>
> 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.
But I'm sure others will disagree.
Mike Deakin