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]]. | ||
| + | |||
| + | =Properties= | ||
| + | [[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
