Note on the Fourier Transform: a summary of important properties

This text is intended as a summary of considerations and motivations for Fourier transform theory intended for students interested in Fourier Theory background. It is not a comprehensive introduction into the subject, one can find such an introduction in the book of Porat, chapters 1-3. The text strives at giving accurate statements on a delicate topic in a field in which the engineering literature is often sloppy, thereby endangering comprehension. Most of the mathematical difficulties students experience with Fourier theory are due to various, sometimes problematic forms of convergence. Accurate proofs require knowledge of Lebesgue integration theory, a topic that is not normally covered in the engineering curriculum. We do not assume such knowledge, the reader will have to accept that in a number of cases we must refer to more advanced mathematical properties, although the properties needed are often intuitive.