# Fourier -- Fourier transform for Weyl algebra

## Synopsis

• Usage:
Fourier A
FourierInverse A
• Inputs:
• A, , , an ideal, or , over the Weyl algebra
• Outputs:
• , , an ideal, or , the Fourier transform of A as a matrix, function, ideal, or chain complex over the Weyl algebra

## Description

The Fourier transform is the automorphism of the Weyl algebra that sends xi to -Di and Di to xi. In order to compute the Fourier transform of the finitely generated module M, we compute the Fourier transform of the matrix A of which M is the cokernel.

 i1 : makeWA(QQ[x,y]) o1 = QQ[x..y, dx, dy] o1 : PolynomialRing, 2 differential variable(s) i2 : A = matrix{{2*x^2+1,y*dy},{9*x*dx, x*y*dx^2}} o2 = | 2x2+1 ydy | | 9xdx xydx^2 | 2 2 o2 : Matrix (QQ[x..y, dx, dy]) <-- (QQ[x..y, dx, dy]) i3 : Fourier A o3 = | 2dx^2+1 -ydy-1 | | -9xdx-9 x2dxdy+2xdy | 2 2 o3 : Matrix (QQ[x..y, dx, dy]) <-- (QQ[x..y, dx, dy]) i4 : L = x^2*dy + y*dy^2 + 3*dx^5*dy 5 2 2 o4 = 3dx dy + x dy + y*dy o4 : QQ[x..y, dx, dy] i5 : Fourier L 5 2 2 o5 = 3x y + y*dx - y dy - 2y o5 : QQ[x..y, dx, dy] i6 : I = ideal(8*x*y*dy^3+y^5, dx^7+5) 5 3 7 o6 = ideal (y + 8x*y*dy , dx + 5) o6 : Ideal of QQ[x..y, dx, dy] i7 : Fourier I 3 5 2 7 o7 = ideal (8y dx*dy - dy + 24y dx, x + 5) o7 : Ideal of QQ[x..y, dx, dy] i8 : C = chainComplex{matrix{{x*dx, y^2+dx}},matrix{{dx*dy},{y^2*dy^3}}} 1 2 1 o8 = (QQ[x..y, dx, dy]) <-- (QQ[x..y, dx, dy]) <-- (QQ[x..y, dx, dy]) 0 1 2 o8 : ChainComplex i9 : FC = Fourier C 1 2 1 o9 = (QQ[x..y, dx, dy]) <-- (QQ[x..y, dx, dy]) <-- (QQ[x..y, dx, dy]) 0 1 2 o9 : ChainComplex i10 : FC.dd 1 2 o10 = 0 : (QQ[x..y, dx, dy]) <--------------------- (QQ[x..y, dx, dy]) : 1 | -xdx-1 dy^2+x | 2 1 1 : (QQ[x..y, dx, dy]) <--------------------------- (QQ[x..y, dx, dy]) : 2 {2} | xy | {2} | y3dy^2+6y2dy+6y | o10 : ChainComplexMap

• WeylAlgebra -- specify differential operators in the ring

## Ways to use Fourier :

• Fourier(ChainComplex)
• Fourier(Ideal)
• Fourier(Matrix)
• Fourier(Module)
• Fourier(RingElement)

## For the programmer

The object Fourier is .