# Eignevalues and Eigenvectors of the Fourier Transform

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"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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