\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 92, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/92\hfil Approximation for the transport problem]
{Approximation in the sense of Kato for the transport problem}

\author[M. A. Cherif, H. Emamirad\hfil EJDE-2009/92\hfilneg]
{Mohamed Amine Cherif, Hassan Emamirad}

\address{Mohamed Amine Cherif \newline
Laboratoire de  Math\'{e}matiques, Universit\'{e} de Poitiers,
Teleport  2, BP 179, 86960 Chassneuil du Poitou, Cedex, France.
\newline
I.P.E.I.S., boite postale 805, Sfax 3000, Tunisie}
\email{mohamedamin.cherif@yahoo.fr}

\address{Hassan Emamirad \newline
Laboratoire de  Math\'{e}matiques, Universit\'{e} de Poitiers.
Teleport  2, BP 179, 86960 Chassneuil du Poitou, Cedex, France }
\email{emamirad@math.univ-poitiers.fr}

\thanks{Submitted July 6, 2009. Published August 6, 2009.}
\subjclass[2000]{65M12, 65J10}
\keywords{Convergence in the Kato sense; transport semigroup}

\begin{abstract}
 Using Chernoff's Theorem, we present an approximation of
 the family $\{S(t): t\ge 0\}$ that converges
 in the sense of Kato to the transport semigroup.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

Let us recall the Chernoff's Theorem as it is given in \cite{Che}.

\begin{theorem}\label{chernoff}
Let $X$ be a Banach space and $\{V(t)\}_{t \ge 0}$ be a family
of contractions on $X$ with $V(0)=I$. Suppose that the
derivative $V'(0)f$ exists for all $f$ in a set $\mathcal{D}$ and
the closure $\Lambda$ of $V'(0)\big|_{\mathcal{D}}$ generates a
$C_0$-semigroup $S(t)$ of contractions. Then, for each
$f\in X$,
\begin{equation}\label{lim}
\lim_{n\to \infty}\|V(\frac tn)^n f-S(t)f\|=0,
\end{equation}
uniformly for $t$ in compact subsets of $\mathbb{R}_+$.
\end{theorem}

We will use the Chernoff's theorem to prove the following result.

\begin{theorem}\label{1}
Let $A$ be the generator of a $C_0$-semigroup $S_0(t)$ such that
$\|S_0(t)\| \le  e^{-\omega t}$ $(\omega \ge 0)$, and $B$ a bounded
perturbation operator such that $\|B\| \le \omega$; thus $A+B$
defined on $D(A)$ generates a $C_0$-semigroup $S(t)$ of
contractions. Then, the conclusion of \eqref{lim} holds
for  $V(t):=S_0(t)+\int_0^t S_0(s)Bds$.
\end{theorem}

\begin{proof} We remark that $V(0)=I$, $V'(0)f = (A+B)f$ for all
$f\in D(A)$ and finally $V(t)$ is a contraction. In fact,
\begin{align*}
\|V(t)\|
&\le \|S_0(t)\|+\|\int_0^t S_0(s)Bds\|\\
&\le  e^{-\omega t}+b\int_0^t  e^{-\omega s}ds
=\big(1-\frac b{\omega}\big) e^{-\omega t} + \frac b{\omega}\le 1,
\end{align*}
where $b=\|B\|$. Since all the assumptions of Theorem \ref{chernoff}
are fulfilled, the conclusion infers from this Theorem.
\end{proof}

In the next section, we define the convergence in the sense of Kato.
In the last section we construct the approximation spaces
convergence in the sense of Kato and we prove that an approximating
family of operators constructed by mean of
$V(t)$ in the transport problem converges in the sense of Kato to
the solution of this problem. This gives a new look to the
transport processes given by  Hejtmanek in
\cite{Hej}. In fact, Hejtmanek used these processes only for
Euler approximation of the transport equation, but we will show
in our forthcoming paper that these processes can be applied not
only to Euler schemes but also to Crank-Nicolson and Predictor-Corrector
algorithms.


