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+math/legendrepoly1D_2.m
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% Generates a 1D orthonormal polynomial base
% polys = legendrepoly1D_2(X,maxorder,w);
% The weighting function has not been tested extensively
% Manuel Guizar - March 10, 2009
% Copyright (c) 2016, Manuel Guizar Sicairos, James R. Fienup, University of Rochester
% All rights reserved.
%
% Redistribution and use in source and binary forms, with or without
% modification, are permitted provided that the following conditions are
% met:
%
% * Redistributions of source code must retain the above copyright
% notice, this list of conditions and the following disclaimer.
% * Redistributions in binary form must reproduce the above copyright
% notice, this list of conditions and the following disclaimer in
% the documentation and/or other materials provided with the distribution
% * Neither the name of the University of Rochester nor the names
% of its contributors may be used to endorse or promote products derived
% from this software without specific prior written permission.
%
% THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
% AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
% IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
% ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
% LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
% CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
% SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
% INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
% CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
% ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
% POSSIBILITY OF SUCH DAMAGE.
function polys = legendrepoly1D_2(X,maxorder,w)
if nargin < 3
w = 1;
end
[nc nr] = size(X);
polys = ones(nc,nr);
%%% Generation of polynomials
for ii = 0:maxorder,
polys(:,:,ii+1) = (X.^(ii));
end
%%% Normalization
for ii = 1:length(polys(1,1,:)),
% polys(:,:,ii) = polys(:,:,ii)/sqrt(sum(sum(abs(polys(:,:,ii)).^2)));
polys(:,:,ii) = polys(:,:,ii)/sqrt(sum(sum(w.*abs(polys(:,:,ii)).^2)));
end
%%% Orthonormalization
for ii = 2:length(polys(1,1,:)),
for jj = 1:ii-1,
polys(:,:,ii) = polys(:,:,ii) - sum(sum(polys(:,:,ii).*polys(:,:,jj).*w))*polys(:,:,jj);
end
polys(:,:,ii) = polys(:,:,ii)/sqrt(sum(sum(w.*polys(:,:,ii).^2)));
end
% polys(:,:,3) = polys(:,:,3) - sum(sum(polys(:,:,3).*polys(:,:,1)))*polys(:,:,1);
% polys(:,:,3) = polys(:,:,3)/sum(sum(polys(:,:,3).^2));