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 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

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) = $
$\mathbb{Z}^2$	$\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

• Robert J. Marks II, Ian A. Gravagne and John M. Davis: A generalized Fourier transform and convolution on time scales (2008)... (next): Section 3