\section{Convergence in the Kato sense}

In this article we give an approximation processus for the
transport equation not only in time but also in space.
For approximation in space we have to recall
the convergence in the sense of Kato (see \cite{Kat}).
 We say that a sequence of Banach
spaces $\{ (X_n,\|.\|_n) : n=1,2,\dots\}$  converges to a Banach
space $(X,\|.\|)$ {\it in the sense of Kato} and we write
$$
X_n \overset{ K} \to X
$$
if for any $n$ there is a linear operator
$P_n \in \mathcal{L} (X,X_n)$ (called an {\it approximating} operator)
satisfying the following
two conditions:
\begin{itemize}
\item[(K1)]
$\lim_{n\to \infty} \|P_nf\|_n = \|f\|$
 for $f\in X$;
\item[(K2)] for each  $ f_n \in X_n$,
 there exists $f^{(n)}\in X$ such that $ f_n = P_n f^{(n)}$
with $\|f^{(n)}\|\le C \|f_n\|_n$ ($C$ is independent of $n$).
\end{itemize}

Let $ X_n \overset {K} \to X$, $B_n \in \mathcal{L}(X_n)$
and $B \in \mathcal{L}(X)$.  We say that $B_n$  converges to $B$
{\it in the sense of Kato} and we write
$ B_n \overset{ K} \to B$
if $\lim_{n\to \infty}\|B_nP_nf - P_nBf
\|_n =0$ for any $f \in X$.  Let $A_n$ and $A$
be the generators of the $C_0$-semigroups $\{T_n(t)\}_{t \geq 0}
\subseteq \mathcal{L}(X_n)$ and $\{T(t)\}_{t \geq 0} \subseteq
 \mathcal{L}(X)$, respectively. Consider the
following three conditions:
\begin{itemize}
\item[(A)] {\bf (Consistency).} There is a complex number
$\lambda$ contained in the resolvent sets $\bigcap_{n\in \mathbb{N}}
\rho(A_n)$ and $\rho(A)$, respectively, such that
$$
(\lambda - A_n)^{-1} \overset{K}{\to} (\lambda -A)^{-1}.
$$
\item[(B)] {\bf (Stability).} There exists a positive constant $M$
and a real number $\omega$ such that
$$
\|T_n(t)\| \le Me^{\omega t},
$$
for any $t\ge 0$ and any $n\in \mathbb{N}$.
\item[(C)] {\bf (Convergence).} For any finite $T > 0$,
$$
{T_n(t)} \overset{ K} \to {T(t)}
$$
uniformly on $[0,T]$, i.e.
\begin{equation}\label{CV}
\lim_{n\to \infty}\sup_{t\in [0,T]}\|T_n(t)P_nf - P_nT(t)f
\|_n =0 \, \, \, \text{ for any   } f \in X.
\end{equation}
\end{itemize}

In \cite{Ush} one can retrieve the standard version of
the Lax equivalence theorem which says that the  conditions
(A) and (B) hold if and only if (C) holds.

\section{Approximation of  transport equation}

Here we consider a matter of particles, constituted of neutrons,
electrons, ions and photons. Each particle moves on a straight line
with constant velocity until it collides with another particle of the
supporting medium resulting in absorption, scattering or
multiplication. The unknown of the transport equation is the
particle density function $u(\mathbf{x},\mathbf{v},t)$. This is a function in the
phase space $(\mathbf{x},\mathbf{v})\in \Omega\times V \subset \mathbb{R}^{2n}$ at the time
$t \ge 0$, which belongs to its natural space $X=L^1(\Omega,V)$.
Actually, $\int_{\Omega\times V} u(\mathbf{x},\mathbf{v},t) \,dx \,dv$ designates the
total number  of particles in the whole space $\Omega\times V$ at
the time $t$. The general form of the transport problem is the
following
\begin{equation} \label{TP}
\begin{gathered}
\frac{\partial u}{\partial t}
=-\mathbf{v}\cdot\nabla {u} -\sigma(\mathbf{x},\mathbf{v})u
+\int_V p(\mathbf{x},\mathbf{v}',\mathbf{v})u(\mathbf{x},\mathbf{v}',t)d\mathbf{v}'\quad  \text{in }\Omega\times V;\\
 u(\mathbf{x},\mathbf{v},t)=0\quad \text{if }\mathbf{x}\cdot \mathbf{v} < 0,\quad \text{for all }
\mathbf{x}\in \partial\Omega;\\
 u(\mathbf{x},\mathbf{v}, 0)= f(\mathbf{x},\mathbf{v})\in X.
\end{gathered}
\end{equation}
In this equation which is known as {linear Boltzmann equation} the
first term of the right hand side $ -\mathbf{v}\cdot\nabla u(\mathbf{x},\mathbf{v},t)$
illustrates the movement of the classical group of the particles in
the absence of the absorption and production interactions. The second
term represents the lost of the particles caused by the diffusion or
absorption at the point $(\mathbf{x},\mathbf{v})$ in the phase space. Finally the
integral of the last term represents the production of the particles
at the point $(\mathbf{x},\mathbf{v})$ in the phase space. The kernel $p(\mathbf{x},\mathbf{v}',\mathbf{v})$
in this integral generates the transition of the states of particles
at the position $\mathbf{x}$ and having the velocity $\mathbf{v}'$ to the particles
at the same position having the velocity $\mathbf{v}$. The velocity space
$V$ is in general a spherical shell in $\mathbb{R}^n$, namely
$$
V=\{ \mathbf{v} \in \mathbb{R}^n: 0\le v_{\rm min} \le |\mathbf{v}|
\le v_{\rm max} \le +\infty\}.
$$

