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The problem has 2 parts.

a)Show that (i) F^2( f )(x)=2*pi*f(-x) and then deduce that (ii) F^4( f )=4*pi*f where F denotes the Fourier Transform

b) Deduce from (i) that for two eigenvalues, all eigenfunctions must be even, and for other two

eigenvalues, all eigenfunctions must be odd and for (ii) deduce that if f is an eigenfunction of F with eigenvalue λ, then λ must take one of four possible

values.

For a)I've tried to use the Inversion Theorem....it seemed appropriate since there is a Fourier transform on one side and a function f multiplied by pi on the other.Obviously I'm doing something wrong because I get a square function under the integral and I don't know how to handle it.I'm assuming that for (ii) maybe I should use induction?!

For b)To be honest I don't have any idea mainly because I haven't figured out a).

This problem bugs me very much so any help will be appreciated.Thank you