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\begin{document}

\title[Recovery of Functions from Inaccurate Information]{How to Recover  Functions
from Inaccurate Information}

\author{G.~G.~Magaril-Il'yaev, K.~Yu.~Osipenko}
\address{Moscow State Institute of Radio Engineering, Electronics and
Automation (Technology University)}
\address{MATI --- Russian State Technological University}
\maketitle

Here two problems are considered: optimal recovery of derivatives of
function from inaccurate information about its spectrum  and optimal
recovery of the solution of the differential equation from
inaccurate information about the initial data.








\section{Optimal recovery of derivatives}

In 1934 Hardy, Littlewood, and P\'olya proved that for all
integers $0<k<n$ the exact inequality
$$\|x^{(k)}\|_{\lt}\le\|x\|_{\lt}^{1-\frac kn}\|x^{(n)}\|_{\lt}^{
\frac kn}$$
holds for all functions $x\in\lt$ for which the $(n-1)$-st derivative is
locally absolute continuous on $\mathbb R$ and $x^{(n)}\in\lt$.

The Hardy--Littlewood--P\'olya inequality may be considered as the solution of the following extremal problem
$$\|x^{(k)}\|_{\lt}\to\max,\quad\|x\|_{\lt}\le\delta_1,\quad
\|x^{(n)}\|_{\lt}\le\delta_2.$$

We consider a slightly different extremal problem which closely connected with problems of signal reconstruction
$$\|x^{(k)}\|_{\lt}\to\max,\quad\|Fx\|_{\Ld}\le\delta,\quad
\|x^{(n)}\|_{\lt}\le1,$$
where $Fx$ is the Fourier transform of $x$ and $\Ds=[-\sigma,\sigma]$, $\sigma>0$. Namely, we consider the problem of optimal recovery of $x^{(k)}$ knowing the Fourier transform of $x$ giving with some error on $\Ds$.

Assume that $x\in\WR$,
$$\WR=\{x\in\lt:x^{(n-1)}\mbox{ is loc. abs. cont. },\ \|x^{(n)}\|_{\lt}\le1\},$$
and for any $x\in\WR$ we know a function $y\in\Ld$ such that
$$\|Fx-y\|_{\Ld}\le\delta.$$

The problem is to recover $x^{(k)}$ knowing $y$.

Any method of recovery is a map $m\colon\Ld\to\lt$. The error of such method is defined as follows
$$e_\sigma(m)=\sup_{\substack {x\in\WR,\ y\in\Ld\\\|Fx-y\|_{\Ld}\le\delta}}
\|x^{(k)}-m(y)\|_{\lt}.$$
We are interested in the value
$$E_{\sigma,2}=\inf_{m\colon\Ld\to\lt}e_\sigma(m),$$
which is called the error of optimal recovery and in the
method $\wm$, for which the infinum is attained that is in
the method $\widehat m$ for which
$$E_{\sigma,2}=e_\sigma(\wm).$$
We call this method the optimal recovery method.


Set
\begin{gather*}
\ws=\left(\dfrac
nk\right)^{\frac1{2(n-k)}}\left(\dfrac{2\pi}{\delta^2}\right)^{\frac1{2n}},\quad
\sigma_0=\min\{\sigma,\ws\},\\
\wa(\xi)=\left(1+\dfrac
n{n-k} \left(\dfrac nk\right)^{\frac
k{n-k}}\left(\dfrac\xi{\sigma_0}\right)^{2n} \right)^{-1}.
\end{gather*}

\begin{theorem}
%\begin{equation}\label{e1}
$$E_{\sigma,2}=\sigma_0^k\sqrt{\dfrac{n-k}{2 \pi n}\left(\dfrac kn\right)^{\frac
k{n-k}}\delta^2+\sigma_0^{2(k-n)}}.$$
%\end{equation}
For all $\alpha$ such that
\begin{equation}\label{ea}
|\alpha(\xi)-\wa(\xi)|\le\sqrt{\wa^2(\xi)+\wa(\xi)
\left(\left(\frac\xi{\sigma_0}\right)^{2(n-k)}-1\right)}
\end{equation}
the methods
%\begin{equation}\label{am}
$$m(y)(t)=\frac1{2\pi}\int_{\Ds}(i\xi)^k\alpha(\xi)y(\xi)e^{i\xi t}\,d\xi$$
%\end{equation}
are optimal.
\end{theorem}

For a fixed error of input data consider the error of optimal recovery
$E_{\sigma,2}$ as a function of $\sigma$. The larger interval $(-\sigma,\sigma)$ we take the less error we have. But beginning with $\ws$ the error $E_\sigma$ does
not change.

