The Fourier transform appears as a fundamental tool in this book, so its main results and properties have been summarized here. Detailed studies of Fourier analysis can be found in the books of Rudin [Rud79] and Yosida [Yos64].
In the following let f be a fast-decaying function in or a function in . The Fourier transform of f is given by
where . The scalar product in is denoted by 〈 f, g 〉L 2 = . We have Parseval’s relation
Thus the Fourier transform can be extended to and even to the space of tempered distributions . By Parseval's relation, the Fourier transform ...