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 Fourier transform of $f$ centered at $s$ by | + | __NOTOC__ |
| − | $$\mathscr{F}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus \mathring{\iota} 2 \pi z}(\tau,s) \Delta \tau,$$ | + | 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]]. | 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= | ||
| + | <center> | ||
| + | {| class="wikitable" | ||
| + | |+Marks-Gravagne-Davis Fourier transform on various time scales | ||
| + | |- | ||
| + | |$\mathbb{T}$ | ||
| + | | | ||
| + | |- | ||
| + | |[[Real_numbers | $\mathbb{R}$]] | ||
| + | |$\mathscr{F}\{f\}(z;s)= \displaystyle\int_{-\infty}^{\infty} f(t)e^{2\pi izt} \mathrm{d}t$ | ||
| + | |- | ||
| + | |[[Integers | $\mathbb{Z}$]] | ||
| + | |$\mathscr{F}\{f\}(z;s) = \displaystyle\sum_{k=-\infty}^{\infty} f(k)e^{2\pi izk} $ | ||
| + | |- | ||
| + | |[[Multiples_of_integers | $h\mathbb{Z}$]] | ||
| + | | $\mathscr{F}\{f\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} f(hk) e^{2\pi i zhk}$ | ||
| + | |- | ||
| + | | [[Square_integers | $\mathbb{Z}^2$]] | ||
| + | | $\mathscr{F}\{f\}(z;s) = $ | ||
| + | |- | ||
| + | |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] | ||
| + | | $\mathscr{F}\{f\}(z;s) = $ | ||
| + | |- | ||
| + | |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]] | ||
| + | | $\mathscr{F}\{f\}(z;s) =$ | ||
| + | |- | ||
| + | |[[Harmonic_numbers | $\mathbb{H}$]] | ||
| + | |$\mathscr{F}\{f\}(z;s) = $ | ||
| + | |} | ||
| + | </center> | ||
| + | |||
| + | =See also= | ||
| + | [[Cuchta-Georgiev Fourier transform]] | ||
=References= | =References= | ||
| − | *{{PaperReference|A generalized Fourier transform and convolution on time scales|2008|Robert J. Marks II|author2=Ian A. Gravagne|author3=John M. Davis|prev=|next=}}: Section 3 | + | *{{PaperReference|A generalized Fourier transform and convolution on time scales|2008|Robert J. Marks II|author2=Ian A. Gravagne|author3=John M. Davis|prev=|next=Marks-Gravagne-Davis Fourier transform as a delta integral with classical exponential kernel}}: Section 3 |
| − | |||
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
