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• ## Re: [harmonic] convolutions of restrictions of functions

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• It is better to read Reiter s book on Classical harmonic analysis and locally compact groups which contains details on this. Shravan ... From: lakshanyamath
Message 1 of 4 , Mar 28, 2011
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 It is better to read Reiter's book on "Classical harmonic analysis and locally compact groups" which contains details on this. Shravan --- On Tue, 22/2/11, lakshanyamath wrote:From: lakshanyamath Subject: [harmonic] convolutions of restrictions of functionsTo: harmonicanalysis@yahoogroups.comDate: Tuesday, 22 February, 2011, 10:07 PM It would be very helpful for me if anyone could explain the following: Let G be a locally compact abelian group and H, a closed subgroup of G. Let f,g be integrable functions on G. Then the convolution of the restrictions of f and g to H, and (f * g) restricted to H, are both integrable functions on H(where f * g is the usual convolution of f and g). Is there any relation between these two functions on H, or between their Fourier transforms? If not in general, atleast when H is an open subgroup.

• Thanks for the answer. Reiter s book, which was mentioned by Mr. Shravan Kumar, relates a few properties of an integrable function on a locally compact
Message 1 of 4 , Apr 13, 2011
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Thanks for the answer. Reiter's book, which was mentioned by Mr. Shravan Kumar, relates a few properties of an integrable function on a locally compact abelian group and its restriction to a closed subgroup. But I could not find an answer to my question in this book. Infact the question arose after reading this book. I request for some more references or details.

--- In harmonicanalysis@yahoogroups.com, shravan kumar <meet_shravan@...> wrote:
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> It is better to read Reiter's book on "Classical harmonic analysis and locally compact groups" which contains details on this.
>
> Shravan
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> --- On Tue, 22/2/11, lakshanyamath <lakshanyamath@...> wrote:
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> From: lakshanyamath <lakshanyamath@...>
> Subject: [harmonic] convolutions of restrictions of functions
> To: harmonicanalysis@yahoogroups.com
> Date: Tuesday, 22 February, 2011, 10:07 PM
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> It would be very helpful for me if anyone could explain the following:
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> Let G be a locally compact abelian group and H, a closed subgroup of G. Let f,g be integrable functions on G. Then the convolution of the restrictions of f and g to H, and (f * g) restricted to H, are both integrable functions on H(where f * g is the usual convolution of f and g).
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> Is there any relation between these two functions on H, or between their Fourier transforms?
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> If not in general, atleast when H is an open subgroup.
>
• My initial guess is those functions(f*g restricted to H and f restrict * g restrict) would differ by a compactly supported function in G.Hence their fourier
Message 1 of 4 , Jun 13, 2011
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My initial guess is those functions(f*g restricted to H and f restrict * g restrict) would differ by a compactly supported function in G.Hence their fourier transforms would vary by a entire, exponential function by paley wiener theorem. please 4give for any mistakes as i just made a guess..shall check into details and try coming up with a more precise ans :)

--- In harmonicanalysis@yahoogroups.com, "lakshanyamath" <lakshanyamath@...> wrote:
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> Thanks for the answer. Reiter's book, which was mentioned by Mr. Shravan Kumar, relates a few properties of an integrable function on a locally compact abelian group and its restriction to a closed subgroup. But I could not find an answer to my question in this book. Infact the question arose after reading this book. I request for some more references or details.
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> --- In harmonicanalysis@yahoogroups.com, shravan kumar <meet_shravan@> wrote:
> >
> > It is better to read Reiter's book on "Classical harmonic analysis and locally compact groups" which contains details on this.
> >
> > Shravan
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> > --- On Tue, 22/2/11, lakshanyamath <lakshanyamath@> wrote:
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> > From: lakshanyamath <lakshanyamath@>
> > Subject: [harmonic] convolutions of restrictions of functions
> > To: harmonicanalysis@yahoogroups.com
> > Date: Tuesday, 22 February, 2011, 10:07 PM
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> > It would be very helpful for me if anyone could explain the following:
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> > Let G be a locally compact abelian group and H, a closed subgroup of G. Let f,g be integrable functions on G. Then the convolution of the restrictions of f and g to H, and (f * g) restricted to H, are both integrable functions on H(where f * g is the usual convolution of f and g).
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> > Is there any relation between these two functions on H, or between their Fourier transforms?
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> > If not in general, atleast when H is an open subgroup.
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>
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