Yuri Kozitsky and Lech Wolowski
A nonlinear dynamical system on the set
of Laguerre entire functions
(358K, PostScrip)
ABSTRACT. A nonlinear modification of the Cauchy problem ${\partial} f
(t, z)/{\partial t}=
\theta D_z f(t,z) + zD^2_z f(t,z)$, $D_z = \partial /\partial z$,
$t\in {I\!\! R}_+ = [0, +\infty)$,
$z\in \hbox{\vrule width 0.6pt height 6pt depth 0pt \hskip -3.5pt}C$,
$\theta \geq 0 $, $f(0,z) = g(z)$, $g\in
{\mathcal L}$ is considered. The set ${\mathcal L}$ consists of
Laguerre entire functions, which one obtains as a closure of the
set of polynomials having real nonpositive zeros only in the
topology of uniform convergence on compact subsets of $ \
\hbox{\vrule width 0.6pt height 6pt depth 0pt \hskip -3.5pt}C$.
The modification means that the time half-line ${I\!\! R}_+$ is
divided into the intervals ${\mathcal I}_n = [(n-1)\tau , n\tau
]$, $n\in {I\!\! N}$, $\tau>0$, and on each ${\mathcal I}_n $ the
evolution is to be described by the above equation but at the
endpoints the function $f(t, z)$ is changed: $f(n\tau , z)
\rightarrow \left[ f\left(n\tau ,
z\delta^{-1-\lambda}\right)\right]^\delta $, with $\lambda>0$ and
an integer $\delta \geq 2$. The resolving operator of such problem
preserves the set ${\mathcal L}$. It is shown that for
$t\rightarrow +\infty$, the asymptotic properties of $f(t, z)$
change considerably when the parameter $\tau$ reaches a threshold
value $\tau_*$. The limit theorems for $\tau < \tau_* $ and for
$\tau = \tau_* $ are proven. Applications, including limit
theorems for weakly and strongly dependent random vectors, are
given.