Cuchta-Georgiev Fourier transform
From timescalewiki
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.
Contents
Properties
Cuchta-Georgiev Fourier transform of delta derivatives
Examples
| $\mathbb{T}$ | |
| $\mathbb{R}$ | $\mathcal{F}\{f\}(z;s)= \displaystyle\int_{-\infty}^{\infty} f(t)e^{-izt} \mathrm{d}t$ |
| $\mathbb{Z}$ | $\mathcal{F}\{f\}(z;s) = \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(k)}{(1+iz)^{k+1-s}}$ |
| $h\mathbb{Z}$ | $\mathcal{F}\{f\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(hk)}{(1+ihz)^{k+1-\frac{s}{h}}}$ |
| $\mathbb{Z}^2$ | $\mathcal{F}\{f\}(z;s) = $ |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | $\mathcal{F}\{f\}(z;s) = $ |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | $\mathcal{F}\{f\}(z;s) =$ |
| $\mathbb{H}$ | $\mathcal{F}\{f\}(z;s) = $ |
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)