In this article. we study the particular feature of the transport
equation in which we replace $\Omega $  with $(-a,a)$
 and we take $V:=[-1,1]$. We assume that
 $\sigma$ is a strictly positive continuous  function with
 \begin{equation}\label{sigma}
0 < s_m\le \sigma(x)\le s_M  \text{ for almost any   } x \in (-a,a)
\end{equation}
and we replace the kernel $p(x,v,v')$ by $\frac 12 p(x)$
which is a positive continuous function
independent of $(v,v')$, such that
\begin{equation}\label{kernel}
0 < \sup_{x\in [-a,a]} p(x)=  k_M.
\end{equation}

With these assumptions the transport problem \eqref{TP}
 can be replaced by the following particular problem
\begin{equation} \label{TP1}
\begin{gathered}
\frac{\partial u}{\partial t}
=-v\cdot\nabla {u} -\sigma(x)u+\frac 12\int_{-1}^1
p(x)u(x,v,t)dv\quad \text{in }(-a,a) \times [-1,1];\\
 u(-a,v \ge 0,t)=0, \quad u(a,v \le 0,t)=0\quad \text{for all }t>0; \\
 u(x,v, 0)= f(\mathbf{x},v)\in L^1((-a,a) \times [-1,1]).
\end{gathered}
\end{equation}

\begin{remark}\label{A} {\rm
We denote the production term
$Af =\frac 12\int_{-1}^1 p(x)f(x,v)dv= p(x)Pf$, with
\begin{equation}\label{P}
Pf =\frac 12\int_{-1}^1 f(x,v)dv,
\end{equation}
which is a rank one projection on  $L^1((-a,a) \times [-1,1])$.
This space being generating we get $\|P\| =1$, and
$\|A\| =k_M$, since $\|A\| \le k_M$ and for the constant
function $p(x)=k_M$ we get the equality.
}\end{remark}

\begin{theorem}\label{sgps}
In the  Banach space $X=L^1((-a,a) \times [-1,1])$ let us
define the operators
$$
T_0f := -v{\partial f}/{\partial x},\quad
T_1f:= T_0f-\sigma(x)f, \quad
\widetilde Tf:=T_0f +Af, \quad
Tf:= T_1f+ Af\,,
$$
where $A$ is defined in Remark \ref{A}.
Any of these operators defined on
$D(T_0):=\{ f\in X: v{\partial f}/{\partial x}\in X, \; f(-a,v \ge 0)=0
 \text{ and }f(a,v \le 0)=0\}$
generates a $C_0$-semigroup which is given respectively by:
\begin{itemize}
\item[(0)] $U_0(t)$ which are  contractions;
\item[(1)]$U_1(t)$ with $\|U_1(t)\|\le  e^{-s_m t}$;
\item[(2)] $V(t)$ with $\|V(t)\|\le  e^{k_M t}$;
\item[(3)] $U(t)$ with $\|U(t)\|\le  e^{(k_M -s_m)t}$.
\end{itemize}
\end{theorem}

\begin{proof}
(0). For $t>0$ such that, $|x-tv| < a$, the semigroup
$U_0(t)f(x,v)=f(x-tv,v)$, satisfies
$\|U_0(t)f\|=\|f\|$ and if $x-tv < -a$ or $x-tv > a$,
 then $U_0(t)f(x,v)=0$.

