Marks-Gravagne-Davis Fourier transform

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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) = $
$h\mathbb{Z}$ $\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