Cuchta-Georgiev Fourier transform

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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.

Examples

Cuchta-Georgiev Fourier transform on various time scales
 $\mathbb{T}$ $\mathbb{R}$ $\mathcal{F}\{f\}(z;s)= \displaystyle\int_{-\infty}^{\infty} f(t)e^{-izt} \mathrm{d}t$ $\mathbb{Z}$ $\mathcal{F}\{f\}(z;s) = \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(k)}{(1+iz)^{k+1-s}}$ $h\mathbb{Z}$ $\mathcal{F}\{f\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(hk)}{(1+ihz)^{k+1-\frac{s}{h}}}$ $\mathbb{Z}^2$ $\mathcal{F}\{f\}(z;s) =$ $\overline{q^{\mathbb{Z}}}, q > 1$ $\mathcal{F}\{f\}(z;s) =$ $\overline{q^{\mathbb{Z}}}, q < 1$ $\mathcal{F}\{f\}(z;s) =$ $\mathbb{H}$ $\mathcal{F}\{f\}(z;s) =$