Difference between revisions of "Marks-Gravagne-Davis Fourier transform"
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=Properties= | =Properties= | ||
[[Marks-Gravagne-Davis Fourier transform as a delta integral with classical exponential kernel]] | [[Marks-Gravagne-Davis Fourier transform as a delta integral with classical exponential kernel]] | ||
| + | |||
| + | =Examples= | ||
| + | *The [[Gaussian_bell | Gaussian bell]] | ||
| + | {| class="wikitable" | ||
| + | |+Time Scale $\Delta$-exponential Functions | ||
| + | |- | ||
| + | |$\mathbb{T}=$ | ||
| + | |$e_p(t,s)=$ | ||
| + | |- | ||
| + | |[[Real_numbers | $\mathbb{R}$]] | ||
| + | |$\mathcal{F}\{f\}(z;s)=$ | ||
| + | |- | ||
| + | |[[Integers | $\mathbb{Z}$]] | ||
| + | |$\mathcal{F}\{f\}(z;s)=$ | ||
| + | |- | ||
| + | |} | ||
| + | |||
=See also= | =See also= | ||
Revision as of 16:12, 15 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 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
- The Gaussian bell
| $\mathbb{T}=$ | $e_p(t,s)=$ |
| $\mathbb{R}$ | $\mathcal{F}\{f\}(z;s)=$ |
| $\mathbb{Z}$ | $\mathcal{F}\{f\}(z;s)=$ |
See also
Cuchta-Georgiev Fourier transform
