Fourier Puzzle

[ptc-1368288]

Puzzle ...

Fourier or else ... what else ?
To answer the question: is it possible to get differently valued points at the same ordinates ?
Answer = YES , simply from Mathcad [blue graph]
The blue graph contains more than needed but for the demo it does not matter. The blue graph is digitized, not too badly reproduced. We extract the spectral and try to recover the function. Is the spectral valid ? Answer = Don't know
Then can't recover the function ? Answer = can't

jmG

[ttokoro]

I find a very old puzzle in Mathcad SNS without solution.  

It can solve by Prime 8. (Of course, old Mathcad also can do all.)