Please see the attached file for the fully formatted problems.
The Fourier transform, call it F, is a linear one-to-one operator from the space of square-integrable functions onto itself. (In fact, we also know that F is an "isometric" mapping, but we will not need this feature in this problem). Indeed,
Note that here x and k are viewed as points on the common domain (?co, oo) of f and F.
(a) Consider the linear operator P and its eigenvalue equation. What are the eigenvahies and the eigenfunctions of F2?
(b) Identify the operator F4? What are its eigenvalues? (c) What are the eigenvalues of F?
Eignevalues and eigenvectors of the Fourier transform are investigated. The solution is detailed and well presented.