Marks-Gravagne-Davis Fourier transform
From timescalewiki
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.
Contents
Properties
Marks-Gravagne-Davis Fourier transform as a delta integral with classical exponential kernel
Examples
| $\mathbb{T}=$ | $e_p(t,s)=$ |
| $\mathbb{R}$ | $\mathcal{F}\{f\}(z;s)=$ |
| $\mathbb{Z}$ | $\mathcal{F}\{f\}(z;s)=$ |
See also
Cuchta-Georgiev Fourier transform
