View source for Marks-Gravagne-Davis Fourier transform

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=
<center>
{| class="wikitable"
|+Marks-Gravagne-Davis Fourier transform on various time scales
|-
|$\mathbb{T}$
|
|-
|[[Real_numbers | $\mathbb{R}$]]
|$\mathscr{F}\{f\}(z;s)= \displaystyle\int_{-\infty}^{\infty} f(t)e^{2\pi izt} \mathrm{d}t$
|-
|[[Integers | $\mathbb{Z}$]]
|$\mathscr{F}\{f\}(z;s) = \displaystyle\sum_{k=-\infty}^{\infty} f(k)e^{2\pi izk} $
|-
|[[Multiples_of_integers | $h\mathbb{Z}$]]
| $\mathscr{F}\{f\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} f(hk) e^{2\pi i zhk}$
|-
| [[Square_integers | $\mathbb{Z}^2$]]
| $\mathscr{F}\{f\}(z;s) = $
|-
|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q &gt; 1$]]
| $\mathscr{F}\{f\}(z;s) = $
|-
|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q &lt; 1$]]
| $\mathscr{F}\{f\}(z;s) =$
|-
|[[Harmonic_numbers | $\mathbb{H}$]]
|$\mathscr{F}\{f\}(z;s) = $
|}
</center>

=See also=
[[Cuchta-Georgiev Fourier transform]]

=References=
*{{PaperReference|A generalized Fourier transform and convolution on time scales|2008|Robert J. Marks II|author2=Ian A. Gravagne|author3=John M. Davis|prev=|next=Marks-Gravagne-Davis Fourier transform as a delta integral with classical exponential kernel}}: Section 3