Genralized Fourier Transform

topsquark

Forum Staff
Jan 2006
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Wellsville, NY
This might actually be Differential Geometry but I'll post it here for now.

I was looking at a paper yesterday about quantization conditions when working on a manifold based on the surface of a sphere. I've never seen that before and it's apparently a lot less intuitive than I would have thought. The paper mentions a "generalized Fourier transform." The best I could do on an internet search is here. But I don't think that's what the paper had in mind because the Wolfram link looks like it's modeled in the usual 3-space, not a more generalized manifold.

Has anyone heard of this? I can quote some sections of the paper if you need it.

Thanks!

-Dan
 

romsek

MHF Helper
Nov 2013
6,663
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California
Looks like it's just the expansion of functions on orthogonal basis functions of any sort, as opposed to just the sinusoids of the Fourier basis.

For example one page mentions a basis of spherical harmonic functions.
 
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topsquark

Forum Staff
Jan 2006
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Wellsville, NY
Looks like it's just the expansion of functions on orthogonal basis functions of any sort, as opposed to just the sinusoids of the Fourier basis.

For example one page mentions a basis of spherical harmonic functions.
Thank you. I was just doing some free association and I might have an educated guess. The Fourier integral transform is usually defined as
\(\displaystyle \psi (p) = \dfrac{1}{\sqrt{2 \pi }} \int _{- \infty}^{ \infty} \psi (x) e^{-( i / \hbar ) ( x p_x + y p_y + z p_z )} ~ dx\)

We can rewrite the exponent as \(\displaystyle -\dfrac{i}{ \hbar } x^i p_i = -\dfrac{i}{ \hbar } x \cdot p\)

Now, the form I'm used to uses the Euclidean metric for the "dot product." But I've also seen the dot product in Minkowski space as \(\displaystyle x \cdot p = t p_t - x p_x - y p_y - z p_z = \eta _{ij}x^i p^j\). So my thought is that, in a general case, we use \(\displaystyle g_{ij} x^i p^j\) with the metric defined on the manifold. So the generalized Fourier transform would simply be
\(\displaystyle \psi (p) = \dfrac{1}{\sqrt{2 \pi }} \int_{-\infty} ^ {\infty} \psi (x) e^{ ( -i/ \hbar ) g_{ij} x^i p^j } ~ dx\)

Does this sound reasonable? (It at least makes sense with romsek's comment.)

Thanks!

-Dan
 
Apr 2015
263
57
Somerset, England
Not sure of your application or if you are really referring to Fourier Transforms of Generalised Functions ?

But here is an official definition.

GFT1.jpg

GFT2.jpg
 
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topsquark

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Jan 2006
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Wellsville, NY
Not sure of your application or if you are really referring to Fourier Transforms of Generalised Functions ?

But here is an official definition.

View attachment 39348

View attachment 39349
Thanks for the article. I admit I'm a little confused by it. It looks like a (linear?) integral transformation. I have no idea why they eventually refer to the form as "Fourier." I would have thought that the kernal would have to be an exponential function but I have no idea how to construct the kernal for this transformation. Maybe I'm missing something.

I really can't tell if that's what I need or not. The paper I'm reading doesn't actually show the calculation, just the results.

Thanks!

-Dan
 
Apr 2015
263
57
Somerset, England
Which is why I am trying to determine your application. I have no experience of topological views of these transforms (on manifolds), but there is some stuff about FT of Generalised Functions of topological manifolds.

EG Fourier Analysis in several Complex Variables

Ehrenpreis





The term Fourier transform has some variability in usage, some use it for a whole class of transforms including Laplace etc.


The extract was from a standard text on operational mathematics by Churchill.


Another reference I do not have is


Generalized Integral Transformations
by


Zemanian


Can you post or PM the relevant bit of the paper and I will see what I can dig up about how it fits in.


Romsek is good to have on your side, he has been helpful to me.


[aside question] Can I use MathML or Latex in the Physics section again yet?


Gosh, looking back I see I last answered a question about Fourier Transforms here back in 2015
and I mentioned a couple of those books then.
 
Last edited:

topsquark

Forum Staff
Jan 2006
11,565
3,453
Wellsville, NY
Which is why I am trying to determine your application. I have no experience of topological views of these transforms (on manifolds), but there is some stuff about FT of Generalised Functions of topological manifolds.

EG Fourier Analysis in several Complex Variables

Ehrenpreis





The term Fourier transform has some variability in usage, some use it for a whole class of transforms including Laplace etc.


The extract was from a standard text on operational mathematics by Churchill.


Another reference I do not have is


Generalized Integral Transformations
by


Zemanian


Can you post or PM the relevant bit of the paper and I will see what I can dig up about how it fits in.


Romsek is good to have on your side, he has been helpful to me.


[aside question] Can I use MathML or Latex in the Physics section again yet?


Gosh, looking back I see I last answered a question about Fourier Transforms here back in 2015
and I mentioned a couple of those books then.
Thanks for the references.

Yes, you can use LaTeX on PHF. The tags are math, not tex.

I'll post the whole paper. What I am looking for is near the top of page 2. Another clue here is probably what the paper refers to as "Pontryagin duality theory." I would look up the authors' pervious paper but I don't have a journal library nearby. I haven't yet got to searching for the Pontryagin duality but I have noted that (so far) I am understanding most of the rest of what I've read (up to and including section 2) so there's no major hurry.

Thanks!

-Dan

View attachment SU(2) sigma.pdf
 
Apr 2015
263
57
Somerset, England
This might actually be Differential Geometry but I'll post it here for now.

I was looking at a paper yesterday about quantization conditions when working on a manifold based on the surface of a sphere. I've never seen that before and it's apparently a lot less intuitive than I would have thought. The paper mentions a "generalized Fourier transform." The best I could do on an internet search is here. But I don't think that's what the paper had in mind because the Wolfram link looks like it's modeled in the usual 3-space, not a more generalized manifold.

Has anyone heard of this? I can quote some sections of the paper if you need it.

Thanks!

-Dan
OK having read the paper, here is my understanding of the situation.

The classic Fourier Transform is based on functions F : R → R and integrals which converge such as the Fourier sine or cosine transforms.

The first level of generalisation was to complex domains which also allowed divergent (infinite) intergals to be taken on board.
This is the level that is mostly used by Physicists and Engineers.
The level allowed the connection to Green's functions and the solution of the simplest forms of Helmholtz equation, amongst other things.

The introduction of Quantum Theory, and in particular the Dirac Delta function necessitated a further generalisation and came at the time of the theory of distributions or generalised functions which supplanted the older real and complex analysis.
The second level came with generalised Green's functions and Helmholtz equations and in theory is more or less the level of generalisation in the paper.

A third level of generalisation came linking Topology and algebra and this is where the Group theory and Noether references arise.

I have about twenty books incorporating the development of these levels and their applications in such wide fields as Functional Analysis, Operational methods, Partial Differential eaqutions, Applied Maths.
Here is a short introduction to the Fourier Transforms of Distributions from one of them on Functional analysis.

genFT1.jpg genFT2.jpggenFT3.jpg
 
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