Convolution Equations

[Kushal Shah]

Hello..

Can someone please suggest some methods to solve convolution equations
of the type:

h(x) 'multiplied by' f(x)=f(x) 'convol' g(x)

where h(x) and g(x) are known functions. Are there analytical methods
to solve such equations in closed form or even approximately? If not,
is it at least possible to solve such equations numerically?

Thanks,
Kushal.

[Ian Parker]

The normal way of dealing with convolution.

http://ccrma.stanford.edu/~jos/st/Convolution_Theorem.html
http://www.dspguide.com/ch18/2.htm

The last reference is about DSP techniques. Click on the reference at
the end and you will get a variety of "go faster" techniques for
implimentation.

The basic fact is you have now replaced a convolution equation with
term by term complex multiplication/division.

- Ian Parker

[Kushal Shah]

On Jan 15, 5:27Êpm, Ian Parker <ianpark...@gmail.com> wrote:
> On 15 Jan, 10:31, Kushal Shah <atmabo...@gmail.com> wrote:> Hello..
>
> > Can someone please suggest some methods to solve convolution equations
> > of the type:
>
> > h(x) 'multiplied by' f(x)=f(x) 'convol' g(x)
>
> > where h(x) and g(x) are known functions. Are there analytical methods
> > to solve such equations in closed form or even approximately? If not,
> > is it at least possible to solve such equations numerically?
>
> The normal way of dealing with convolution.
>

> http://ccrma.stanford.edu/~jos/st/Convolution_Theorem.htmlhttp://www.dspguide.com/ch18/2.htm

>
> The last reference is about DSP techniques. Click on the reference at
> the end and you will get a variety of "go faster" techniques for
> implimentation.
>
> The basic fact is you have now replaced a convolution equation with
> term by term complex multiplication/division.
>

> Ê - Ian Parker

Hello Ian,

The links you have given are nice. But they only give a way of doing
the convolution. This will work only when one already knows the
function 'f(x)'. But in the question I posted, f(x) is unknown. What I
need is a way to solve the equation to find out 'f(x)'. So, the usual
ideas on carrying out the convolution will not work.

Thanks,
Kushal.

[Dmitry Shintyakov]

> The basic fact is you have now replaced a convolution equation with
> term by term complex multiplication/division.

Yes, Fourier transform converts convolution to multiplication, but in
the same time, it converts multiplication to convolution. Thus there
is not much sense in applying Fourier transform to the Kushan Shan's
equation.

By the way, that can be interpreted as problem of finding
eigenfunctions of linear operator T:

T[f(x)] = convolute(f(x), g(x)) / h(x)

[Ian Parker]

> ? T[f(x)] = convolute(f(x), g(x)) / h(x)

I still think you will get more tractable matrices if you do a Fourier
transform.

- Ian Parker

[G. A. Edgar]

In article <gkppnv$1kl9$1...@dizzy.math.ohio-state.edu>, Dmitry Shintyakov
<shint...@gmail.com> wrote:

There is an extensive theory of "singular integral equations" that will
include this one.

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/

[Ilya Zakharevich]

[A complimentary Cc of this posting was sent to
G. A. Edgar
<ed...@math.ohio-state.edu.invalid>], who wrote in article <gkq0hj$2c7d$1...@dizzy.math.ohio-state.edu>:

> > T[f(x)] = convolute(f(x), g(x)) / h(x)
>
> There is an extensive theory of "singular integral equations" that will
> include this one.

I would note that the "equation" is "singular" only for very special
values of g and h. E.g., if g = delta', one gets an ODE for f...

Yours,
Ilya