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

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<center>
 
<center>
 
{| class="wikitable"
 
{| class="wikitable"
−
|+Time Scale $\Delta$-exponential Functions
+
|+Time Scale foo Functions
 
|-
 
|-
−
|$\mathbb{T}=$
+
|$\mathbb{T}$
−
|$e_p(t,s)=$
+
|
 
|-
 
|-
 
|[[Real_numbers | $\mathbb{R}$]]
 
|[[Real_numbers | $\mathbb{R}$]]
−
|$\mathcal{F}\{f\}(z;s)=$
+
|$foo(t)= $
 
|-
 
|-
 
|[[Integers | $\mathbb{Z}$]]
 
|[[Integers | $\mathbb{Z}$]]
−
|$\mathcal{F}\{f\}(z;s)=$  
+
|$foo(t) = $
 
|-
 
|-
 +
|[[Multiples_of_integers | $h\mathbb{Z}$]]
 +
| $foo(t) = $
 +
|-
 +
| [[Square_integers | $\mathbb{Z}^2$]]
 +
| $foo(t) = $
 +
|-
 +
|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q &gt; 1$]]
 +
| $foo(t) = $
 +
|-
 +
|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q &lt; 1$]]
 +
| $foo(t) =$
 +
|-
 +
|[[Harmonic_numbers | $\mathbb{H}$]]
 +
|$foo(t) = $
 +
|}</pre>
 +
generates
 +
{| class="wikitable"
 +
|+Time Scale foo Functions
 +
|-
 +
|$\mathbb{T}$
 +
|
 +
|-
 +
|[[Real_numbers | $\mathbb{R}$]]
 +
|$foo(t)=  $
 +
|-
 +
|[[Integers | $\mathbb{Z}$]]
 +
|$foo(t) = $
 +
|-
 +
|[[Multiples_of_integers | $h\mathbb{Z}$]]
 +
| $foo(t) = $
 +
|-
 +
| [[Square_integers | $\mathbb{Z}^2$]]
 +
| $foo(t) = $
 +
|-
 +
|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q &gt; 1$]]
 +
| $foo(t) = $
 +
|-
 +
|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q &lt; 1$]]
 +
| $foo(t) =$
 +
|-
 +
|[[Harmonic_numbers | $\mathbb{H}$]]
 +
|$foo(t) = $
 
|}
 
|}
 
</center>
 
</center>

Revision as of 16:13, 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

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

Examples

Time Scale foo Functions
$\mathbb{T}$
$\mathbb{R}$ $foo(t)= $
$\mathbb{Z}$ $foo(t) = $
$h\mathbb{Z}$ $foo(t) = $
$\mathbb{Z}^2$ $foo(t) = $
$\overline{q^{\mathbb{Z}}}, q > 1$ $foo(t) = $
$\overline{q^{\mathbb{Z}}}, q < 1$ $foo(t) =$
$\mathbb{H}$ $foo(t) = $
</pre>

generates

Time Scale foo Functions
$\mathbb{T}$
$\mathbb{R}$ $foo(t)= $
$\mathbb{Z}$ $foo(t) = $
$h\mathbb{Z}$ $foo(t) = $
$\mathbb{Z}^2$ $foo(t) = $
$\overline{q^{\mathbb{Z}}}, q > 1$ $foo(t) = $
$\overline{q^{\mathbb{Z}}}, q < 1$ $foo(t) =$
$\mathbb{H}$ $foo(t) = $

See also

Cuchta-Georgiev Fourier transform

References

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