looking for references on "Generalized/Complex Fourier Transform"?

Hi all,

I recently got to know there is a type of Fourier Transform called
"Generalized or Complex Fourier Transform". It extends the frequency
domain variable to a strip on the complex plane. There is regularity
associated with the transform in this domain. I am not familiar with
the complex/generalized Fourier Transform though I knew the Fourier
transform with the transform domain variable being a real variable. I
am interested in learning more about these generalized transforms. And
"regularity" sounds unheard of for me.

Moreover, I am interested in the operations of these generalized FTs,
such as Parsavel identity, etc. Are the normal Fourier transform
properties and theorems completely carry over to the
generalized/complexe Fourier transforms?

I also want to know about when is a function FT transformable and when
is the transform invertible? In real Fourier transforms we've learned
the transformability conditions and we also know there is distribution
theory that generalizes things, but how does the distribution theory
interplay with the generalized/complex Foureir transforms?

Furthermore, I am interested in inverting the generalized/complex
Foureir transforms on the complex plane, i.e., how to invert the
generalized/complex Fourier transform by doing contour integral? I have
searched a few complex analysis books but I'd like to see a book or
reference notes/articles/papers that talk about the inverting of
Fourier transforms extensively and in great details. Because I met with
a lot difficulty in inverting transforms.

Could anybody please give me some pointers? Thanks a lot!

Luna.

Luna Moon wrote:
> Hi all,
>
> I recently got to know there is a type of Fourier Transform called
> "Generalized or Complex Fourier Transform". It extends the frequency
> domain variable to a strip on the complex plane. There is regularity
> associated with the transform in this domain.

Seems to me as if you have come across a seriously obfuscated
description of the Discrete Fourier Transform, DFT. The best
mathematical coverage of the DFT I have seen, s Oppenheim and
Schafer's "Digital Signal Processing" from 1975. Oppenheim and
Schafer have (co)authored numerous books with nearly identical
titles, make sure to get the one from 1975.

Rune

On 18 Aug 2006 21:10:06 -0700, "Luna Moon" <lunamoonmoon@gmail.com>
wrote:

A place to start would be the theorems of Paley&Weiner,
for example in Rudin "Real and Complex Analysis".

>Hi all,
>
>I recently got to know there is a type of Fourier Transform called
>"Generalized or Complex Fourier Transform". It extends the frequency
>domain variable to a strip on the complex plane. There is regularity
>associated with the transform in this domain. I am not familiar with
>the complex/generalized Fourier Transform though I knew the Fourier
>transform with the transform domain variable being a real variable. I
>am interested in learning more about these generalized transforms. And
>"regularity" sounds unheard of for me.
>
>Moreover, I am interested in the operations of these generalized FTs,
>such as Parsavel identity, etc. Are the normal Fourier transform
>properties and theorems completely carry over to the
>generalized/complexe Fourier transforms?
>
>I also want to know about when is a function FT transformable and when
>is the transform invertible? In real Fourier transforms we've learned
>the transformability conditions and we also know there is distribution
>theory that generalizes things, but how does the distribution theory
>interplay with the generalized/complex Foureir transforms?
>
>Furthermore, I am interested in inverting the generalized/complex
>Foureir transforms on the complex plane, i.e., how to invert the
>generalized/complex Fourier transform by doing contour integral? I have
>searched a few complex analysis books but I'd like to see a book or
>reference notes/articles/papers that talk about the inverting of
>Fourier transforms extensively and in great details. Because I met with
>a lot difficulty in inverting transforms.
>
>Could anybody please give me some pointers? Thanks a lot!
>
>Luna.


************************

David C. Ullrich

On 18 Aug 2006 21:47:56 -0700, "Rune Allnor" <allnor@tele.ntnu.no>
wrote:

>
>Luna Moon wrote:
>> Hi all,
>>
>> I recently got to know there is a type of Fourier Transform called
>> "Generalized or Complex Fourier Transform". It extends the frequency
>> domain variable to a strip on the complex plane. There is regularity
>> associated with the transform in this domain.
>
>Seems to me as if you have come across a seriously obfuscated
>description of the Discrete Fourier Transform, DFT.

Doesn't seem that way to me. Maybe (s)he has just come
across a description of something you're not familiar with?

> The best
>mathematical coverage of the DFT I have seen, s Oppenheim and
>Schafer's "Digital Signal Processing" from 1975. Oppenheim and
>Schafer have (co)authored numerous books with nearly identical
>titles, make sure to get the one from 1975.
>
>Rune


************************

David C. Ullrich

Luna Moon wrote:

> I recently got to know there is a type of Fourier Transform
> called "Generalized or Complex Fourier Transform". It extends
> the frequency domain variable to a strip on the complex plane.

Maybe it's the same as the bilateral Laplace transform?


Martin

-- 
Quidquid latine scriptum sit, altum viditur.

