It is well known that a function can be decomposed uniquely into the sum of an odd and an even function.
This notion can be extended to the unique decomposition into the sum of four functions – two of which are
even and two odd. These four functions are eigenvectors of the Fourier Transform with four different
eigenvalues. That is, the Fourier transform of each of the four components is simply that component multiplied
by the corresponding eigenvalue. Some eigenvectors of the discrete Fourier transform of particular
interest find application in coding, communication and imaging. Some of the underlying mathematics
goes back to the times of Carl Friedrich Gauss.
%0 Journal Article
%1 horn10
%A Horn, Berthold K. P.
%D 2010
%I Taylor & Francis
%J Transactions of the Royal Society of South Africa
%K circulant dft eigenvalues fourier linear.algebra matrix
%N 2
%P 100--106
%R 10.1080/0035919X.2010.510665
%T Interesting Eigenvectors of the Fourier Transform
%V 65
%X It is well known that a function can be decomposed uniquely into the sum of an odd and an even function.
This notion can be extended to the unique decomposition into the sum of four functions – two of which are
even and two odd. These four functions are eigenvectors of the Fourier Transform with four different
eigenvalues. That is, the Fourier transform of each of the four components is simply that component multiplied
by the corresponding eigenvalue. Some eigenvectors of the discrete Fourier transform of particular
interest find application in coding, communication and imaging. Some of the underlying mathematics
goes back to the times of Carl Friedrich Gauss.
@article{horn10,
abstract = {It is well known that a function can be decomposed uniquely into the sum of an odd and an even function.
This notion can be extended to the unique decomposition into the sum of four functions – two of which are
even and two odd. These four functions are eigenvectors of the Fourier Transform with four different
eigenvalues. That is, the Fourier transform of each of the four components is simply that component multiplied
by the corresponding eigenvalue. Some eigenvectors of the discrete Fourier transform of particular
interest find application in coding, communication and imaging. Some of the underlying mathematics
goes back to the times of Carl Friedrich Gauss.},
added-at = {2015-09-02T16:07:01.000+0200},
author = {Horn, Berthold K. P.},
biburl = {https://www.bibsonomy.org/bibtex/274aad8555a41572792da275660698a76/ytyoun},
doi = {10.1080/0035919X.2010.510665},
interhash = {e9ae6bf5788195ce236be90d0eaadb88},
intrahash = {74aad8555a41572792da275660698a76},
journal = {Transactions of the Royal Society of South Africa},
keywords = {circulant dft eigenvalues fourier linear.algebra matrix},
number = 2,
pages = {100--106},
publisher = {Taylor \& Francis},
timestamp = {2015-11-23T10:09:19.000+0100},
title = {Interesting Eigenvectors of the {Fourier} Transform},
volume = 65,
year = 2010
}