| ##### 7.5.5.0. fourier Procedure from library `dmodapp.lib` (see dmodapp_lib). Usage: fourier(I[,v]); I an ideal, v an optional intvec Return: ideal Purpose: computes the Fourier transform of an ideal in a Weyl algebra Assume: The basering is the n-th Weyl algebra over a field of characteristic 0 and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 holds, i.e. the sequence of variables is given by x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator belonging to x(i). Note: The Fourier automorphism is defined by mapping x(i) to -D(i) and D(i) to x(i). If v is an intvec with entries ranging from 1 to n, the Fourier transform of I restricted to the variables given by v is computed. Example:
    LIB "dmodapp.lib";
    ring r = 0,(x,y,Dx,Dy),dp;
    def D2 = Weyl();
    setring D2;
    ideal I = x*Dx+2*y*Dy+2, x^2*Dx+y*Dx+2*x;
    intvec v = 2;
    fourier(I,v);
    ==> _[1]=x*Dx-2*y*Dy
    ==> _[2]=x^2*Dx-Dx*Dy+2*x
    fourier(I);
    ==> _[1]=-x*Dx-2*y*Dy-1
    ==> _[2]=x*Dx^2-x*Dy
See also: inverseFourier.

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