The Riemann–Liouville transform is defined on \((0,\infty)\times\mathbb{R}\) by \(D= \frac{\partial}{\partial x}\) and \(\Xi =\frac{\partial^2}{\partial r^2}+\frac{2\alpha+1}{r}\frac{\partial}{\partial r}+D^2\). One associates translation operators \(\tau_{(\xi_1,\xi_2)}\) defined by \[\tau_{(r,x)}f(s,y)=\frac{\Gamma(\alpha+1)}{\sqrt{\pi}\Gamma(\alpha+1/2)} \int_0^\pi f(\sqrt{r^2+s^2+2rs\cos\theta},x+y)\, \sin^{2\alpha}(\theta)\, d\theta\, . \] Observing that eigenfunctions \(\varphi_{\lambda_0,\lambda}\) of the system \(Du=-i\lambda u\); \(\Xi u=-\lambda_0^2 u\) with initial values \(u(0,0)=1\) and \(\frac{\partial u}{\partial x}(0,x)=0\) have the form \(\varphi_{\lambda_0,\lambda}=j_\alpha(f\sqrt{\lambda+\lambda_0^2}) \, e^{i\lambda x}\) for the modified Bessel function \(j_\alpha\), one has \(\tau_{(r,x)}\varphi_{\lambda_0,\lambda}(s,y)=\varphi_{\lambda_0,\lambda}(r,x)\varphi_{\lambda_0,\lambda}(s,y)\). Convolution is defined as \((f\ast g)(r,x)=\int_0^\infty\int_{-\infty}^\infty \tau_{(r,-x)}(f^\vee)(s,y) g(s,y) \, d\nu_\alpha (s,y)\) (\(f^\vee(s,y)=f(s,-y)\)), with respect to the measure \(d\nu_\alpha(r,x)=\frac{r^{2\alpha+1}}{2^\alpha\Gamma(\alpha+1)}\otimes\frac{dx}{\sqrt{2\pi}}\) on \((0,\infty)\times \mathbb{R}\). This convolution is shown to satisfy the standard Young’s inequality.
The Fourier transform \(\mathcal{F}_\alpha(f)(\lambda_0,\lambda)\) is defined by integration against \(\varphi_{\lambda_0,\lambda}\) with respect to \(d\nu_\alpha\) with a corresponding inverse \(\widetilde{\mathcal{F}}_\alpha\). A Fourier inversion formula is proved for \(L^1(d\nu_\alpha)\) along with a Plancherel formula and a Fourier convolution-product formula.
A modulation can be defined by \(M_{(\xi_1,\xi_2)}f=\widetilde{\mathcal{F}}_\alpha\Bigl(\sqrt{\tau_{(\xi_1,\xi_2)}(|\widetilde{\mathcal{F}}_\alpha(f)|^2)}\Bigr)\). Then an analogue of the short-time Fourier transform can be defined as \(V_g(f)(r,x)=f\ast M_{(\xi_1,\xi_2)}(g)(r,-x)\).
Window functions are defined and it is shown that the following Plancherel-type formula extends to this windowed Fourier transform: \[\|\mathcal{V}_g(f)\|_{2,\nu_\alpha\otimes\nu_\alpha}=\|f\|_{2,\nu_\alpha}\|g\|_{2,\nu_\alpha}\, . \] An inversion formula for \(\mathcal{V}_g(f)\) is proved that is analogous to the known formula for the standard windowed Fourier transform on \(L^2(\mathbb{R}^n)\), and the following analogue of the Heisenberg inequality is established: \[\||(r,x)|\mathcal{V}_g(f)\|_{2,\nu_\alpha\otimes\nu_\alpha} \||(\xi_1,\xi_2)|\mathcal{V}_g(f)\|_{2,\nu_\alpha\otimes\nu_\alpha} \geq\frac{2\alpha+3}{2} \|f\|_{2,\nu_\alpha}^2\|g\|_{2,\nu_\alpha}^2\, . \] Other variations are investigated on sets of finite measure.