Difference between revisions of "Cuchta-Georgiev Fourier transform"

From timescalewiki
Jump to: navigation, search
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 a function. 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 [[circle minus]] operation and $e_{\ominus iz}$ denotes the [[delta exponential]].

Revision as of 16:26, 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 circle minus operation and $e_{\ominus iz}$ denotes the delta exponential.

See also

Marks-Gravagne-Davis Fourier transform

References