Difference between revisions of "Shifting problem"

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(Properties)
(Properties)
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[[Delta integral of certain shift of f is delta integral of f]]<br />
 
[[Delta integral of certain shift of f is delta integral of f]]<br />
 
[[Delta partial derivative of shift along diagonal]]<br />
 
[[Delta partial derivative of shift along diagonal]]<br />
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> Define $u_a(t)= \left\{\begin{array}{ll} 0 &; t < a \\
 
1 &; t \geq a \end{array} \right..$ Then
 
$$\mathscr{L}_{\mathbb{T}}\{u_s \hat{f}(\cdot,s) \}(z) = e_{\ominus z}(s,t_0)\mathscr{L}_{\mathbb{T}}\{f\}(z).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong>  █
 
</div>
 
</div>
 
  
 
=Examples=
 
=Examples=

Revision as of 14:38, 21 January 2023

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
Delta partial derivative of shift along diagonal

Examples

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

See also

Convolution