In the present paper we recall the main points of the Fourier Transform developments, in particular the historical origin of the inversion formula. Hence we construct explicit examples of functions in different zones of the range of the Fourier transform in \(L^1\). These can be used as exercises in a basic course of signal processing or harmonic analysis.
"A Didactic Note on Classic Function Spaces and the Fourier Transform." Real Anal. Exchange 44 (1) 229 - 246, 2019. https://doi.org/10.14321/realanalexch.44.1.0229