Difference between revisions of "Marks-Gravagne-Davis Fourier transform"
From timescalewiki
(→Examples) |
(→Examples) |
||
| Line 9: | Line 9: | ||
<center> | <center> | ||
{| class="wikitable" | {| class="wikitable" | ||
| − | |+ | + | |+Marks-Gravagne-Davis Fourier transform on various time scales |
|- | |- | ||
|$\mathbb{T}$ | |$\mathbb{T}$ | ||
Revision as of 16:15, 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.
Contents
Properties
Marks-Gravagne-Davis Fourier transform as a delta integral with classical exponential kernel
Examples
| $\mathbb{T}$ | |
| $\mathbb{R}$ | $\mathscr{F}\{f\}(z;s)= $ |
| $\mathbb{Z}$ | $\mathscr{F}\{f\}(z;s) = $ |
| $h\mathbb{Z}$ | $\mathscr{F}\{f\}(z;s) = $ |
| $\mathbb{Z}^2$ | $\mathscr{F}\{f\}(z;s) = $ |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | $\mathscr{F}\{f\}(z;s) = $ |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | $\mathscr{F}\{f\}(z;s) =$ |
| $\mathbb{H}$ | $\mathscr{F}\{f\}(z;s) = $ |
See also
Cuchta-Georgiev Fourier transform
