Difference between revisions of "Marks-Gravagne-Davis Fourier transform"

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$$\mathscr{F}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus \mathring{\iota} 2 \pi z}(\tau,s) \Delta \tau,$$
 
$$\mathscr{F}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus \mathring{\iota} 2 \pi z}(\tau,s) \Delta \tau,$$
 
where $\ominus$ denotes the [[circle minus]] operation, $e_{\ominus \mathring{\iota}2 \pi z}$ denotes the [[delta exponential]], and $\mathring{\iota}$ denotes the [[Hilger pure imaginary]].
 
where $\ominus$ denotes the [[circle minus]] operation, $e_{\ominus \mathring{\iota}2 \pi z}$ denotes the [[delta exponential]], and $\mathring{\iota}$ denotes the [[Hilger pure imaginary]].
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=Properties=
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[[Expression as a delta integral with classical exponential kernel]]
  
 
=See also=
 
=See also=

Revision as of 16:03, 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 a function. Define the Fourier transform of $f$ centered at $s$ by $$\mathscr{F}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus \mathring{\iota} 2 \pi z}(\tau,s) \Delta \tau,$$ where $\ominus$ denotes the circle minus operation, $e_{\ominus \mathring{\iota}2 \pi z}$ denotes the delta exponential, and $\mathring{\iota}$ denotes the Hilger pure imaginary.

Properties

Expression as a delta integral with classical exponential kernel

See also

Cuchta-Georgiev Fourier transform

References

[1]