Numerical Differentiation using Fourier Transform

• To: mathgroup at smc.vnet.net
• Subject: [mg32941] Numerical Differentiation using Fourier Transform
• From: m.p.croucher at sheffield.ac.uk (Mike)
• Date: Wed, 20 Feb 2002 01:26:25 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Hi

I need to differentiate a function that i only know numerically and
for various reasons I have been asked to do this via a Fourier
Transform Method.  Before doing it on the horrible function I am
dealing with, I thought I would get to grips with the method using
much easier functions.

So say I want to differentiate

E^(-x^2)

w.r.t x using Fourier Transforms : analytically i could proceed as
follows

FourierTransform[E^(-x^2), x, w]] = 1/(Sqrt[2]*E^(w^2/4))

multiply this by I*w and taking the inverse Transform yields

(-2*x)/E^x^2

as expected so i am along the right lines here.  Now I assume that I
only know this function numerically and see if i can reproduce this
result.  First I made a table of values :

test = Table[E^(-x^2), {x, 0.01, 20, 0.01}];

and find it's DFT using

Fourier[test]

now in order to find the derivative I need to multiply this by I*w and
take the inverse transform.  My problem is how to do this?  what are
my values of w?  The DFT looks very different to the one I found
analytically and to be honest I don't have a clue of whats going on.
Any help would be appreciated - Thanks

Mike

```

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