Uniform sampling along characteristics for solutions of the telegraph system

For each function $a : \mathbb{R} \mapsto \mathbb{C}$ with integrable modulus on $\mathbb{R}$, we define the {exponential telegraphic} function as

$$ a_{\mathbb{T}} (x,y):=
\int_{\mathbb{R}}
\ a (t) \ \exp{(i x t + i y/t)}\ d t \ ,\ \ \ x, y \in \mathbb{R} \ . $$

Every exponential telegraphic function is a continuous solution on $\mathbb{R}^{2}$ of the partial differential equation $U_{xy}+U=0$ with two independent real variables $x, y$. Conversely, for each continuous solution $w$ of the equation $U_{xy}+U=0$ on a convex compact subset $K$ in $ \mathbb{R}^{2}$ with nonempty interior, there exists an exponential telegraphic function $a_{\mathbb{T}} = a_{\mathbb{T}} (w, K)$ which coincides with $w$ on $K$ whenever $w_{x}$ and $w_{y}$ are continuous on $K$. Exponential telegraphic functions have first been studied in 2011, see [1] where it is proved that each such function can be recoverable sampled at the points $(0,\pi n), (\pi n,0)$, $n\! \in\! \mathbb{Z}\!:=\! \{...,-1,0,1,...\}$, lying on two characteristics $x=0$ and $y=0$ of the equation $U_{xy}+U=0$. In other words, it follows from $a_{\mathbb{T}} (\pi n,0) = a_{\mathbb{T}} (0,\pi n) = 0$, $n \in \mathbb{Z}$, that $a_{\mathbb{T}} (x,y)=0$ for every $x, y \in\mathbb{R}$. In this work, we provide a new proof of the fact that $a_{\mathbb{T}} (\pi n,0) = a_{\mathbb{T}} (0,-\pi n) = 0$ for all $n \in \mathbb{N}_{0}\!:=\! \{0,1,2,...\}$, implies $a_{\mathbb{T}} (x,-y)=0$ for each $x, y \geq 0$ (cp. [2]), which means possibility to restore each exponential telegraphic function in the quadrant $[0,+\infty)\times (-\infty, 0]$ by its values at the points $(0,-\pi n), (\pi n,0)$, $n\! \in\! \mathbb{N}_{0}$. We apply these results to continuously differentiable one time by each variable solutions $ v(t, x)$ and $i(t, x)$ of the telegraph system
$$ \left\{ \begin{aligned} i_{x} (t, x) + C \cdot v_{t}(t, x) + G \cdot v(t, x) = 0 \, ,\quad & R - \text{resistance},\, L - \text{inductance},\, & D := L G - C R \neq 0, \\ v_{x}(t, x) + L \cdot i_{t}(t, x) + R \cdot i(t, x) = 0 \, ,\quad& C - \text{capacitance},\, G - \text{leakance},\, & t \geq 0, \ x \in \mathbb{R},\\ \end{aligned} \right. $$

with the additional restriction of the existence $T > 0$ satisfying $ v(t, 0) = i(t, 0)= 0$, $t \geq T$. It follows that such $ v$ and $i$ in the angle $|x| \leq t/\sqrt{LC}$, $t \geq 0$, $ x \in \mathbb{R}$ between the two characteristics $x = \pm t/\sqrt{LC}$ are uniquely determined by the values of $v$ or $i$ at the points $(2\pi n L C / |D| \, , \ \pm 2\pi n \sqrt{L C} / |D|)$, $n \in \mathbb{N}_{0}$, lying on these characteristics.