Difference between revisions of "Marks-Gravagne-Davis Fourier transform"
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|[[Real_numbers | $\mathbb{R}$]] | |[[Real_numbers | $\mathbb{R}$]] | ||
| − | |$ | + | |$\mathscr{F}\{f\}(z;s)= $ |
|- | |- | ||
|[[Integers | $\mathbb{Z}$]] | |[[Integers | $\mathbb{Z}$]] | ||
| − | |$ | + | |$\mathscr{F}\{f\}(z;s) = $ |
|- | |- | ||
|[[Multiples_of_integers | $h\mathbb{Z}$]] | |[[Multiples_of_integers | $h\mathbb{Z}$]] | ||
| − | | $ | + | | $\mathscr{F}\{f\}(z;s) = $ |
|- | |- | ||
| [[Square_integers | $\mathbb{Z}^2$]] | | [[Square_integers | $\mathbb{Z}^2$]] | ||
| − | | $ | + | | $\mathscr{F}\{f\}(z;s) = $ |
|- | |- | ||
|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] | |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] | ||
| − | | $ | + | | $\mathscr{F}\{f\}(z;s) = $ |
|- | |- | ||
|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]] | |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]] | ||
| − | | $ | + | | $\mathscr{F}\{f\}(z;s) =$ |
|- | |- | ||
|[[Harmonic_numbers | $\mathbb{H}$]] | |[[Harmonic_numbers | $\mathbb{H}$]] | ||
| − | |$ | + | |$\mathscr{F}\{f\}(z;s) = $ |
|} | |} | ||
</center> | </center> | ||
Revision as of 16:14, 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
| $\mathbb{T}$ | |
| $\mathbb{R}$ | $\mathscr{F}\{f\}(z;s)= $ |
| $\mathbb{Z}$ | $\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
