Can anyone help with these questions?
a. Use the Inversion Theorem to show that (F2f)(x) = 2πf(−x).
b. Deduce that F4f = 4π2f.
c. From (b) deduce that, if f is an eigenfunction of F with eigenvalue λ, then λ must take one of four possible values.
d. From (a) deduce that, for two of these eigenvalues, all eigenfunctions must be even, and for the other two eigenvalues, all eigenfunctions must be odd.
e. Use the result of Problem 13 to show that exp(−x2/2) is an even eigenfunction, and write down the corresponding eigenvalue.
f. Similarly, use the result of Problem 14 to write down an odd eigenfunction corresponding to a second eigenvalue.
g. Continuing, find an eigenfunction for each of the two remaining eigenvalues.