Eignevalues and eigenvectors of the fourier transform

Question

Problems:

Eignevalues and Eigenvectors of the Fourier Transform

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,

                        F : L²(-∞,∞) → L²(-∞,∞)

ƒ(x) _˜→ F[ƒ](k) Ξ 1/√2∏ ∫^∞_-∞ e^-ikx ƒ(x)dx Ξ F(k)

Note that here x and k are viewed as points on the common domain (-∞,∞) of ƒ and F.

(a) Consider the linear operator F² and its eigenvalue equation.

                                      F²ƒ = λƒ

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?