# Generalisation of the Fourier Transform

If we define a Phase-shift operator as follows $$\Phi_{(a,b)}[f(x)] = e^{i b (x - a /2)} f(x-a)$$ and recall the following property of the Fourier transform $$F[e^{i a (x+ b /2)} f(x+b)] = e^{ib (\omega - a/2)}\hat{f}(\omega-a)$$ we see that these operators satisfy the relation $$\Phi_{(-b,a)} = F^{-1} \circ \Phi_{(a,b)} \circ F$$ or rather, that conjugation by the Fourier transform switches the action of phase multiplication and a shift.

It's not really a ''switch'' though, its a $$\pi/2$$ rotation in the argument space $$(a,b)$$. I.e., using $$R_\theta = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$$ we find $$\Phi_{R_{\pi/2}(a,b)} = F^{-1} \circ \Phi_{(a,b)} \circ F.$$ Is anything known about the operators $$T_\theta$$ which satisfy corresponding relations for different rotations $$\Phi_{R_\theta(a,b)} = T_\theta^{-1} \circ \Phi_{(a,b)} \circ T_\theta.$$ Do these have a name? Or known properties? Can $$T_\theta[f]$$ be calculated for generic functions and generic $$\theta$$? Or even for special non-trivial $$\theta$$ other than integer multiples of $$\pi/2$$?

I can see that for $$\theta=2\pi/n$$ that $$T_\theta^n = \mathrm{identity}$$ and that $$T_\theta$$ should be a linear transformation, and I am interested in what other properties they may be known to have.