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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, B, relative to the basis B found in part b).
d) FIND the secular determinant, the eigenvalues and the corresponding eigenvectors of 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?