Difference between revisions of "Shifting problem"
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| − | 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 $t,s \in \mathbb{T}$: | + | __NOTOC__ |
| + | 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 $t,s \in \mathbb{T}$: | ||
$$\left\{ \begin{array}{ll} | $$\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, \\ | \dfrac{\partial \hat{f}}{\Delta t}(t,\sigma(s))=-\dfrac{\partial \hat{f}}{\Delta s}(t,s)& ; t \geq s \geq t_0, \\ | ||
| Line 5: | Line 6: | ||
\end{array} \right.$$ | \end{array} \right.$$ | ||
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= | ||
| + | [[Delta integral of certain shift of f is delta integral of f]]<br /> | ||
| + | [[Delta partial derivative of shift along diagonal]]<br /> | ||
| + | |||
| + | =Examples= | ||
| + | <center> | ||
| + | {| class="wikitable" | ||
| + | |+Time Scales Shift | ||
| + | |- | ||
| + | | $\mathbb{T}$ | ||
| + | | $\hat{f}(t,s)=$ | ||
| + | |- | ||
| + | |[[Real_numbers | $\mathbb{R}$]] | ||
| + | |$f(t-s)$ | ||
| + | |- | ||
| + | |[[Integers | $\mathbb{Z}$]] | ||
| + | |$f(t-s+t_0)$ | ||
| + | |- | ||
| + | |[[Multiples_of_integers | $h\mathbb{Z}$]] | ||
| + | | | ||
| + | |- | ||
| + | | [[Square_integers | $\mathbb{Z}^2$]] | ||
| + | | | ||
| + | |- | ||
| + | |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] | ||
| + | | | ||
| + | |- | ||
| + | |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]] | ||
| + | | | ||
| + | |- | ||
| + | |[[Harmonic_numbers | $\mathbb{H}$]] | ||
| + | | | ||
| + | |} | ||
| + | </center> | ||
| + | |||
| + | =See also= | ||
| + | [[Unilateral convolution]] <br /> | ||
| + | [[Unilateral Laplace transform]]<br /> | ||
| + | |||
| + | =References= | ||
| + | *{{PaperReference|The convolution on time scales|2007|Martin Bohner|author2=Gusein Sh. Guseinov|prev=|next=}}: Definition 2.1 | ||
Latest revision as of 14:51, 21 January 2023
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
| $\mathbb{T}$ | $\hat{f}(t,s)=$ |
| $\mathbb{R}$ | $f(t-s)$ |
| $\mathbb{Z}$ | $f(t-s+t_0)$ |
| $h\mathbb{Z}$ | |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
See also
Unilateral convolution
Unilateral Laplace transform
References
- Martin Bohner and Gusein Sh. Guseinov: The convolution on time scales (2007): Definition 2.1