\begin{figure}[h]
$$\begin{picture}(300,170)
\put(0,10){\vector(1,0){200}}
\put(10,0){\vector(0,1){170}}
\put(188,0){$\sigma$}
\put(-14,160){$E_{\sigma,2}$}
\put(116,-2){$\ws$}
\qbezier(20,170)(40,50)(120,50)
\put(120,50){\line(1,0){60}}
\put(120,10){\line(0,1){40}}
\end{picture}$$
\caption{}\label{A}
\end{figure}


Consequently, for $\sigma>\ws$ the observed information becomes partially redundant. To avoid this case the following condition
\begin{equation}\label{up}
\delta^2\sigma^{2n}\le2\pi\left(\frac nk\right)^{\frac n{n-k}}
\end{equation}
should hold. This inequality may be considered as some ``uncertain
principle''.

From Theorem~1 we may obtain the following family of optimal methods.

\begin{corollary}
For all
$$0\le\theta\le\left(\frac{n-k}n\right)^{\frac1{2k}}
\left(\frac kn\right)^{\frac1{2(n-k)}}$$
the methods
\begin{multline*}
m(y)(t)=\frac1{2\pi}\int_{|\xi|\le\theta\sigma_0}(i\xi)^ky(\xi)
e^{i\xi t}\,d\xi\\
+\frac1{2\pi}\int_{\theta\sigma_0\le|\xi|\le\sigma_0}(i\xi)^k\alpha(\xi)
y(\xi)e^{i\xi t}\,d\xi,
\end{multline*}
%\end{equation}
where $\alpha$ any function satisfying \eqref{ea}, are optimal.
\end{corollary}

Note that obtained methods do not smooth the input data on the interval $[-\theta\sigma_0,\theta\sigma_0]$.

Now let us consider the case when the error of the input data is measured in $L_\infty$-norm.
Let $S$ be the Schwartz space of rapidly decreasing infinitely
differentiable functions on $\mathbb R$, $S'$ the dual space of
distributions, and $F\colon S'\to S'$ the Fourier transform. Let
$$C_\infty^n=\{\,x\in X_\infty^n:\|x^{(n)}\|_{\lt}\le1\,\},$$
where
$$X_\infty^n=\{\,x\in S':Fx\in\li,\ x^{(n)}\in\lt\,\}.$$
Assume that for any $x\in C_\infty$ we know a function $y\in\li$ such that
$$\|Fx-y\|_{\lis}\le\delta.$$

The problem again is to recover $x^{(k)}$ knowing $y$.
Now we are interested in the value
$$E_{\sigma,\infty}=\inf_{m\colon\lis\to\lt}\sup_{\substack {x\in C_\infty^n,\ y\in\lis\\
\|Fx-y\|_{\lis}\le\delta}}
\|x^{(k)}-m(y)\|_{\lt}$$
and in the optimal method $\wm$, that is, in the method for which the lower bound ia attained.

\begin{theorem}\label{22}
Set
$$\ws_\infty=(\pi(2n+1))^{\frac1{2n+1}}\delta^{-\frac2{2n+1}},\quad\widetilde\sigma_0=
\min(\sigma,\ws).$$
Then
$$E_{\sigma,\infty}=\begin{cases}\sqrt{\sigma^{-2(
n-k)}+\dfrac{2\delta^2(n-k)}{\pi(2k+1)(2n+1)}\sigma^{2k+1}},&\sigma<\ws_\infty,
\\[15pt]
\sqrt{\dfrac{2n+1}{2k+1}}\left(\dfrac1{\pi(2n+1)}\right)^{\frac{n-k}{2n+1}}
\delta^{\frac{2(n-k)}{2n+1}},&\sigma\ge\ws_\infty,\end{cases}$$
and the method
%\begin{equation}\label{ss}
$$\wm(y)(t)=\frac1{2\pi}\int_{-\widetilde\sigma_0}^{\widetilde\sigma_0}(i\xi)^k\biggl(1-\left(
\frac\xi{\widetilde\sigma_0}\right)^{2(n-k)}\biggr)y(\xi)e^{i\xi t}\,d\xi$$
%\end{equation}
is optimal.
\end{theorem}

It follows from this theorem that for a given $\delta$, starting from $\ws_\infty$,
further extension of the interval on which the Fourier transform of a
function from $C_\infty^n$ is given with error $\delta$ does not result in a decrease in
the recovery error. In other words, if the
relation
\begin{equation}\label{pn}
\delta^2\sigma^{2n+1}\le\pi(2n+1)
\end{equation}
between the input data and the size of the interval on which the data is
measured is violated, then the available information turns out to be
redundant. Inequality \eqref{pn} is an analog of uncertain
principle \eqref{up} in this case.

\section{Optimal recovery of the solution of the heat equation}

Now we consider the problem of optimal recovery of the solution of the heat equation
from inaccurate observations of the solution at the time moments $t_1,\ldots,t_n$.

