Difference between revisions of "Shifting problem"
From timescalewiki
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The solution $\hat{f}$ of the shifting problem is called the shift of $f$ (also called the delay of $f$). | The solution $\hat{f}$ of the shifting problem is called the shift of $f$ (also called the delay of $f$). | ||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\displaystyle\int_{t_0}^t \hat{f}(t,\sigma(\xi))\Delta \xi=\displaystyle\int_{t_0}^t f(\xi) \Delta \xi,$$ | ||
| + | where $\hat{f}$ denotes the solution of the [[shifting problem]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | =Examples= | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Time Scale Shift | |+Time Scale Shift | ||
|- | |- | ||
| − | |$\mathbb{T}$ | Shift | + | |$\mathbb{T}$ | Shift $\hat{f}(t,s)$ |
| | | | ||
|- | |- | ||
| Line 16: | Line 27: | ||
|- | |- | ||
|[[Integers | $\mathbb{Z}$]] | |[[Integers | $\mathbb{Z}$]] | ||
| − | | | + | |$\hat{f}(t,s)=f(t-s+t_0)$ |
|- | |- | ||
|[[Multiples_of_integers | $h\mathbb{Z}$]] | |[[Multiples_of_integers | $h\mathbb{Z}$]] | ||
Revision as of 18:42, 8 February 2016
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
Theorem: The following formula holds: $$\displaystyle\int_{t_0}^t \hat{f}(t,\sigma(\xi))\Delta \xi=\displaystyle\int_{t_0}^t f(\xi) \Delta \xi,$$ where $\hat{f}$ denotes the solution of the shifting problem.
Proof: █
Examples
| Shift $\hat{f}(t,s)$ | |
| $\mathbb{R}$ | $\hat{f}(t,s)=f(t-s)$ |
| $\mathbb{Z}$ | $\hat{f}(t,s)=f(t-s+t_0)$ |
| $h\mathbb{Z}$ | |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
