> > Who could be the author of the first rigorous proof of the inversion > formula for the Fourier Transform? > Or is it considered as a mere by-product of Dirichlet's ideas? > (although there is no quotation of it in Dirichlet's work)
It's some years since I looked into this, and in the end it is a question bound to have a less than definite answer. However, when I did look into these things in detail, I formed the view that the first rigorous proof was Pringsheim's in DDMV 16 (1907), pp. 2-16.