*Comment*  
The [inverse Fourier transform][1] of $\phi_{1,2}$ looks like this  
[![inv fourier][2]][2]

Is there any reason to think there exist $a,b$ where it has different behavior?


  [1]: https://en.wikipedia.org/wiki/Fourier_transform#Definition
  [2]: https://i.sstatic.net/qocrI.jpg