\documentclass[runningheads]{llncs}
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\begin{document}
\begin{contribution}
\title{Linear Hamiltonian Systems -- Continous~versus~Discrete}
\titlerunning{Linear Hamiltonian Systems}% abbreviated title (for running head)
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\author{Ond\v rej \surname{Do\v sl\'y}}%
\authorrunning{Ond\v rej \surname{Do\v sl\'y}}% abbreviated author list (for running head)

% modified list of authors for the TOC (add the affiliations)
\tocauthor{Ond\v rej Do\v sl\'y (Masaryk University Brno)}

\institute{Department of Mathematics, Masaryk University, 
Jan\'a\v ckovo n\'am. 2a,\\
662\,95 Brno, The Czech Republic\\
\email{dosly@math.muni.cz}\\
\homepage{http://www.math.muni.cz/\~{}ndosly}
}

\maketitle

\begin{abstract} 
Oscillatory properties of discrete and continuous linear Hamiltonian systems
are compared. In particular, possibilities to establish discrete versions
of the so-called reciprocity principle and trigonometric transformation
are examined.
\end{abstract}

\begin{AMSclassification} \subject{34C10}, \subject{39A10}
\end{AMSclassification}

\begin{keywords} \keyword{Hamiltonian system}, \keyword{reciprocity principle}
\end{keywords}

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#1\\#2\end{array}\right)$}}
\newcommand{\Mat}[4]{\mbox{$\left(\begin{array}{cc}
#1&#2\\#3&#4\end{array}\right)$}}

\section{Introduction}

The aim of this contribution is to investigate relationship between
oscillatory pro\-per\-ties of the continuous (=~differential) linear
Hamiltonian system (further LHS)
$$
x'=A(t)x+B(t)u,\quad u'=C(t)x-A^T(t)u, \leqno{{\rm (H}_c\rm{)}}
$$
and its difference (= discrete) counterpart
$$
\Delta x_k=A_kx_{k+1}+B_ku_k,\quad \Delta u_k=C_kx_{k+1}-A^T_ku_k.
\leqno{{\rm (H}_d{\rm)}}
$$
We suppose that $t\in I=[a,b]\subset \bbbr$, $k\in [0,N]\cap \bbbn$,
$N\in \bbbn$, both in continuous and discrete case $A,B,C$ are
$n\times n$ matrices, $B,C$ are symmetric, i. e. $B=B^T,$ $C=C^T$.
Moreover,
in the discrete case we suppose that the
matrix $(I-A_k)$ is nonsingular and its inverse we denote by $\tilde A_k$.

Linear Hamiltonian systems cover a large variety of linear equations.
For example, the even order, self-adjoint, differential equation
\begin{equation}
\sum_{\nu=0}^{n}(-1)^{\nu}(r_\nu(t)y^{(\nu)})^{(\nu)}=0\label{1.1}
\end{equation}
may be written as LHS (H$_c$) using the substitution
$$ x=\left(\begin{array}{c} y\\ y'\\ \vdots\\y^{(n-1)}\\
\end{array}\right),\quad
u=\left(\begin{array}{c} (-1)^{n-1}(r_ny^{(n)})^{(n-1)}+\dots+r_1y'\\
\vdots\\
-(r_ny^{(n)})'+r_{n-1}y^{(n-1)}\\
r_ny^{(n)}\\ \end{array}\right).
$$
Then $(x,u)$ solves (H$_c$) with $A,B,C$ given by
$$
\begin{array}{c} B(t)={\rm diag}\{0,\dots,0,r^{-1}_n(t)\},
\quad
C(t)={\rm diag}\{r_0(t),\dots,r_{n-1}(t)\},\\
A=A_{ij}=\left\{\begin{array}{l} 1,\quad {\rm if }\ j=i+1,\ i=1,\dots,n-1,\\
        0,\quad {\rm elsewhere.}
\end{array}\right.
\end{array}
$$
Another example is the second order system
\begin{equation}
\left[R(t)x'+Q(t)x\right]'-\left[Q^T(t)x'+P(t)x\right]=0
\label{1.4}\end{equation}
with $n\times n$ matrices $P,Q,R$, whereby $P,R$ are symmetric and $R$ nonsingular.
Putting $u=R(t)x'+Q(t)x$, the pair of $n$-vectors $(x,u)$ solves (H$_c$) with
$B=R^{-1}$, $A=-R^{-1}Q,$ $C=P-Q^TR^{-1}Q.$
Discrete analogies of (\ref{1.1}) and (\ref{1.4}) can be written in the form
(H$_d$) using essentially the same substitutions as in the continuous case.

