Fourier Transforms

ExampleFrom Example 1 of §3.2.1, it is known that the FT of the function fx exp−a|x|, where2a2aa is a real constant and a 0isf̂k . What is the FT of gx ?a 2 k 2 a 2 x 2Using the terminology of the question, we have gx f̂x and the result above states thatthe FT of f̂x is 2f−k. Thus the FT of gx is 2f−k, i.e.as |−k| |k|.ĝk 2 f−k 2 exp−a|−k| 2 exp−a|k|3.2.5 Raleigh / Plancheral TheoremsThese results were originally due to Raleigh and rigorously proved by Plancherel; they arethe FT equivalent of Parseval’s theorem in Fourier Series. Let fx and gx be real- orcomplex-valued functions and both absolutely integrable. Consider the integral of theproduct of fx and the complex conjugate gx, employing (3.2.1) and (3.2.9) gives− −−fx gx dx fx 1 2 gk e −ikx dk dx− 12− fx gk e ikx dk dx− 1 2 gk− fx e ikx dx dk−fx gx dx 1 2 f k gk dk (3.2.11)−A special case occurs when f g and (3.2.11) becomes−|fx| 2 dx 1 22 f k dk (3.2.12)−This relationship is referred to as the Conservation of Energy, whereas (3.2.11) sometimescalled the Power Relation.ExampleThe FT of the real-valued function fx exp−a|x| has already been shown in one of theexamples above to be 2a/a 2 k 2 . Applying (3.2.12) gives−|e −a|x| | 2 dx 12−2aa 2 k 22dk.The first integral can be evaluated10