Is there any way to prove the equation of fractional Fourier transform of a function (the integral of the fractional kernel function), given that the nth Fourier transform is just the composition of the Fourier transform of a function n times, through fractional integration?
So if we had: $$(\mathcal{F}\circ\mathcal{F}\circ\dots)(f(t))=\mathcal{F}^a(f(t))$$ Then could we use the Riemann-Liouville integral in some way? $$\mathcal{F}^a(f(t))=\mathcal{F}^{a-1}\circ\int_{-\infty}^{\infty} f(\xi)e^{-2\pi i \xi t}d\xi$$ $$=\mathcal{F}^{a-2}\circ\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(\xi)(e^{-2\pi i \xi t})^2d\xi d\xi$$ $$\dots=\mathcal{I}^a_\xi (f(\xi)e^{-2\pi i \xi t a})$$ $$=\dfrac{1}{\Gamma (a)} \int_{0}^{\xi}f(u)e^{-2\pi i u t a}(\xi -u)^{a-1}du$$