Difference between revisions of "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 Marks-Gravagne-Davis Fourier transform of $f$ centered at $s$ by | 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 | ||
$$\mathscr{F}_{\mathbb{T}}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus \mathring{\iota} 2 \pi z}(\tau,s) \Delta \tau,$$ | $$\mathscr{F}_{\mathbb{T}}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus \mathring{\iota} 2 \pi z}(\tau,s) \Delta \tau,$$ | ||
Latest revision as of 14:11, 28 January 2023
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
| $\mathbb{T}$ | |
| $\mathbb{R}$ | $\mathscr{F}\{f\}(z;s)= \displaystyle\int_{-\infty}^{\infty} f(t)e^{2\pi izt} \mathrm{d}t$ |
| $\mathbb{Z}$ | $\mathscr{F}\{f\}(z;s) = \displaystyle\sum_{k=-\infty}^{\infty} f(k)e^{2\pi izk} $ |
| $h\mathbb{Z}$ | $\mathscr{F}\{f\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} f(hk) e^{2\pi i zhk}$ |
| $\mathbb{Z}^2$ | $\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