Let $u$ be the solution of the heat equation in $\mathbb R^d$
%\begin{equation}\label{rex}
\begin{align*}
&u_t=\Delta u,\\
&u_{\big|t=0}=f(x),\quad f\in\ld.
\end{align*}
Assume that we know functions $y_j\in\ld$, $j=1,\ldots,n$, such
that
$$\|u(t_j,\cdot)-y_j(\cdot)\|_{\ld}\le\delta_j,\quad j=1,\ldots,n.$$
What is the best way to use this information to recover the
temperature distribution at the time $\tau\ne t_j$, $1\le
j\le n$, that is to recover the function $u(\tau,\cdot)$?

We admit as recovery methods arbitrary maps $m\colon(\ld)^n\to\ld$. For a fixed method $m$
the quantity
\begin{multline*}
e_\tau(\ld,\delta,m)
=\sup_{\substack{f,y_1,\ldots,y_n\in\ld\\
\|u(t_j,\cdot)-y_j(\cdot)\|_{\ld}\le\delta_j,\ j=1,\ldots,n}}
\|u(\tau,\cdot)-m(y)(\cdot)\|_{\ld},\\
\end{multline*}
where $u$ is the solution of the heat equation with the initial function $f$,
$\delta=(\delta_1,\ldots,\delta_n)$, and $y=(y_1,\ldots,y_n)$, is called the error of the
method $m$.

We are interested in the value
$$E_\tau(\ld,\delta)=\inf_{m\colon(\ld)^n\to\ld}e_\tau(\ld,\delta,m),$$
which is called the error of optimal recovery and in the
method $\wm$, for which the infinum is attained that is in
the method $\widehat m$ for which
$$E_\tau(\ld,\delta)=e_\tau(\ld,\delta,\wm).$$
We call this method the optimal recovery method.

To formulate the result we consider the set
$$M=\co\{\,(t_j,\log1/\delta_j),\,\,1\le j\le n\,\}+\{\,(t,0)\mid\,\,t\ge0\,\},$$
where $\co A$ is a convex hull of $A$. Define the function
$\theta(t)$, $t\in[t_1,\infty)$ as follows
$$\theta(t)=\max\{\,y:(t,y)\in M\,\}.$$
It is clear that $\theta$ is a polygonal line on $[t_1,\infty)$. Let $t_{s_j}$, $j=1,\ldots,r$,
be points of break of $\theta$.

For $\tau\in(t_{s_j},t_{s_{j+1}})$ put
$$
\wl_{s_j}=\frac{t_{s_{j+1}}-\tau}{t_{s_{j+1}}-t_{s_j}}\left(\frac{\delta_{s_{
j+1}}}{\delta_{s_j}}\right)^{\frac{2(\tau-t_{s_j})}{t_{s_{j+1}}-t_{s_j}}},\quad
\wl_{s_{j+1}}=\frac{\tau-t_{s_j}}{t_{s_{j+1}}-t_{s_j}}\left(\frac{\delta_{s_j}
}{\delta_{s_{j+1}}}\right)^{\frac{2(t_{s_{j+1}}-\tau)}{t_{s_{j+1}}-t_{s_j}}
}
$$



%\begin{align*}
%\wl_{s_j}&=\frac{t_{s_{j+1}}-\tau}{t_{s_{j+1}}-t_{s_j}}\left(\frac{\delta_{s_{
%j+1}}}{\delta_{s_j}}\right)^{\frac{2(\tau-t_{s_j})}{t_{s_{j+1}}-t_{s_j}}},\\
%\wl_{s_{j+1}} &=\frac{\tau-t_{s_j}}{t_{s_{j+1}}-t_{s_j}}\left(\frac{\delta_{s_j}
%}{\delta_{s_{j+1}}}\right)^{\frac{2(t_{s_{j+1}}-\tau)}{t_{s_{j+1}}-t_{s_j}}
%}.
%Here two problems are surveyed
%\end{align*}

\begin{theorem}
For all $\tau\ge t_1$ $$E_\tau(\ld,\delta)=e^{-\theta(\tau)}.$$
If $\tau\in(t_{s_j},t_{s_{j+1}})$, then for all $\gamma_j$ such that
\begin{equation}\label{gg}
\wl_{s_{j+1}}|\gamma_j(\xi)|^2e^{2|\xi|^2(t_{s_j}-\tau)}+\wl_{s_j}
|1-\gamma_j(\xi)|^2e^{2|\xi|^2(t_{s_{j+1}}-\tau)}\le\wl_{s_j}\wl_{s_{j+1}},
\end{equation}
all methods
$$m(y)(t)=(K_j*y_{s_j})(t)+(L_{j+1}*y_{s_{j+1}})(t),$$
where
$$FK_j(\xi)=\gamma_j(\xi)e^{|\xi|^2(t_{s_j}-\tau)},\quad FL_{j+1}(\xi)=(1-\gamma_j(\xi))e^{|\xi|^2(t_{s_{j+1}}-\tau)},$$
are optimal.

