@JohannesHahn it is not directly related to a research problem (as so many question on this page are), but it can be used to get the fundamental solution of Laplace's equation very straightforwardly by fourier transform methods without the use of distribution theory or tedious calculations and i considered this as intersting.

@JohannesHahn as I explained in the question. It is a problem that you can state without distribution theory, but it seems that a solution to the problem requires these methods cause I could not find a proof of this with elementary methods

thanks, I got your point and now I understood your answer. thank you very much.