The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view the Fourier transform as a projection of a vector from the Hilbert space of square-integrable functions on a certain orthonormal basis.
It seems, though, that the Dirac delta-function can be naturally introduced in non-standard calculus. It is a hyperreal function that is $1/dx$ in the interval $[-dx/2, dx/2]$ and zero everywhere else. Here, $dx$ is an infinitesimal hyperreal. But i could not find any treatments of the Fourier transform through non-standard calculus (though it seems to be fruitful and straightforward).
So, my question is: are there any works that treat the Fourier transform through non-standard calculus? Or, if not, are there any intrinsic technical difficulties? And, finally, if it can be done, is it possible to formalize the Fourier transform as a projection of a vector in a linear space of hyperreal functions onto an orthonormal basis?