The traditional approach to Fourier analysis emphasizes the fact, that Lebesgue integration is needed in order to properly define the Fourier transform
of functions, and L2(R) is needed to formulate Plancherel's theorem. If one wants to go beyond this the basic principles of the theory of topological vector spaces are required in order to introduce (the Fourier transform for) tempered distributions. We want to explain, that there is an alternative, building on simple principles of basic functional analysis (Banach spaces and their duals) which allows to incorporate the essentials of these two worlds.

Starting from linear algebra and with a couple of MATLAB experiments one can easily demonstrate the basic properties of the DFT (realized via the FFT routine). After introducing the STFT (short time Fourier transform or spectrogram, which in a way is also part of the Windows Media Player) one can design the Banach space of test functions S_0(Rd), which is well suited for the purpose of Conceptual Harmonic Analysis: the corresponding theory of generalized functions (we call them ``mild distributions’’) allows to build a unified approach to Fourier Analysis, but provides also a new perspective on the connections between the different settings (e.g. viewing the Fourier Transform properly as a limit of Fourier series expansions).