Shifting problem

Let $\mathbb{T}$ be a time scale with $\sup \mathbb{T}=\infty$, $t_0 \in \mathbb{T}$, and $f \colon [t_0,\infty) \cap \mathbb{T} \rightarrow \mathbb{C}$. The shifting problem is the following partial dynamic equation for $t,s \in \mathbb{T}$: $$\left\{ \begin{array}{ll} \dfrac{\partial \hat{f}}{\Delta t}(t,\sigma(s))=-\dfrac{\partial \hat{f}}{\Delta s}(t,s)& ; t \geq s \geq t_0, \\ \hat{f}(t,t_0)=f(t)&; t \geq t_0. \end{array} \right.$$ The solution $\hat{f}$ of the shifting problem is called the shift of $f$ (also called the delay of $f$).

Properties

Delta integral of certain shift of f is delta integral of f
Delta partial derivative of shift along diagonal

Examples

Time Scales Shift

$\mathbb{T}$	$\hat{f}(t,s)=$
$\mathbb{R}$	$f(t-s)$
$\mathbb{Z}$	$f(t-s+t_0)$
$h\mathbb{Z}$
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$
$\mathbb{H}$

See also

Unilateral convolution
Unilateral Laplace transform

References

• Martin Bohner and Gusein Sh. Guseinov: The convolution on time scales (2007): Definition 2.1