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    Eignevalues and Eigenvectors of the Fourier Transform

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    Problem attached.

    "Eigenvalues and Eigenvectors of the Fourier Transform"
    Recall that the Fourier transform F is a linear one-to-one transformation from L2 (?cc, cc) onto itself.
    Let .. be an element of L2(?cc,cc).
    Let..= , the Fourier transform of.., be defined by
    It is clear that
    are square-integrable functions, i.e. elements of L2(?oo, cc)
    Consider the SUBSPACE W C L2(?oo, oo) spanned by these vectors, namely
    W = span{...} C L2(?co, cc).
    a) SHOW that W is finite dimensional. What is dim W?
    b) EXHIBIT a basis for W.
    c) It is evident that F is a (unitary) transformation on W.
    FIND the representation matrix of F, [7]B, relative to the basis B found in part b).
    d) FIND the secular determinant, the eigenvalues and the corresponding eigenvectors of [7]B.
    e) For W, EXHIBIT an alternative basis which consists entirely of eigenvectors of F, each one labelled by its respective eigenvalue.
    f) What can you say about the eigenvalues of F viewed as a transformation on L2(?oo, oo) as compared to [F]B which acts on a finite-dimensional vector space?

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    Solution Summary

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