Difference between revisions of "Cuchta-Georgiev Fourier transform of delta derivatives"
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==Theorem== | ==Theorem== | ||
If $f$ is $k$-times [[delta derivative|delta differentiable]] and for all $\ell \in \{0,\ldots,k-1\}$, $\displaystyle\lim_{t \rightarrow \pm \infty} f^{\Delta^{\ell}}(t)e_{\ominus iz}(t,s)=0$, then | If $f$ is $k$-times [[delta derivative|delta differentiable]] and for all $\ell \in \{0,\ldots,k-1\}$, $\displaystyle\lim_{t \rightarrow \pm \infty} f^{\Delta^{\ell}}(t)e_{\ominus iz}(t,s)=0$, then | ||
| − | $$\mathcal{F}_{\mathbb{T}}\left\{f^{\Delta^k}\right\}(z;s) = (iz)^k \mathcal{F}_{\mathbb{T}}\{f\}(z;s) | + | $$\mathcal{F}_{\mathbb{T}}\left\{f^{\Delta^k}\right\}(z;s) = (iz)^k \mathcal{F}_{\mathbb{T}}\{f\}(z;s),$$ |
| + | where $\mathcal{F}_{\mathbb{T}}$ denotes the [[Cuchta-Georgiev Fourier transform]]. | ||
==Proof== | ==Proof== | ||
==References== | ==References== | ||
| − | *{{PaperReference| | + | *{{PaperReference|Analysis of the bilateral Laplace transform on time scales with applications|2021|Tom Cuchta|author2=Svetlin Georgiev|prev=|next=}}: Theorem 11 |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 16:50, 15 January 2023
Theorem
If $f$ is $k$-times delta differentiable and for all $\ell \in \{0,\ldots,k-1\}$, $\displaystyle\lim_{t \rightarrow \pm \infty} f^{\Delta^{\ell}}(t)e_{\ominus iz}(t,s)=0$, then $$\mathcal{F}_{\mathbb{T}}\left\{f^{\Delta^k}\right\}(z;s) = (iz)^k \mathcal{F}_{\mathbb{T}}\{f\}(z;s),$$ where $\mathcal{F}_{\mathbb{T}}$ denotes the Cuchta-Georgiev Fourier transform.
