34] investigated the classical and minimax extrapolation,
interpolation and filtering problems for sequences and processes with $n$th
stationary increments. Investigation of the mean-square optimal linear
estimation problems for functionals of stationary stochastic sequences and
processes with missing observations is started in the papers by Moklyachuk and
Sidei [67], [68]. The minimax estimation problems for functionals of a
generalization of stationary processes – harmonizable stable stochastic
sequences and processes is started in the papers by Moklyachuk and Ostapenko
[65], [66]. For the results for functionals of random fields see the book by
Moklyachuk and Shchestyuk [69].

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