Difference between revisions of "Cuchta-Georgiev Fourier transform"
From timescalewiki
| Line 1: | Line 1: | ||
Let $\mathbb{T}$ be a [[time scale]] and let $s \in \mathbb{T}$. Let $f \colon \mathbb{T} \rightarrow \mathbb{C}$ be [[regulated]]. Define the Cuchta-Georgiev Fourier transform of $f$ centered at $s$ by | Let $\mathbb{T}$ be a [[time scale]] and let $s \in \mathbb{T}$. Let $f \colon \mathbb{T} \rightarrow \mathbb{C}$ be [[regulated]]. Define the Cuchta-Georgiev Fourier transform of $f$ centered at $s$ by | ||
$$\mathcal{F}_{\mathbb{T}}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus iz}(\sigma(t),\tau) \Delta \tau,$$ | $$\mathcal{F}_{\mathbb{T}}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus iz}(\sigma(t),\tau) \Delta \tau,$$ | ||
| − | where $\ominus$ denotes the [[circle minus]] operation and $e_{\ominus iz}$ denotes the [[delta exponential]]. | + | where $\ominus$ denotes the [[forward circle minus]] operation and $e_{\ominus iz}$ denotes the [[delta exponential]]. |
=Properties= | =Properties= | ||
Revision as of 16:42, 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 regulated. Define the Cuchta-Georgiev Fourier transform of $f$ centered at $s$ by $$\mathcal{F}_{\mathbb{T}}\{f\}(z;s)=\displaystyle\int_{\mathbb{T}} f(\tau)e_{\ominus iz}(\sigma(t),\tau) \Delta \tau,$$ where $\ominus$ denotes the forward circle minus operation and $e_{\ominus iz}$ denotes the delta exponential.
Properties
Cuchta-Georgiev Fourier transform of delta derivatives
See also
Marks-Gravagne-Davis Fourier transform
References
- Tom Cuchta and Svetlin Georgiev: Analysis of the bilateral Laplace transform on time scales with applications (2021): Definition 4.1 (15)
