analogue of Fourier transform where i is replaced by $sqrt{i}$

Question

For some reason complicated to explain, I am interested in the operator $T$ which to some test function $\Phi$ associates $$T\Phi(x)=\int_\mathbb{R} e^{jxw}\Phi(v)dv$$ where $j=(1+\imath)/\sqrt{2}$. Has anyone seen such a transform? Or is there a general class of transforms from which it is a particular case?

Answer 1

Well, $$ T\Phi(x)=\mathcal F\Phi\Bigl(\frac{j}{i}\,cx\Bigr). $$