What is the relation between the domain of Fractional Fourier transform and time and frequency?

Question

This refers to Wikipedia's page on Fractional Fourier Transform.
In the definition section it clearly writes that:

The FrFT argument $u$ is neither a spatial one $x$ nor a frequency $\xi$. We will see why it can be interpreted as linear combination of both coordinates $(x,\xi)$

But I did not see the relation between the domain of Fractional Fourier transform ($u$) and space and frequency $(x,\xi)$. I can understand that domain '$u$' is a linear combination of '$(x,\xi)$' but my question what is the relation? Can anyone point out the exact relation?

Answer 1

I think it is more correct to say that the $u$ axis is a linear combination of the $x$ and $\xi$ axes:

$$(x, \xi) = (1, 0)x + (0, 1)\xi = \left((1, 0)\cos(\alpha) + (0, 1)\sin(\alpha)\right)u,$$

that is:

$$ x = u\cos(\alpha)\\ \xi = u\sin(\alpha).$$

The definition of the fractional Fourier transform from that Wikipedia page is:

$$\mathcal{F}_\alpha[f](u) = \sqrt{1-i\cot(\alpha)} e^{i \pi \cot(\alpha) u^2} \int_{-\infty}^\infty e^{-i2\pi (\csc(\alpha) u x - \frac{\cot(\alpha)}{2} x^2)} f(x)\, \mathrm{d}x.$$

If you use another variable, $y$, for the integral, and substitute $u = \sqrt{x^2 + \xi^2}$ and $\alpha = \text{atan}(\xi, x)$, then the fractional Fourier transform can be written as:

$$\mathcal{F}[f](x, \xi) = e^{i \pi \frac{\xi}{x} (x^2+\xi^2)} \sqrt{1-\frac{i \xi}{x}}\int^\infty_{-\infty}e^{i \pi \frac{y}{x} \left(\xi y-2 x^{\frac{1}{\xi}} \sqrt{\xi^2+x^2}\right)} f(y)\, \text{d}y.$$