Difference between revisions of "Cuchta-Georgiev Fourier transform"

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=Properties=
 
=Properties=
 
[[Cuchta-Georgiev Fourier transform of delta derivatives]]
 
[[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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|+Marks-Gravagne-Davis Fourier transform on various time scales
 +
|-
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|$\mathbb{T}$
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|
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|-
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|[[Real_numbers | $\mathbb{R}$]]
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|$\mathcal{F}\{f\}(z;s)= \displaystyle\int_{-\infty}^{\infty} f(t)e^{2\pi izt} \mathrm{d}t$
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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} f(k)e^{2\pi izk} $
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|-
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|[[Multiples_of_integers | $h\mathbb{Z}$]]
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| $\mathcal{F}\{f\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} f(hk) e^{2\pi i zhk}$
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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>
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=See also=
 
=See also=

Revision as of 16:44, 15 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

Marks-Gravagne-Davis Fourier transform on various time scales
$\mathbb{T}$
$\mathbb{R}$ $\mathcal{F}\{f\}(z;s)= \displaystyle\int_{-\infty}^{\infty} f(t)e^{2\pi izt} \mathrm{d}t$
$\mathbb{Z}$ $\mathcal{F}\{f\}(z;s) = \displaystyle\sum_{k=-\infty}^{\infty} f(k)e^{2\pi izk} $
$h\mathbb{Z}$ $\mathcal{F}\{f\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} f(hk) e^{2\pi i zhk}$
$\mathbb{Z}^2$ $\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

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