Difference between revisions of "Shifting problem"
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\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 $f$). | ||
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| + | {| class="wikitable" | ||
| + | |+Time Scale Shift | ||
| + | |- | ||
| + | |$\mathbb{T}$ | Shift | ||
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| + | |[[Real_numbers | $\mathbb{R}$]] | ||
| + | |$\hat{f}(t,s)=f(t-s)$ | ||
| + | |- | ||
| + | |[[Integers | $\mathbb{Z}$]] | ||
| + | | | ||
| + | |- | ||
| + | |[[Multiples_of_integers | $h\mathbb{Z}$]] | ||
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| + | |- | ||
| + | | [[Square_integers | $\mathbb{Z}^2$]] | ||
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| + | |- | ||
| + | |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] | ||
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| + | |- | ||
| + | |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]] | ||
| + | | | ||
| + | |- | ||
| + | |[[Harmonic_numbers | $\mathbb{H}$]] | ||
| + | | | ||
| + | |} | ||
Revision as of 14:35, 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 $f$).
| Shift | |
| $\mathbb{R}$ | $\hat{f}(t,s)=f(t-s)$ |
| $\mathbb{Z}$ | |
| $h\mathbb{Z}$ | |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
