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.


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