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.

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

Started by ●August 19, 2006

Reply by ●August 19, 20062006-08-19

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

Reply by ●August 19, 20062006-08-19

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

Reply by ●August 19, 20062006-08-19

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

Reply by ●August 19, 20062006-08-19

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.

Reply by ●August 19, 20062006-08-19

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;

Reply by ●August 20, 20062006-08-20

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!

Reply by ●August 20, 20062006-08-20

"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

Reply by ●August 20, 20062006-08-20

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?

Reply by ●August 21, 20062006-08-21

Luna Moon wrote:> Thanks a lot folks! I did find the keywords in Rudin's book. And Idid> find that book mentined above. These are very good starting points! > thanks a lot! > > I now understand that this is actually a bilateral laplacetransform.> 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