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function [V]=ref_hermbasis(L)
%-*- texinfo -*-
%@deftypefn {Function} ref_hermbasis
%@verbatim
%REF_HERMBASIS Orthonormal basis of discrete Hermite functions.
% Usage: V=hermbasis(L);
%
% HERMBASIS(L) will compute an orthonormal basis of discrete Hermite
% functions of length L. The vectors are returned as columns in the
% output.
%
% All the vectors in the output are eigenvectors of the discrete Fourier
% transform, and resemble samplings of the continuous Hermite functions
% to some degree.
%
%
% References:
% H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay. The Fractional Fourier
% Transform. John Wiley and Sons, 2001.
%
%
%@end verbatim
%@strong{Url}: @url{http://ltfat.github.io/doc/reference/ref_hermbasis.html}
%@seealso{dft, pherm}
%@end deftypefn
% Copyright (C) 2005-2016 Peter L. Soendergaard <peter@sonderport.dk>.
% This file is part of LTFAT version 2.2.0
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program. If not, see <http://www.gnu.org/licenses/>.
% AUTHOR : Peter L. Soendergaard
% TESTING: TEST_HERMBASIS
% REFERENCE: OK
% Create tridiagonal sparse matrix
A=ones(L,3);
A(:,2)=(2*cos((0:L-1)*2*pi/L)-4).';
H=spdiags(A,-1:1,L,L);
H(1,L)=1;
H(L,1)=1;
H=H*pi/(i*2*pi)^2;
% Blow it to a full matrix, and use the linear algebra
% implementation. This always works.
[V,D]=eig(full(H));
% If L is not a factor of 4, then all the eigenvalues of the tridiagonal
% matrix are distinct. If L IS a factor of 4, then one eigenvalue has
% multiplicity 2, and we must split the eigenspace belonging to this
% eigenvalue into a an even and an odd subspace.
if mod(L,4)==0
x=V(:,L/2);
x_e=(x+involute(x))/2;
x_o=(x-involute(x))/2;
x_e=x_e/norm(x_e);
x_o=x_o/norm(x_o);
V(:,L/2)=x_o;
V(:,L/2+1)=x_e;
end;
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