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

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Eignevalues and Eigenvectors: Example Problem

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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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