Investigation of oscillatory properties of (H$_c$) has a relatively long
history and was initiated by the paper of Morse~\cite{12} from~1930.
Since that time oscillation theory of (H$_c$) attracted a considerable attention
and the results of this investigation up to seventies can be found in
the monograph of Reid~\cite{13}.
In~1995 Kratz~\cite{11} published another comprehensive monograph which
in addition to the classical results contains also the results achieved in the
period 1980--95.

In contrast to the continuous case, oscillation theory of discrete
systems (H$_d$) is much less developed and the fundamental result of
this theory, a discrete version of the so-called Roundabout Theorem,
was established only very recently by Bohner~\cite{5}. This paper
accomplished the effort of several mathematicians in the last decade to
prove the discrete Roundabout Theorem in its full generality, see~\cite{3}.

Here we concentrate our attention to two problems of oscillation theory
both of (H$_c$) and (H$_d$); to the so-called reciprocity principle
and to trigonometric transformation. The paper is organized as follows.
In the next section we recall basic facts of oscillation theory of
continuous systems (H$_c$) and we formulate continuous versions of
trigonometric transformation and reciprocity principle. In the last
section we point out what difficulties one meets when trying to ''discretize''
these results, i.\,e., to extend them to (H$_d$).

\section{Continuous Hamiltonian systems}

We start with basic concepts of oscillation theory of (H$_c$).

\begin{definition} Two points $t_1,t_2$ are said to be {\it conjugate}
relative to (H$_c$) if there exists a solution $(x,u)$ such that
$x(t_1)=0=x(t_2)$ and $x(t)\not\equiv 0$ in $[t_1,t_2]$. System (H$_c$) is
said to be {\it conjugate} in an interval $[a,b]$ if there exist $t_1,t_2\in
[a,b]$ which are conjugate relative to (H$_c$), in the opposite case (H$_c$)
is said to be {\it disconjugate}. System (H$_c$) is said to be
{\it oscillatory} if for every $c\in \bbbr$ this system is conjugate
in $[c,\infty)$, in the opposite case (H$_c$) is said to be
{\it nonoscillatory}.
\end{definition}

As mentioned in the previous section, principal statement concerning
oscillatory
pro\-perties of (H$_c$) is the so-called Reid Roundabout 
Theorem~\cite{13}. Before formulating it, we recall some very elementary
properties of solutions of Hamiltonian systems.

Simultaneously with (H$_c$) we consider its matrix analogy
\begin{equation}
X'=A(t)X+B(t)U,\quad U'=C(t)X-A^T(t)U,
\label{2.1}
\end{equation}
where $X,U$ are $n\times n$ matrices. If $(X,U),(\tilde X,\tilde U)$ are two
solutions of~(\ref{2.1}) then the ''Wronskian-type'' identity
$X^T\tilde U-U^T\tilde X\equiv K$ holds, where $K$ is a constant $n\times n$
matrix. A solution $(X,U)$ of (\ref{2.1}) is said to be {\it conjoined}
if $X^TU$ is symmetric and it is said to be {\it conjoined basis}
if, moreover, rank $(X^T,U^T)=n$.

\begin{proposition} \label{p21} (Reid~\cite{13}). The following statements
are equivalent:
\begin{itemize}
\item[(i)] System (H$_c$) is disconjugate in the interval $[a,b]$.
\item[(ii)] The quadratic functional
$${\cal F}(x,u)=\int_a^b [u^T(t)B(t)u(t)+x^T(t)C(t)x(t)]dt$$
is positive for every nontrivial $(x,u)$ satisfying
$x'=A(t)x+B(t)u$ and $x(a)=0=x(b)$.
\item[(iii)] The solution $(X,U)$ of (\ref{2.1}) given by the initial
condition $X(a)=0,U(a)=I$ satisfies $\det X(t)\ne 0$, $t\in (a,b]$.
\item[(iv)] There exists a conjoined basis $(X,U)$ of (\ref{2.1}) such that
$X(t)$ is nonsingular for $t\in [a,b]$.
\item[(v)] There exists a symmetric matrix $Q$ which for $t\in [a,b]$
solves the Riccati matrix differential equation
$$Q'=-C(t)+A^T(t)Q+QA(t)+QB(t)Q$$
related to (H$_c$) by the substitution $Q=UX^{-1}$.
\end{itemize}
\end{proposition}

For a better understanding of this statement we suggest the reader
to see (H$_c$) as a rewritten second order equation 
(compare~(\ref{1.1}) with
$n=1$) and to compare this statement with the well known results
of oscillation theory of second order equations.