Luna Moon wrote:
> Hi all,
>
> I recently got to know there is a type of Fourier Transform called
> "Generalized or Complex Fourier Transform". It extends the frequency
> domain variable to a strip on the complex plane. There is regularity
> associated with the transform in this domain. I am not familiar with
> the complex/generalized Fourier Transform though I knew the Fourier
> transform with the transform domain variable being a real variable. I
> am interested in learning more about these generalized transforms. And
> "regularity" sounds unheard of for me.
>
> Moreover, I am interested in the operations of these generalized FTs,
> such as Parsavel identity, etc. Are the normal Fourier transform
> properties and theorems completely carry over to the
> generalized/complexe Fourier transforms?
>
> I also want to know about when is a function FT transformable and when
> is the transform invertible? In real Fourier transforms we've learned
> the transformability conditions and we also know there is distribution
> theory that generalizes things, but how does the distribution theory
> interplay with the generalized/complex Foureir transforms?
>
> Furthermore, I am interested in inverting the generalized/complex
> Foureir transforms on the complex plane, i.e., how to invert the
> generalized/complex Fourier transform by doing contour integral? I have
> searched a few complex analysis books but I'd like to see a book or
> reference notes/articles/papers that talk about the inverting of
> Fourier transforms extensively and in great details. Because I met with
> a lot difficulty in inverting transforms.
>
> Could anybody please give me some pointers? Thanks a lot!
>
> Luna.

In addition to the references people have already given you, perhaps I
could make a comment. Take the Fourier transform in 1d. You can always
treat the resulting Fourier transform as a function of a complex
frequency, but if you try and invert it by integrating over the whole
complex plane you will get an infinity.

Perhaps you are thinking of the following: the poles of the Fourier
transform as a function of complex frequency are closely related to the
rate of decay of the original function. Basically, the further the pole
is from the real axis, the faster the function decays. You should be
able to find information about this topic quite easily, perhaps in the
references you have already been given.

illywhacker;

Thanks a lot folks! I did find the keywords in Rudin's book. And I did
find that book mentined above. These are very good starting points!
thanks a lot!

I now understand that this is actually a bilateral laplace transform.
So if you have more pointers/references on laplace transform, please
kindly let me know! Thanks a lot!

"Luna Moon" <lunamoonmoon@gmail.com> writes:

> Thanks a lot folks! I did find the keywords in Rudin's book. And I did
> find that book mentined above. These are very good starting points!
> thanks a lot!
>
> I now understand that this is actually a bilateral laplace transform.
> So if you have more pointers/references on laplace transform, please
> kindly let me know! Thanks a lot!

My copy of Spiegel's "Applied Differential Equations" is a beloved
tome on this and other areas of mathematics. It's a bit dated, but the
math hasn't changed too much in a couple of hundred years.
-- 
%  Randy Yates                  % "Remember the good old 1980's, when 
%% Fuquay-Varina, NC            %  things were so uncomplicated?"
%%% 919-577-9882                % 'Ticket To The Moon' 
%%%% <yates@ieee.org>           % *Time*, Electric Light Orchestra
http://home.earthlink.net/~yatescr

Omega Cubed wrote:
>
> A lot of things you mentioned here fits into the framework of Lebesgue
> integrability, and I am not sure if it plays too well with the
> framework of holomorphic functions and complex analysis. But perhaps I
> am just not familiar enough with the area. At some parts of your
> questions here are answered in the reference above.
>

Exactly! You said it. The integrablility still plays along with
holomorphic functions and the complex analysis. Now my headache is on
the interplay between the Lebesgue integrability , regular functions
and complex analysis, and the theory of the DISTRIBUTIONS.

The integrability now has two folds of meaning: 1. by restricting the
region of convergence/integration, we can define integrability on the
complex plane using generalized/complex Fourier Analysis; 2. by using
the theory of the distributions, I can integrate quite a few other
functions that result in generalized functions or distributions.

It is interesting to compare these two types of Fourier integrations,
which type can integrate a broader class of functions? And how to
rigorously use them in my derivations and interchangebly... And see how
do they interplay with each other.

For example, my preliminary thought is: let's say I want to find a
Fourier Transform of a complicated function f(t),

Using generalized function and distribution theory, FT{f(t)} can be
reduced to

FT{g(t)}+FT{h(t)}+a few other things,  and then I begin to use
Complex/Generalized FT and do contour integral in complex plane to see
if it is better to work out FT{g(t)} that way, etc...

Any more thoughts and more pointers?

Luna Moon wrote:
> Thanks a lot folks! I did find the keywords in Rudin's book. And I
did
> find that book mentined above. These are very good starting points!
> thanks a lot!
>
> I now understand that this is actually a bilateral laplace
transform.
> So if you have more pointers/references on laplace transform, please
> kindly let me know! Thanks a lot!

You should not think of Laplace and Fourier transforms as essentially
different, but try to take into account their relationship by change
of variable etc.. You seem to interested in the special case of
"probability distributions", but you might also do well to scan the
pure maths section of a good university library for advanced Fourier
transform stuff. One old reference I found for the "probability
distributions" case is:
Kawata T (1972) Fourier Analysis in Probability Theory, Academic
Press, NY.

David Jones