Difference between revisions of "Marks-Gravagne-Davis Fourier Transform"
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(Tom moved page Marks-Gravagne-Davis Fourier Transform to Marks-Gravagne-Davis 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 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]]. | ||
| + | |||
| + | =References= | ||
| + | [http://web.ecs.baylor.edu/faculty/gravagnei/archived/Fourier.pdf] | ||
Latest revision as of 15:53, 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.