Now we state some results concerning transformations of LHS.
These results we need for formulation both of trigonometric
transformation and reciprocity principle. A $2n\times 2n$ matrix
${\cal R}$ is said to be {\it symplectic} if
${\cal R}^T{\cal J}{\cal R}={\cal J},$ where
${\cal J}=\Mat{0}{I} {-I}{0},$
$I$ being the $n\times n$ identity matrix. If ${\cal R}=\mat{H\ \ K}{M\ \ N}$,
where $H,K,M,N$ are $n\times n$ matrices then ${\cal R}$ is symplectic
if and only if
\begin{equation}\label{2.2}
H^TK=K^TH,\ M^TN=N^TM,\ H^TN-K^TM=I.
\end{equation}

Consider the transformation
\begin{equation}\label{2.3}
\mat{x}{u}={\cal R}(t)\mat{y}{z},\ {\rm where}\quad
{\cal R}(t)=\Mat {H(t)}{M(t)}{K(t)}{N(t)}
\end{equation}
is symplectic and continuously differentiable. Then the variables
$y,z$ satisfy the LHS
$$
y'=\bar A(t)y+\bar B(t)z,\quad z'=\bar C(t)y-\bar A^T(t)z,
\leqno{{\rm (}\bar H_c{\rm)}}
$$
where
%the matrices $\bar A,\bar B,\bar C$ can be computed explicitly
%(but these explicit formulas are not importatnt at this moment).
\begin{eqnarray*}
\bar A&=&N^T[-H' +AH+BK]-M^T[-K'+CH-A^TK],\\
\bar B&=&N^T[-M'+AM+BN]+M^T[-N'+CM-A^TN],\\
\bar C&=&-K^T[-H'+AH+BK]+H^T[-N'+CM-A^TN].
\end{eqnarray*}

Observe that in the case $M(t)\equiv 0$ transformation~(\ref{2.3}) preserves
oscillatory properties of transformed systems since then $H(t)$ is nonsingular
(compare~(\ref{2.2})), hence $t_1,t_2$ are conjugate relative to (H$_c)$
if and only if they are conjugate relative to ($\bar H_c$). Consequently,
transformation~(\ref{2.3}) with $M(t)\equiv 0$ is the powerful tool for
investigation of oscillatory properties of (H$_c$). This system is transformed
into an ''easier'' system and from oscillatory properties of this ''easy''
system we deduce oscillatory properties of (H$_c$).
One of such ''easy systems'' is the so-called trigonometric system.

\begin{theorem} \label{t21} (Do\v sl\'y~\cite{7}). 
There exist continuously differentiable
$n\times n$ matrices $H,K$ such that $H$ is nonsingular, $H^TK\equiv K^TH$
and the transformation
$$
\mat{x}{u}=\Mat{H(t)}{0}{K(t)}{(H^T(t))^{-1}}\mat{y}{z}
$$
transforms (H$_c$) into the trigonometric system
\begin{equation}\label{2.4}
y'=Q(t)z,\quad z'=-Q(t)y,
\end{equation}
where $Q$ is a nonnegative definite symmetric $n\times n$ matrix.
\end{theorem}

The trigonometric system was introduced in~\cite{4} in connection with
the Pr\"ufer-type transformation for (H$_c$) and the terminology
{\it trigonometric system} is justified by the fact that in the
scalar case, i. e., when $y,z,q$ are scalar quantities, then
$$(y_1,z_1)=(\sin\int^t\!\! q(s)\,ds,\cos \int^t\!\! q(s)\,ds),\quad
(y_2,z_2)=(\cos \int^t\!\! q(s)\,ds,-\sin \int^t \!\!q(s)\,ds)$$
form the basis of the solution space of (\ref{2.4}). In the $n$-dimensional
case system (\ref{2.4}) cannot be in general solved explicitly, but it may be
proved that its solutions have many properties which in the
scalar case reduce to the well-known trigonometric identities
and these properties we may use to study properties of (H$_c$).

