While classical Fourier Analysis has its basis in integration theory, using the Lebesgue integral in order to define the Fourier transform of an integrable function as a pointwise defined, continuous function, already the inverse Fourier transforms requires some approximations.
The same is true for the proof of the Plancherel Theorem, showing that $L^2(R^d)$ is mapped isometrically onto itself by the Fourier (extended) transform. One then usually goes on to define topological vector spaces of test functions, such as the Schwartz space, in order to then extend the Fourier transform to e.g. tempered distributions in the sense of Laurent Schwartz. But this is a highly non-trivial path, and at the end it is not even convincing when it comes to the discussion of the approximation of the true Fourier transform of an integrable function by means of FFT-based algorithms.

Time-frequency analysis and in particular Gabor Analysis require other function spaces in order to e.g. describe continuous dependency of the dual Gabor atom on the lattice constants in the time-frequency plane. There is a Banach space of test functions, the so-called Segal algebra $S_0(R^d)$, which is much easier to handle than the Schwartz space. Together with the Hilbert space $L^2(R^d)$ and the dual space $S_0'(R^d)$ the form a Banach Gelfand triple which conveniently describes the Fourier transform in all its variants, and $w*$-convergence helps to give various transitions (e.g. from the periodic case to the non-periodic case) a correct mathematical correct way.

A similar talk will be given in New Delhi in January 2018. hgfei