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]].
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=Properties=
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[[Cuchta-Georgiev Fourier transform of delta derivatives]]
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=Examples=
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<center>
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{| class="wikitable"
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Cuchta-Georgiev Fourier transform on various time scales
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|-
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|$\mathbb{T}$
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|
 +
|-
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|[[Real_numbers | $\mathbb{R}$]]
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|$\mathcal{F}\{f\}(z;s)= \displaystyle\int_{-\infty}^{\infty} f(t)e^{-izt} \mathrm{d}t$
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|-
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|[[Integers | $\mathbb{Z}$]]
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|$\mathcal{F}\{f\}(z;s) = \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(k)}{(1+iz)^{k+1-s}}$
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|-
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|[[Multiples_of_integers | $h\mathbb{Z}$]]
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| $\mathcal{F}\{f\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(hk)}{(1+ihz)^{k+1-\frac{s}{h}}}$
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|-
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| [[Square_integers | $\mathbb{Z}^2$]]
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| $\mathcal{F}\{f\}(z;s) = $
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|-
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|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q &gt; 1$]]
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| $\mathcal{F}\{f\}(z;s) = $
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|-
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|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q &lt; 1$]]
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| $\mathcal{F}\{f\}(z;s) =$
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|-
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|[[Harmonic_numbers | $\mathbb{H}$]]
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|$\mathcal{F}\{f\}(z;s) = $
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|}
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</center>
  
 
=See also=
 
=See also=
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=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)
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[[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.

Properties

Cuchta-Georgiev Fourier transform of delta derivatives

Examples

Cuchta-Georgiev Fourier transform on various time scales
$\mathbb{T}$
$\mathbb{R}$ $\mathcal{F}\{f\}(z;s)= \displaystyle\int_{-\infty}^{\infty} f(t)e^{-izt} \mathrm{d}t$
$\mathbb{Z}$ $\mathcal{F}\{f\}(z;s) = \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(k)}{(1+iz)^{k+1-s}}$
$h\mathbb{Z}$ $\mathcal{F}\{f\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(hk)}{(1+ihz)^{k+1-\frac{s}{h}}}$
$\mathbb{Z}^2$ $\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