Now turn our attention to the reciprocity principle for LHS. We start
with the following elementary example. Consider the second order
equation
\begin{equation}\label{2.5}
(r(t)y')'+p(t)y=0
\end{equation}
with {\it positive} coefficients $r,p$. If we denote $z=r(t)y'$ then this
function verifies the so-called reciprocal equation
\begin{equation}\label{2.6}
\left({1\over p(t)}z'\right)'+{1\over r(t)}z=0.
\end{equation}
Using an elementary argument it is easy to see that a solution $y$ of
(\ref{2.5}) oscillates if and only if its derivatives $y'$ oscillates,
i.\,e., (\ref{2.5}) is oscillatory if and only if (\ref{2.6}) is oscillatory.

Reciprocity principle concerns extension of this statement to (H$_c$).

\begin{theorem}\label{t22} (Ahlbrandt~\cite{1}). Suppose that $B(t)\geq 0$,
$C(t)\leq 0$ (this means that $B$ is nonnegative definite and
$C$ nonpositive definite) for large $t$ and both system
(H$_c$) and its reciprocal system
$$
y'=-A^T(t)y-C(t)z,\quad z'=-B(t)y+A(t)z
\leqno{{\rm (R}_c{\rm )}}$$
are eventually controllable (i.e., the trivial
solution $(x,u)=(0,0)$ is the only solution of (H$_c$) for which one
of the components $x,u$ is eventually vanishing). Then (H$_c$) is oscillatory
if and only if (R$_c$) is oscillatory.
\end{theorem}

Obviously, this statement is a generalization of the relationship between
(\ref{2.5}) and (\ref{2.6}) and claims, roughly speaking, that (H$_c$)
is oscillatory
with respect to the first component $x$ if and only if it is oscillatory
with respect to the second component $u$ (compare Definition 1).
Indeed, (R$_c$) results from (H$_c$) upon the transformation
\begin{equation}\label{2.7}
\mat{x}{u}=\mat {0\quad I}{ -I \quad 0} \mat{y}{z}=
{\cal J}\mat{y}{z}
\end{equation}
which essentially only reverses the order of equations in (H$_c$).

\section{Discrete Hamiltonian systems}

Similar to the continuous case, we start with the definition of basic concepts.

\begin{definition} We say that the interval $(k,k+1]$,
$k\in \bbbn$, contains a {\it generalized zero} of a solution
$(x,u)$ of (H$_d$) if $x_k\ne 0$, there exists $c\in \bbbr^n$
such that
$$x_{k+1}=\tilde A_kB_kc,\ \ \ {\rm and } \ \ \ c^TB_k^\dagger(I-A_k)x_k\leq 0.$$
System (H$_d$) is said to be {\it disconjugate} in an interval
$[n,m]$ if any solution of (H$_d$) has at most one generalized zero
in $[n,m+1]$ and, moreover, any solution satisfying $x_n=0$ has no generalized
zero in $(n,m+1]$, in the
opposite case (H$_d$) is said to be {\it conjugate} in $[n,m]$.
System (H$_d$) is said to be {\it nonoscillatory} if there exists
$n\in \bbbn$ such that (H$_d$) is disconjugate on $[n,m]$ for every
$m>n$, in the opposite case (H$_d$) is said to be {\it oscillatory}.
\end{definition}

In the above definition ${}^\dagger$ denotes the Moore-Penrose generalized inverse matrix,
for an $n\times n$ matrix $V$ its generalized inverse $V^\dagger$
is the (unique) $n\times n$ matrix such that $VV^\dagger$, $V^\dagger V$
are symmetric and $V^\dagger VV^\dagger=V^\dagger$,
$VV^\dagger V=V$.

Basic oscillatory properties of discrete Hamiltonian systems
are summarized in the discrete version of Roundabout Theorem.

\begin{proposition}\label{p31} (Bohner~\cite{6}). The following statements
are equivalent:
\begin{itemize}
\item[(i)] System (H$_d$) is disconjugate in the interval $[0,N]$, $N\in
\bbbn.$
\item[(ii)] The discrete quadratic functional
$${\cal F}(x,u)=\sum_{k=0}^{N}\left\{u_k^TB_ku_k+x^T_{k+1}C_kx_{k+1}\right\}$$
is positive for every nontrivial $(x,u)$ satisfying
$\Delta x_k=A_kx_{k+1}+B_ku_k$ and $x_0=0=x_{N+1}$.
\item[(iii)] The matrix solution $(X,U)$ of (H$_d$) given by the initial
condition $X_0=0$, $U_0=I$ satisfies
$${\rm Ker\,}X_{k+1}\subseteq {\rm Ker\,}X_k\ \ {\rm and }\ \
X_kX_{k+1}^\dagger \tilde A_kB_k\geq 0,\ k=1,\dots,N.$$
\item[(iv)] There exists a conjoined basis $(X,U)$ of (H$_d$)
(this is defined in the same way as for (H$_c$)) such that
$X_k$ are nonsingular and
$X_kX_{k+1}^{-1} \tilde A_kB_k\geq 0,\ k=0,\dots,N.$
\item[(v)] There exist symmetric matrices $Q_k$ such that
$(I+B_kQ_k)$ are nonsingular, $(I+B_kQ_k)^{-1}B_k\geq 0$, and verify
the discrete Riccati matrix difference equation
$$Q_{k+1}=C_k+(I-A_k^T)Q_k(I+B_kQ_k)^{-1}(I-A_k),$$
$k=0,\dots,N$.
\end{itemize}
\end{proposition}

