# Fourier -- Fourier transform for Weyl algebra

## Synopsis

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

## Description

The Fourier transform is the automorphism of the Weyl algebra which sends xi to -Di and Di to xi.

 i1 : makeWA(QQ[x,y]) o1 = QQ[x..y, dx, dy] o1 : PolynomialRing, 2 differential variables i2 : L = x^2*dy + y*dy^2 + 3*dx^5*dy 5 2 2 o2 = 3dx dy + x dy + y*dy o2 : QQ[x..y, dx, dy] i3 : Fourier L 5 2 2 o3 = 3x y + y*dx - y dy - 2y o3 : QQ[x..y, dx, dy]