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

• The Gaussian bell

Time Scale $\Delta$-exponential Functions

$\mathbb{T}=$	$e_p(t,s)=$
$\mathbb{R}$	$\mathcal{F}\{f\}(z;s)=$
$\mathbb{Z}$	$\mathcal{F}\{f\}(z;s)=$

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