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

From timescalewiki
Jump to: navigation, search
(Examples)
 
(5 intermediate revisions by the same user not shown)
Line 1: Line 1:
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
+
__NOTOC__
$$\mathscr{F}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus \mathring{\iota} 2 \pi z}(\tau,s) \Delta \tau,$$
+
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 Marks-Gravagne-Davis Fourier transform of $f$ centered at $s$ by
 +
$$\mathscr{F}_{\mathbb{T}}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus \mathring{\iota} 2 \pi z}(\tau,s) \Delta \tau,$$
 
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]].
  
Line 15: Line 16:
 
|-
 
|-
 
|[[Real_numbers | $\mathbb{R}$]]
 
|[[Real_numbers | $\mathbb{R}$]]
|$\mathscr{F}\{f\}(z;s)= $
+
|$\mathscr{F}\{f\}(z;s)= \displaystyle\int_{-\infty}^{\infty} f(t)e^{2\pi izt} \mathrm{d}t$
 
|-
 
|-
 
|[[Integers | $\mathbb{Z}$]]
 
|[[Integers | $\mathbb{Z}$]]
|$\mathscr{F}\{f\}(z;s) = $
+
|$\mathscr{F}\{f\}(z;s) = \displaystyle\sum_{k=-\infty}^{\infty} f(k)e^{2\pi izk} $
 
|-
 
|-
 
|[[Multiples_of_integers | $h\mathbb{Z}$]]
 
|[[Multiples_of_integers | $h\mathbb{Z}$]]
| $\mathscr{F}\{f\}(z;s) = $
+
| $\mathscr{F}\{f\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} f(hk) e^{2\pi i zhk}$
 
|-
 
|-
 
| [[Square_integers | $\mathbb{Z}^2$]]
 
| [[Square_integers | $\mathbb{Z}^2$]]

Latest revision as of 14:11, 28 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 Marks-Gravagne-Davis Fourier transform of $f$ centered at $s$ by $$\mathscr{F}_{\mathbb{T}}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus \mathring{\iota} 2 \pi z}(\tau,s) \Delta \tau,$$ 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

Marks-Gravagne-Davis Fourier transform as a delta integral with classical exponential kernel

Examples

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

References