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Re: [harmonic] convolutions of restrictions of functions

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  • shravan kumar
    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 <lakshanyamath@...> wrote:

      From: lakshanyamath <lakshanyamath@...>
      Subject: [harmonic] convolutions of restrictions of functions
      To: harmonicanalysis@yahoogroups.com
      Date: 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.


    • lakshanyamath
      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 2 of 4 , Apr 13 3:46 AM
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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:
        >
        > 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 <lakshanyamath@...> wrote:
        >
        > 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.
        >
      • opticsabru
        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 3 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
          > >
          > > --- On Tue, 22/2/11, lakshanyamath <lakshanyamath@> wrote:
          > >
          > > 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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          > >  
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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).
          > >
          > > 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.
          > >
          >
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