Shifting problem

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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 $f$).


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


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

See also

Unilateral convolution
Unilateral Laplace transform