Difference between revisions of "Cuchta-Georgiev Fourier transform"
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Let $\mathbb{T}$ be a [[time scale]] and let $s \in \mathbb{T}$. Let $f \colon \mathbb{T} \rightarrow \mathbb{C}$ be [[regulated]]. Define the Cuchta-Georgiev Fourier transform of $f$ centered at $s$ by | Let $\mathbb{T}$ be a [[time scale]] and let $s \in \mathbb{T}$. Let $f \colon \mathbb{T} \rightarrow \mathbb{C}$ be [[regulated]]. Define the Cuchta-Georgiev Fourier transform of $f$ centered at $s$ by | ||
$$\mathcal{F}_{\mathbb{T}}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus iz}(\sigma(t),\tau) \Delta \tau,$$ | $$\mathcal{F}_{\mathbb{T}}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus iz}(\sigma(t),\tau) \Delta \tau,$$ | ||
| − | where $\ominus$ denotes the [[circle minus]] operation and $e_{\ominus iz}$ denotes the [[delta exponential]]. | + | where $\ominus$ denotes the [[forward circle minus]] operation and $e_{\ominus iz}$ denotes the [[delta exponential]]. |
=Properties= | =Properties= | ||
[[Cuchta-Georgiev Fourier transform of delta derivatives]] | [[Cuchta-Georgiev Fourier transform of delta derivatives]] | ||
| + | |||
| + | =Examples= | ||
| + | <center> | ||
| + | {| class="wikitable" | ||
| + | Cuchta-Georgiev Fourier transform on various time scales | ||
| + | |- | ||
| + | |$\mathbb{T}$ | ||
| + | | | ||
| + | |- | ||
| + | |[[Real_numbers | $\mathbb{R}$]] | ||
| + | |$\mathcal{F}\{f\}(z;s)= \displaystyle\int_{-\infty}^{\infty} f(t)e^{-izt} \mathrm{d}t$ | ||
| + | |- | ||
| + | |[[Integers | $\mathbb{Z}$]] | ||
| + | |$\mathcal{F}\{f\}(z;s) = \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(k)}{(1+iz)^{k+1-s}}$ | ||
| + | |- | ||
| + | |[[Multiples_of_integers | $h\mathbb{Z}$]] | ||
| + | | $\mathcal{F}\{f\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(hk)}{(1+ihz)^{k+1-\frac{s}{h}}}$ | ||
| + | |- | ||
| + | | [[Square_integers | $\mathbb{Z}^2$]] | ||
| + | | $\mathcal{F}\{f\}(z;s) = $ | ||
| + | |- | ||
| + | |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] | ||
| + | | $\mathcal{F}\{f\}(z;s) = $ | ||
| + | |- | ||
| + | |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]] | ||
| + | | $\mathcal{F}\{f\}(z;s) =$ | ||
| + | |- | ||
| + | |[[Harmonic_numbers | $\mathbb{H}$]] | ||
| + | |$\mathcal{F}\{f\}(z;s) = $ | ||
| + | |} | ||
| + | </center> | ||
=See also= | =See also= | ||
| Line 11: | Line 42: | ||
=References= | =References= | ||
*{{PaperReference|Analysis of the bilateral Laplace transform on time scales with applications|2021|Tom Cuchta|author2=Svetlin Georgiev|prev=|next=}}: Definition 4.1 (15) | *{{PaperReference|Analysis of the bilateral Laplace transform on time scales with applications|2021|Tom Cuchta|author2=Svetlin Georgiev|prev=|next=}}: Definition 4.1 (15) | ||
| + | |||
| + | [[Category:Definition]] | ||
Latest revision as of 20:18, 22 January 2023
Let $\mathbb{T}$ be a time scale and let $s \in \mathbb{T}$. Let $f \colon \mathbb{T} \rightarrow \mathbb{C}$ be regulated. Define the Cuchta-Georgiev Fourier transform of $f$ centered at $s$ by $$\mathcal{F}_{\mathbb{T}}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus iz}(\sigma(t),\tau) \Delta \tau,$$ where $\ominus$ denotes the forward circle minus operation and $e_{\ominus iz}$ denotes the delta exponential.
Contents
Properties
Cuchta-Georgiev Fourier transform of delta derivatives
Examples
| $\mathbb{T}$ | |
| $\mathbb{R}$ | $\mathcal{F}\{f\}(z;s)= \displaystyle\int_{-\infty}^{\infty} f(t)e^{-izt} \mathrm{d}t$ |
| $\mathbb{Z}$ | $\mathcal{F}\{f\}(z;s) = \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(k)}{(1+iz)^{k+1-s}}$ |
| $h\mathbb{Z}$ | $\mathcal{F}\{f\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(hk)}{(1+ihz)^{k+1-\frac{s}{h}}}$ |
| $\mathbb{Z}^2$ | $\mathcal{F}\{f\}(z;s) = $ |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | $\mathcal{F}\{f\}(z;s) = $ |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | $\mathcal{F}\{f\}(z;s) =$ |
| $\mathbb{H}$ | $\mathcal{F}\{f\}(z;s) = $ |
See also
Marks-Gravagne-Davis Fourier transform
References
- Tom Cuchta and Svetlin Georgiev: Analysis of the bilateral Laplace transform on time scales with applications (2021): Definition 4.1 (15)