(1). The $C_0$-semigroup generated by $T_1$ is
\begin{equation}\label{U_1}
[U_1(t)f](\mathbf{x},\mathbf{v}):= e^{-\int_0^t
\sigma(\mathbf{x}-s\mathbf{v})ds}f(\mathbf{x}-t\mathbf{v},\mathbf{v})
\end{equation}
and
$$
\int_{-a}^a\int_{-1}^1|[U_1(t)f](\mathbf{x},\mathbf{v})|\,dx\,dv
 \le  e^{-ts_m}\int_{-a}^a\int_{-1}^1|f(\mathbf{x}-t\mathbf{v},\mathbf{v})
|\,dx\,dv.
$$

(2). For $V(t)$ we will use the Dyson-Phillips formula:
$$
V_0(t)=U_0(t), \quad  V(t):= \sum_{n=0}^\infty V_n(t),
$$
where
$$
V_{n+1}(t) =\int_0^t V_0(t-s)AV_n(s) ds.
$$
Suppose that $\|V_n(s)\| \le (k_M s)^n/n!$, then by induction we get
\begin{align*}
\|V_{n+1}(t)f\|&\le  \int_0^t \|V_0(t-s)AV_n(s)f\| ds\\
&\le  \int_0^t \|AV_n(s)f\| ds
 \le  \int_0^t k_M \frac{(k_M s)^n}{n!}\|f\| ds\\
&=\frac{(k_M s)^{n+1}}{(n+1)!}\|f\|.
\end{align*}
in which we have used Remark \ref{A}. Consequently,
$$
\|V(t)\|\le  \sum_{n=0}^\infty \|V_n(t)\|
 \le \sum_{n=0}^\infty \frac{(k_M t)^n}{n!}= e^{k_M t}.
$$

(3). We argue as in \textbf{(2)}, but we replace the Dyson-Phillips
formula by
$U(t):= \sum_{n=1}^\infty U_n(t)$ and we deduce by induction
for $\|U_{n+1}(t)\| \le  e^{-ts_m}(k_M t)^n/n!$
that
$$
\|U(t)\|\le  \sum_{n=1}^\infty \|U_n(t)\|
\le \sum_{n=1}^\infty  e^{-ts_m}\frac{(k_M t)^{n-1}}{(n-1)!}
= e^{(k_M -s_m)t}.
$$
\end{proof}

Let us define the approximating spaces
 $X_n$ in this special case.
 We divide the phase space $(-a,a)\times [-1,1]$ into a finite
 number of cells by chopping the $x$ interval $(-a,a)$ into $2m_n$
 equal parts and the $v$ interval $[-1,1]$ into $2\mu_n$ equal
 parts; $h_n$ and $k_n$ are the lengths of these parts, that is,
 $$
h_n=\frac{a}{m_n},\quad  k_n=\frac{1}{\mu_n}.
$$
 Then each cell can be labeled by a pair of integers
$(i,j)\in \mathcal{N}$, where
$$
\mathcal{N} :=\{(i,j):
 i=-m_n,\dots,-1,0,1,\dots,m_n.\; j=-\mu_n,\dots,-1,0,1,\dots,\mu_n\}.
$$
The number  of the particles in cell
$\gamma(i,j)=[{ih_n}, (i+1)h_n]\times[jk_n,(j+1)k_n]$ is written
$\xi_{i,j}$.

We define the set of all real vectors $\xi_{i,j}$ as the Banach
space $X_n$ with the norm
$$
\|\xi\|_n = \sum_{i,j} |\xi_{i,j}|, \quad \xi\in X_n\,.
$$
At this point let us prove that the approximating space $X_n$
converges in the sense of Kato to $X$.

\begin{lemma}\label{P_n}
For $P_nf=\{\xi_{i,j}: (i,j)\in \mathcal{N}\}$ where
$$
\xi_{i,j}=\int_{ih_n}^{(i+1)h_n}\int_{jk_n}^{(j+1)k_n}f(x,v)\,dx\,dv,
$$
we have
\begin{itemize}
\item[(i)] $\|P_nf\|_n =\|f\|$ for all $0\le f\in X$;
\item[(ii)] $\|P_n\|_{\mathcal{L}(X,X_n)}=1$;
\item[(iii)] $\lim_{n\to \infty} \|P_nf\|_n = \|f\|
\quad \text{ for any  } f\in X$.
\end{itemize}
\end{lemma}

\begin{proof}
(i) For every $f(x,v) \ge 0$, we get
$$
\|P_nf\|_n =\sum_{i,j} \int_{ih_n}^{(i+1)h_n}
\int_{jk_n}^{(j+1)k_n}f(x,v)\,dx\,dv=\|f\|.
$$

 (ii) Since $\|P_nf\|_n \le \|f\|$,
(ii) follows from (i).

