RE: L2 inner product. Integrate and Conjugate?

• To: mathgroup at smc.vnet.net
• Subject: [mg36339] RE: [mg36308] L2 inner product. Integrate and Conjugate?
• From: "David Park" <djmp at earthlink.net>
• Date: Mon, 2 Sep 2002 04:08:48 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Andreas,

inner[f_Function, g_Function] :=
Integrate[ComplexExpand[Conjugate[f[x]]]*g[x], {x, -1/2, 1/2}]

inner[# &, # &]
1/12

I'm not too knowledgable about using complex functions in Mathematica but
sometimes I think that "ComplexExpand" should be renamed "ComplexSimplify".
One very often needs it.

David Park

From: Andreas Dietrich [mailto:andreas.dietrich at physik.uni-marburg.de]
To: mathgroup at smc.vnet.net

Hello.

I am trying to implement the inner product in the space of
complex-valued, square integrable functions over [-1/2,1/2], which can
be expressed in Mathematica code as

inner[f_Function,g_function]:=Integrate[Conjugate[f[x]]*g[x],{x,-1/2,1/2}]

This is simple enough. Problem is, Mathematica seamingly cannot
evaluate the Integral for even the simplest of functions:
In[10]:=inner[#&,#&]

Out[10]:=\!\(\[Integral]\_\(-\(1\/2\)\)\%\(1\/2\)\(x\ Conjugate[
x]\) \[DifferentialD]x\)

As you see, the Integrate returns unevaluated. It works fine if I
remove the Conjugate. Unfortunately the Conjugate is needed for
positive definiteness.

Various variants with Composition, Re and Im etc. don't work either.

This should be a So how do I get Integrate to work with Conjugate?

I use Mathematica 4.1.2.0 on Linux/i386.

Thank you,
Andreas

--
True Pleasure in this society is more dangerous than bank robbery.

```

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