Shifting problem
Let $\mathbb{T}$ be a time scale, $t_0 \in \mathbb{T}$, 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
Theorem: Suppose that $\hat{f}$ has partial $\Delta$-derivatives of all orders. Then $$\dfrac{\partial^k \hat{f}}{\Delta^k t} (t,t)=f^{\Delta^k}(t_0).$$
Proof: █
Theorem: Define $u_a(t)= \left\{\begin{array}{ll} 0 &; t < a \\ 1 &; t \geq a \end{array} \right..$ Then $$\mathscr{L}_{\mathbb{T}}\{u_s \hat{f}(\cdot,s) \}(z) = e_{\ominus z}(s,t_0)\mathscr{L}_{\mathbb{T}}\{f\}(z).$$
Proof: █
Examples
| Shift $\hat{f}(t,s)$ | |
| $\mathbb{R}$ | $\hat{f}(t,s)=f(t-s)$ |
| $\mathbb{Z}$ | $\hat{f}(t,s)=f(t-s+t_0)$ |
| $h\mathbb{Z}$ | |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