Concerning transformations of discrete LHS (H$_d$), the situation is not
so easy as in the continuous case. In the discrete case we must suppose that
the matrices $(I-A_k)$ are nonsingular and this assumption must satisfy also
the system after transformation. To ensure this, we need an extra assumption
as shows the next theorem.

\begin{theorem} \label{t31} (Do\v sl\'y~\cite{8}). 
Let ${\cal R}_k$ be a $2n \times 2n$
symplectic matrix consisting of $n \times n$ matrices
${\cal R}_k=\mat{H_k\ \  M_k}{K_k\ \ N_k}$
such that the matrix
\begin{equation}
\Mat{H_k+B_kK_k} {(I-A_k)M_{k+1}}{(I-A_k^T)K_k} {N_{k+1}-C_kM_{k+1}}
\label{3.1}
\end{equation}
is nonsingular and denote
$\mat{ D_k \ F_k}{ E_k \ G_k}$
its inverse.
The transformation
\begin{equation}\label{3.2}
 \mat{x}{u}={\cal R}_k\mat{y}{z}
\end{equation}
transforms (H$_d$) into the system
$$
\Delta y_k=\bar A_k y_{k+1}+\bar B_k z_k,\quad
\Delta z_k=\bar C_ky_{k+1}-\bar A_k^Tz_k,
\leqno{{\rm(\bar H}_d{\rm )}}
$$
where
\begin{eqnarray*}
\bar A_k&=&D_k(-\Delta H_k+A_kH_{k+1}+B_kK_k)+F_k(-\Delta K_k+C_kH_{k+1}-
A_k^TK_k),\\
\bar B_k&=&D_k(-\Delta M_k+A_kM_{k+1}+B_kN_k)+F_k(-\Delta N_k+C_kM_{k+1}-
A_k^TN_k),\\
\bar C_k&=&E_k(-\Delta H_k+A_kH_{k+1}+B_kK_k)+G_k(-\Delta K_k+C_kH_{k+1}-
A_k^TK_k),
\end{eqnarray*}
in particular, the matrices $\bar B_k,\bar C_k$ are symmetric and $(I-\bar A_k)$
are nonsingular, i.e., ($\bar H_d$) is again a difference LHS.
\end{theorem}

Having now in disposal the above given statements, we may try to extend
reciprocity principle and trigonometric transformation to discrete
systems. Let us start with reciprocity principle. If we apply
transformation (\ref{3.2}) with ${\cal R}={\cal J}$ (this transformation in the
continuous case relates (H$_c$) and (R$_c$)) to (H$_d$), it is easy to see
that the assumption of Theorem \ref{t31} concerning nonsingularity
of the matrix
in (\ref{3.1}) is not generally satisfied, i.e. the resulting
(reciprocal) system
$$
\Delta y_k=-A^T_ky_k-C_kz_{k+1},\quad
\Delta z_k=-B_ky_k+A_kz_{k+1}
\leqno{{\rm (R}_d{\rm)}}$$
is the system of a different kind than (H$_d$). In fact, the variable $x$
which defines oscillatory properties of (H$_d$) appears in the
right-hand-side of this system with indices $k+1$, whereas the variable
$y$ which should define oscillations of (R$_d$) appears there with
indices $k$. For this reason, Definition 2 and Proposition \ref{p31} do not
apply to (R$_d$). However, as suggests the equivalence between oscillatory
properties of the pair of second order equations
$\Delta (r_k\Delta x_k)+p_kx_{k+1}=0$ and
$\Delta (p_k^{-1}\Delta z_k)+r_{k+1}^{-1}z_{k+1}=0$, which follows
using the same argumentation as in the continuous case, we may expect
some kind of similarity between oscillatory properties of (H$_d$)
and~(R$_d$).

