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<head><title>Fourier -- Fourier transform for Weyl algebra</title>
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<div><h1>Fourier -- Fourier transform for Weyl algebra</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>Fourier A</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>A</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, or <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, or<span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span> of the same type as <tt>A</tt> -- the Fourier transform of <tt>A</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>The Fourier transform is the automorphism of the Weyl algebra which sends <em>x<sub>i</sub></em> to <em>D<sub>i</sub></em> and <em>D<sub>i</sub></em> to <em>-x<sub>i</sub></em>.<table class="examples"><tr><td><pre>i1 : W = QQ[x,y,Dx,Dy, WeylAlgebra => {x=>Dx,y=>Dy}]
o1 = W
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : L = x^2*Dy + y*Dy^2 + 3*Dx^5*Dy
5 2 2
o2 = 3Dx Dy + x Dy + y*Dy
o2 : W</pre>
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<tr><td><pre>i3 : Fourier L
5 2 2
o3 = 3x y + y*Dx - y Dy - 2y
o3 : W</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="../../Macaulay2Doc/html/___Weyl__Algebra.html" title="name for an optional argument">WeylAlgebra</a> -- name for an optional argument</span></li>
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<div class="waystouse"><h2>Ways to use <tt>Fourier</tt> :</h2>
<ul><li>Fourier(Ideal)</li>
<li>Fourier(Matrix)</li>
<li>Fourier(RingElement)</li>
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