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    Eignevalues and Eigenvectors of the Fourier Transform

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    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?

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    Solution Summary

    Eignevalues and eigenvectors of the Fourier transform are investigated. The solution is detailed and well presented.