 (iii) Let $f\in C(\Omega\times V)$ the space of the continuous
functions on $\Omega\times V$. For any $\varepsilon >0$, there exists a
large $N >>1$, such that for
$n\ge N$ there exists a collection $\Gamma$ of small cells
$\gamma(i,j)$  so  that on each $\gamma(i,j) \in \Gamma$, $f$
has a constant sign and
$$
\Big|\int_{(-a,a)\times [-1,1]} |f(x,v)|\,dx\,dv
 -\sum_{\gamma(i,j)\in \Gamma} \int_{\gamma(i,j)}
 |f(x,v)|\,dx\,dv\Big| <\varepsilon,
$$
 which implies (iii) for the continuous functions and the
assumption follows from the density of $ C(\Omega\times V)$
in $X=L^1(\Omega,V)$.
\end{proof}

The condition (K1) follows from Lemma \ref{P_n}(iii)
and for the condition  (K2) we denote by $\chi_{i,j}$ the
characteristic function of the cell $\gamma(i,j)$, and
for any $\{\xi_{i,j}\}\in X_n$ we define $f^{(n)}\in X$
as $f^{(n)}(x)= \sum_{i,j} \frac{\xi_{i,j}}{h_nk_n}\chi_{i,j}$
and we have
$$
\int_{(-a,a)\times [-1,1]} |f^{(n)}(x)|\,dx\,dv
\le \sum_{i,j} \int_{\gamma(i,j)}
 \big|\frac{\xi_{i,j}}{h_nk_n}\chi_{i,j}\big|\,dx\,dv
= \sum_{i,j} |\xi_{i,j}|,
$$
since $\int_{\gamma(i,j)} \frac{\chi_{i,j}}{h_nk_n}\,dx\,dv= 1$.

In this section we consider the system \eqref{TP1}, with the notation
of Remark \ref{A}, $Af=pPf$, where $P$ is the projection defined in
\eqref{P}.

Here, we do not have at our disposition an explicit expression
of the semigroup as
$U_0(t)f(x,v)=f(x-tv,v)$ or
$U_1(t)f(x,v)= e^{-\int_{0}^{t}\sigma(x-sv)ds}f(x-tv,v)$,
but  we can introduce the operator
\begin{align}
[V(t)f](x,v)
&:= e^{-\int_{0}^{t}\sigma (x-sv)ds}f(x-tv, v)\nonumber\\
&\quad  +\frac{1}{2}\int_{0}^{t} e^{-\int_{0}^{s}\sigma (x
-rv)dr}p(x-sv)\int_{-1}^{1}f(x-sv,v^{\prime })dv^{\prime }ds\label{V1}\\
&=U_1(t)f+\int_0^tU_1(s)pPfds=U_1(t)f+\int_0^tU_1(s)Afds.\label{V2}
\end{align}
The operator $V(t)$ is not itself a semigroup as $U_0(t)$ or
$U_1(t)$, but it can act as the operator function $V(t)$
in Chernoff's theorem (Theorem \ref{chernoff}).

We approximate this operator by
\begin{equation}\label{U_n}
U_n(k\tau_n):=U_{1,n}(t)(I+\tau_n A_n)^k,
\end{equation}
where
\begin{equation}\label{A_n}
[A_n \xi]_{i,j}:=\frac {k_np_i}2\sum_{l=-\mu_n}^{\mu_n-1}\xi_{i,l},
\end{equation}
for every $j$, $-\mu_n\leq j\leq \mu_n-1$, with
$p_i= p(\theta)$, $\theta \in [ih_n,(i+1)h_n)$.
(In Remarks \ref{u_n} \textbf{(a)} below, we will explain the precise
feature of this approximation).

Now, let $U(t)$ be the transport semigroup defined in
Theorem \ref{sgps}.

