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Re: Problem with FourierParameters

  • To: mathgroup at smc.vnet.net
  • Subject: [mg87125] Re: [mg86998] Problem with FourierParameters
  • From: Peter Breitfeld <phbrf at t-online.de>
  • Date: Tue, 1 Apr 2008 05:41:42 -0500 (EST)
  • References: <200803280817.DAA04820@smc.vnet.net> <Pine.LNX.4.63.0803311739200.27272@wopr.wolfram.com>

Am 1. Apr 2008 um 00:55 schrieb Devendra Kapadia:
> On Fri, 28 Mar 2008, Peter Breitfeld wrote:
>
>> I get a wrong result using FourierParameters:
>>
>> FourierTransform[f, x, w]
>> InverseFourierTransform[%, w, x]
>>
>> gives
>>
>> 1/2 Sqrt[\[Pi]/2] (Sign[1 - w] + Sign[1 + w])
>> Sin[x]/x
>>
>> which is what I expected. But with the setting of FourierParameters:
>>
>> FourierTransform[f, x, w, FourierParameters -> {1, -1}]
>> InverseFourierTransform[%, w, x, FourierParameters -> {1, -1}]
>>
>> gives
>>
>> 1/2 \[Pi] Sign[1 - w] + 1/2 \[Pi] Sign[1 + w]     <--- This is OK
>> 0                                                 <--- Bug?
>>
>> Other functions e.g. f=1/(1+x^2) transform correctly in both cases.
>>
>> Gruss Peter
>>
> Hello Peter,
>
> The answer given by InverseFourierTransform for your example with
> FourierParameters -> {1, -1} is indeed incorrect.
>
> We use symbolic integration for computing the inverse transform
> in both the cases given above. The incorrect answer appears to
> be caused by the fact that the output from FourierTransform in
> the second case is not in factored form. Hence, a partial
> workaround for the problem is to use Factor in the call
> to InverseFourierTransform, as shown below:
>
> ===========================
>
> In[2]:= FourierTransform[Sin[x]/x, x, w, FourierParameters -> {1, -1}]
>
>         Pi Sign[1 - w]   Pi Sign[1 + w]
> Out[2]= -------------- + --------------
>               2                2
>
> In[3]:= InverseFourierTransform[Factor[%], w, x,
>            FourierParameters -> {1, -1}]
>
>         Sin[x]
> Out[3]= ------
>           x
>
> =====================
>
> Thank you for reporting the problem. We are sorry for the  
> inconvenience
> caused by it.
>
> Sincerely,
>
> Devendra Kapadia,
> Wolfram Research, Inc.

Dear Ms Kapadia,

Thank you for this hint.
I experimented a bit with different FourierParameters and it seems  
that the wrong answer always comes up, if the first parameter (a) ist  
not 0. Can you confirm this?

Sincerly,

Peter Breitfeld
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