Re: L2 inner product. Integrate and Conjugate?

• To: mathgroup at smc.vnet.net
• Subject: [mg36335] Re: [mg36308] L2 inner product. Integrate and Conjugate?
• From: BobHanlon at aol.com
• Date: Mon, 2 Sep 2002 04:08:43 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```In a message dated 8/31/02 3:14:33 AM, andreas.dietrich at physik.uni-marburg.d=
e
writes:

> 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.
>

Would including Simplify or FullSimplify provide the results that youn want?

inner[f_Function,g_Function]:=

Integrate[Simplify[Conjugate[f[x]]*g[x], Element[x, Reals]],
{x, -1/2, 1/2}];

inner[#&,#&]

1/12

Bob Hanlon
Chantilly, VA   USA

```

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