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Re: [harmonic] convolutions of restrictions of functions

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  • 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 1 of 4 , Apr 13, 2011
      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 2 of 4 , Jun 13, 2011
        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.
        >
        >
        > --- 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
        > >
        > >
        > >
        > >
        > >
        > >
        > >
        > >  
        > >
        > >
        > >
        > >
        > >
        > >
        > >
        > >
        > >
        > > 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.
        > >
        >
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