\begin{theorem}\label{kato2}
Under the assumption  $2k_M < s_m $, we have the convergence
of $U_{n}(t)$ to $U(t)$ in the sense of Kato.
\end{theorem}

\begin{proof}
We have to prove that
\begin{equation}\label{katocon}\|U_{n}(t)P_nf-P_nU(t)f\|_n  \rightarrow 0
 ,\end{equation}
as $n\to \infty$.
First we prove that
\begin{equation}\label{SnUn}
U_{n}(k\tau_n)P_nf=P_nV(\tau_n)^kf.
\end{equation}
In fact,
\begin{align*}
P_nV(\tau_n)f
&= P_n\Big[ e^{-\int_{0}^{\tau_n}\sigma(x-sv) ds}f(x-\tau_nv,v)\\
&\quad+\frac{1}{2}\int_{0}^{\tau_n} e^{-\int_{0}^{s}\sigma(x-rv) dr}p(x-sv)\int_{-1}^{1}f(x-sv,v^{\prime })dv^{\prime }ds\Big] \\
&=\exp (-\tau _{n}\sigma_{i-j})\xi _{i-j,j}+ \frac{k_n\tau_n}{2}p_{i-j} e^{-\tau _{n}\sigma_{i-j}}
\sum_{l=-\mu_n}^{\mu_n-1}\xi_{i-j,l}\\
&=\big[U_{1,n}(\tau _{n})(I+\tau _{n}A_n)\xi\big]_{i,j}\\
&=U_{1,n}(\tau _{n})(I+\tau _{n}A_n)P_nf=U_n(\tau_n)P_nf.
\end{align*}
Hence, by taking $g=V(\tau_n)f$, we obtain
$$
P_nV(\tau_n)^2 f = P_nV(\tau_n)g=U_n(\tau_n)P_ng=U_n(\tau_n)^2P_nf,
$$
and by induction we retrieve \eqref{SnUn}.
Once the identity \eqref{SnUn} is proven, we replace
$U_{n}(t)P_nf$ by $P_nV(\tau_n)^nf$ in \eqref{katocon}
and we use the isometric character of $P_n$ (see Lemma \ref{P_n}),
then  we get
$$
\|U_{n}(t)P_nf-P_nU(t)f\|_n = \|V(t/n)^nf-U(t)f\|.
$$
Now, if $\omega=s_m-k_M$, thanks to Theorem \ref{sgps}(3), $U(t)$
satisfies $\|U(t)\|\le e^{-\omega t}$, and since $2k_M< s_m$,
we get $k_M<\omega$. So we can replace in Theorem \ref{1},
$S_0(t)$ by $U_1(t)$ and $B$ by the production operator
$A$, the formula \eqref{V2} show that we can use this
Theorem to prove that \eqref{katocon} holds.
\end{proof}

\begin{remark}\label{u_n} \rm
\begin{itemize}
\item[(a)] We can approximate the integral
$\int_0^{t} \sigma(ih_n-sjk_n)ds$ by $\mathbf{\sigma}_{i,j}^{(n)}$, where
\begin{equation}\label{bsg}
\mathbf{\sigma}_{i,j}^{(l)}:=\tau_n\sum_{k=1}^l\sigma(ih_n-jk\tau_nk_n).
\end{equation}
In this case the approximation of $U_1$ given by
\eqref{U_1} would be
$$
 U_{1,n}(t)= \exp\big(-\mathbf{\sigma}_{i,j}^{(n)}\big)
f(ih_n-nj\tau_n k_n,jk_n),
$$
where $\sigma_{i-kj}= \sigma(h_n(i-kj))$.
Replacing $f(ih_n-jn\tau_n k_n,jk_n)$ by $\xi_{i-nj,j}$ as before we
get
\begin{equation}\label{paag}
[U_{1,n}(t)\xi]_{i,j}=
\exp\big(-\mathbf{\sigma}_{i,j}^{(n)}\big)\xi_{i-nj,j}.
\end{equation}
So
$[U_{1,n}(\tau_n)\xi]_{i,j}=  e^{-\tau_n\sigma_{i-j}}\xi_{i-j,j}$.

\item[(b)]  We note that by taking $k=n$, $U_n(t)$
given in \eqref{U_n}, can be written as
$$
U_n(t)=U_{1,n}(t)\Big(\sum_{k=0}^n C_n^k(\tau_nA_n)^k\big).
$$
Hence
$$
[U_{n}(t)\xi]_{i,j}=[U_{1,n}(t)\xi]_{i,j}+U_{1,n}(t)
\Big(\sum_{k=1}^n C_n^k(\tau_np_i)^k\Big)
\frac{k_n}2\sum_{l=-\mu_n}^{\mu_n-1}\xi_{i,l}.
$$
\end{itemize}
\end{remark}

\begin{thebibliography}{00}

\bibitem{Che}  P. R. Chernoff;
\emph{Note on product formulas for operator semigroups.}
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\bibitem{Hej}  H. J. Hejtmanek;
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