- View Source
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:*<lakshanyamath@...>*

From: lakshanyamath <lakshanyamath@...>

Subject: [harmonic] convolutions of restrictions of functions

To: harmonicanalysis@yahoogroups.com

Date: Tuesday, 22 February, 2011, 10:07 PMIt 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.

- View SourceThanks 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?

>

> If not in general, atleast when H is an open subgroup.

> - View SourceMy 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

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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?

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> > If not in general, atleast when H is an open subgroup.

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>