Difference between revisions of "Marks-Gravagne-Davis Fourier transform as a delta integral with classical exponential kernel"
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==Theorem== | ==Theorem== | ||
If $0 \in \mathbb{T}$, then the [[Marks-Gravagne-Davis Fourier transform]] obeys | If $0 \in \mathbb{T}$, then the [[Marks-Gravagne-Davis Fourier transform]] obeys | ||
| − | $$\mathscr{F}\{f\}(z;0) = \displaystyle\int_{-\infty}^{\infty} f(t)e^{-2i\pi zt} \Delta t.$$ | + | $$\mathscr{F}_{\mathbb{T}}\{f\}(z;0) = \displaystyle\int_{-\infty}^{\infty} f(t)e^{-2i\pi zt} \Delta t.$$ |
==Proof== | ==Proof== | ||
==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=Marks-Gravagne-Davis Fourier transform|next=}}: Section 3 (3.1) | |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 16:43, 15 January 2023
Theorem
If $0 \in \mathbb{T}$, then the Marks-Gravagne-Davis Fourier transform obeys $$\mathscr{F}_{\mathbb{T}}\{f\}(z;0) = \displaystyle\int_{-\infty}^{\infty} f(t)e^{-2i\pi zt} \Delta t.$$