In studying the relationship between (H$_d$) and (R$_d$), the principal role
play the so-called {\it symplectic systems}, i.\,e. systems of the form
$$\mat{x_{k+1}}{u_{k+1}}=S_k\mat{x_k}{u_k}, \quad
S_k=\mat{{\cal A}_k\ \ {\cal B}_k}{{\cal C}_k\ \ {\cal D}_k},
\leqno{(S)}
$$
where $S_k$ are symplectic $2n\times 2n$ matrices. Expanding forward
differences in (H$_d$) and (R$_d$), it is not difficult to see that these
systems are symplectic systems.
Oscillation theory of symplectic systems was established in~\cite{7}
and fundamental definition is the following:

\begin{definition} We say that the interval $(k,k+1]$
contains the {\it generalized zero} of a solution $(x,u)$ of (S)
if $x_k\ne 0$, there exists $c\in\bbbr^n$ such that
$$x_{k+1}={\cal B}_kc\quad {\rm and }\quad c^T{\cal B}_k^\dagger x_k\leq 0.$$
Oscillation and nonoscillation of symplectic systems are defined
via generalized zeros in the same way as for Hamiltonian systems.
Applying these definitions to (H$_d$) and (R$_d$) we get the following
discrete version of the reciprocity principle.
\end{definition}

\begin{theorem} \label{t32} (Do\v sl\'y-Bohner~\cite{6}). Suppose
that $B_k\geq 0$, $C_k\leq 0$, for large $k$, and both (H$_d$) and
(R$_d$) are eventually controllable. Then (H$_d$) is oscillatory
if and only if (R$_d$) is oscillatory.
\end{theorem}

Now let us pass to trigonometric transformation. In the continuous case
this transformation may be regarded as a transformation which transforms
(H$_c$) into a system which complies with its reciprocal system.
This observation shows that to find a discrete analogy of this statement
will not be easy. As ''the first approximation'' of the problem consider the
pair of second order equations
$$
x''+p(t)x=0\ \ {\rm and }\ \ \Delta ^2x_k=p_kx_{k+1}=0.
\leqno{{\rm (E}_{c,d}{\rm )}}
$$
Let $x_1,x_2$ be solutions of (E$_c$) and consider the transformation
$x=\sqrt{x_1^2+x_2^2}\,y$. This is just the trigonometric transformation
of (E$_{c}$) which transforms this equation into the equation
$$(q^{-1}(t)y')'+q(t)y=0$$
with $q=(x_1^2+x_2^2)^{-2}$, its solutions are
$y_1=\sin\int_{t_0}^tq(s)\,ds$, $y_2=\cos\int_{t_0}^tq(s)\,ds$.
The function $\alpha(t)=\int_{t_0}^t (x_1^2+x_2^2)^{-2}\,ds$ is sometimes
called the {\it zero-counting} function since each its crossing of a multiple
of $\pi$ gives a zero point of any solution of (E$_c$) satisfying $y(t_0)=0$.

Let $y_k,z_k$ be solutions of (E$_d$) and let
$h_k=\sqrt{y_k^2+z_k^2}$. If we try to modify the method used for (E$_c$)
to discrete equation (E$_d$), after some computation we find
that the only ''discrete candidate'' which could play the role of the
function $x_1^2+x_2^2$, is the sequence
$y_ky_{k+1}+z_kz_{k+1}$. However, in contrast to the continuous
case where the function $x_1^2+x_2^2$ is positive and hence
$\alpha$ is increasing, the sequence $y_ky_{k+1}+z_kz_{k+1}$ may change
its sign, may even be identically zero as shows the simple equation
$\Delta^2 x_k+2x_{k+1}=0$. This leads to the following conclusion:

{\it No discrete analogy of the trigonometric transformation is known
even for the simple second order equation (E$_d$).}

Our investigation of relations between oscillation theories of
discrete and continuous systems we may conclude with the following
brief summary.
In some problems, mostly based on equivalence of (i) and (ii) in the Roundabout
Theorem, one can find full analogy between discrete and continuous
statements. On the other hand, in some problems one needs to establish
completely new methods (oscillation theory of symplectic systems in
extension of the reciprocity principle) and, in some cases, till now no
discrete analogy of ''continuous'' statements is known as in the case
of trigonometric transformation.

Supported by the Grant No.~201/96/0410 of the Czech Grant Agency (Prague)

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\end{thebibliography}
\end{contribution}
\end{document}