For $\tau=t_{s_j}$, $j=1,\ldots,r$, methods $m(y)(t)=y_{s_j}(t)$ are optimal and for $\tau>t_{s_r}$ the method
$$m(y)=F^{-1}(e^{-|\xi|^2(\tau-t_{s_r})}Fy_{s_r}(\xi))(x)$$
is optimal.
\end{theorem}

Condition \eqref{gg}
may be rewritten in the form
$$\left|\gamma_j(\xi)-\frac{\mu_1}{\mu_1+\mu_2}\right|\le
\frac{\sqrt{\mu_1\mu_2}\sqrt{\mu_1+\mu_2-1}}{\mu_1+\mu_2},$$
where
$$\mu_1=\wl_{s_j}e^{-2|\xi|^2(t_{s_j}-\tau)},\quad\mu_2=\wl_{s_{j+1}}
e^{-2|\xi|^2(t_{s_{j+1}}-\tau)}.$$
It can be shown that $\mu_1+\mu_2\ge1$ for all $\xi\in\mathbb R^d$. Thus, $\gamma_j(\xi)$ may be chosen from the interval
$$\left[\frac{\mu_1}{\mu_1+\mu_2}-\frac{\sqrt{\mu_1\mu_2}\sqrt{\mu_1+\mu_2-1}}
{\mu_1+\mu_2},\frac{\mu_1}{\mu_1+\mu_2}+\frac{\sqrt{\mu_1\mu_2}\sqrt{\mu_1+\mu_2-1}}
{\mu_1+\mu_2}\right].$$

Note that optimal method of recovery uses not more than two observations.
To find these observation we have to construct the set $M$ and the polygonal line
$\theta$. Then we have to find the nearest points of break of
$\theta$ to the point $\tau$. The observations at these points are those that
use in optimal method of recovery.


Note also that we can make more precise points of observation which are not on the
polygonal line. Suppose that for some $t_m$, $t_{s_j}<t_m<t_{s_{j+1}}$ and
$$\theta(t_m)>\log1/\delta_m.$$
Then optimal recovery method gives the error less than $\delta_m$. Indeed
$$\|u(t_m,\cdot)-\wm(y)(\cdot)\|_{\ld}\le e^{-\theta(t_m)}<\delta_m.$$

\begin{thebibliography}{11}
\bibitem{MO1} Magaril-Il'yaev~G.~G., Osipenko~K.~Yu. Optimal recovery of functions and their
derivatives from Fourier coefficients prescribed with an error, {\it Mat. Sb.} {\bf193} (2002),
79--100; English transl. in {\it Sbornic: Mathematics} {\bf193} (2002),
387--407.
\bibitem{MO2} Magaril-Il'yaev~G.~G., Osipenko~K.~Yu. Optimal recovery of functions and their derivatives from inaccurate
information about a spectrum and inequalities for derivatives, {\it Funkc.
analiz i ego prilozh.} {\bf37} (2003), 51--64; English transl. in {\it
Funct. Anal and Its Appl.}, {\bf37} (2003), 203--214.
\bibitem{MO3} Magaril-Il'yaev~G.~G., Osipenko~K.~Yu. Optimal recovery of the solution of the
heat equation from inaccurate data, {\it Mat. Sb.}, {\bf200}, 5 (2009) 37--54;
English transl. in {\it Sbornic: Mathematics} 200 (2009), 665--682.
\bibitem{MO4} Magaril-Il'yaev~G.~G., Osipenko~K.~Yu. On recovery of convolution type operators
from inaccurate information, {\it Trudy Mat. Inst. Steklov}, {\bf269} (2010), 181--192.
\bibitem{MO5} Magaril-Il'yaev~G.~G., Osipenko~K.~Yu. On optimal harmonic synthesis from inaccurate spectral data, {\it Funkc.
analiz i ego prilozh.}, 44:3 (2010), 76--79; English transl. in {\it Funct.
Anal and Its Appl.}, 44:3 (2010), 223--225.
\bibitem{MO6} Magaril-Il'yaev~G.~G., Osipenko~K.~Yu. Hardy--Littlewood--P\'olya inequality and
recovery of derivatives from inaccurate data, {\it Dokl. Akad. Nauk}, {\bf438}, 3
(2011), 300--302; English transl. in {\it Dokl. Math.}, {\bf83}, 3 (2011).

\end{thebibliography}


\end{document}
