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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/algebra fourier.spad}
+\author{James Davenport}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{domain FCOMP FourierComponent}
+<<domain FCOMP FourierComponent>>=
+)abbrev domain FCOMP FourierComponent
+++ Author: James Davenport
+++ Date Created: 17 April 1992
+++ Date Last Updated: 12 June 1992
+++ Basic Functions:
+++ Related Constructors:
+++ Also See:
+++ AMS Classifications:
+++ Keywords:
+++ References:
+++ Description:
+FourierComponent(E:OrderedSet):
+       OrderedSet with
+         sin: E -> $
+         ++ sin(x) makes a sin kernel for use in Fourier series
+         cos: E -> $
+         ++ cos(x) makes a cos kernel for use in Fourier series
+         sin?: $ -> Boolean
+         ++ sin?(x) returns true if term is a sin, otherwise false
+         argument: $ -> E
+         ++ argument(x) returns the argument of a given sin/cos expressions
+    ==
+  add
+   --representations
+   Rep:=Record(SinIfTrue:Boolean, arg:E)
+   e:E
+   x,y:$
+   sin e == [true,e]
+   cos e == [false,e]
+   sin? x == x.SinIfTrue
+   argument x == x.arg
+   coerce(x):OutputForm ==
+     hconcat((if x.SinIfTrue then "sin" else "cos")::OutputForm,
+              bracket((x.arg)::OutputForm))
+   x<y ==
+     x.arg < y.arg => true
+     y.arg < x.arg => false
+     x.SinIfTrue => false
+     y.SinIfTrue
+
+@
+\section{domain FSERIES FourierSeries}
+<<domain FSERIES FourierSeries>>=
+)abbrev domain FSERIES FourierSeries
+++ Author: James Davenport
+++ Date Created: 17 April 1992
+++ Date Last Updated:
+++ Basic Functions:
+++ Related Constructors:
+++ Also See:
+++ AMS Classifications:
+++ Keywords:
+++ References:
+++ Description:
+FourierSeries(R:Join(CommutativeRing,Algebra(Fraction Integer)),
+              E:Join(OrderedSet,AbelianGroup)):
+       Algebra(R) with
+         if E has canonical and R has canonical then canonical
+         coerce: R -> $
+         ++ coerce(r) converts coefficients into Fourier Series
+         coerce: FourierComponent(E) -> $
+         ++ coerce(c) converts sin/cos terms into Fourier Series
+         makeSin: (E,R) -> $
+         ++ makeSin(e,r) makes a sin expression with given argument and coefficient
+         makeCos: (E,R) -> $
+         ++ makeCos(e,r) makes a sin expression with given argument and coefficient
+    == FreeModule(R,FourierComponent(E))
+  add
+   --representations
+   Term := Record(k:FourierComponent(E),c:R)
+   Rep  := List Term
+   multiply : (Term,Term) -> $
+   w,x1,x2:$
+   t1,t2:Term
+   n:NonNegativeInteger
+   z:Integer
+   e:FourierComponent(E)
+   a:E
+   r:R
+   1 == [[cos 0,1]]
+   coerce e ==
+      sin? e and zero? argument e => 0
+      if argument e < 0  then
+           not sin? e => e:=cos(- argument e)
+           return [[sin(- argument e),-1]]
+      [[e,1]]
+   multiply(t1,t2) ==
+     r:=(t1.c*t2.c)*(1/2)
+     s1:=argument t1.k
+     s2:=argument t2.k
+     sum:=s1+s2
+     diff:=s1-s2
+     sin? t1.k =>
+       sin? t2.k =>
+         makeCos(diff,r) + makeCos(sum,-r)
+       makeSin(sum,r) + makeSin(diff,r)
+     sin? t2.k =>
+       makeSin(sum,r) + makeSin(diff,r)
+     makeCos(diff,r) + makeCos(sum,r)
+   x1*x2 ==
+     null x1 => 0
+     null x2 => 0
+     +/[+/[multiply(t1,t2) for t2 in x2] for t1 in x1]
+   makeCos(a,r) ==
+      a<0 => [[cos(-a),r]]
+      [[cos a,r]]
+   makeSin(a,r) ==
+      zero? a => []
+      a<0 => [[sin(-a),-r]]
+      [[sin a,r]]
+
+@
+\section{License}
+<<license>>=
+--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
+--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 Numerical ALgorithms Group Ltd. 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.
+@
+<<*>>=
+<<license>>
+
+<<domain FCOMP FourierComponent>>
+<<domain FSERIES FourierSeries>>
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}