Jump to content

Wikipedia:Reference desk/Archives/Mathematics/2017 September 4

From Wikipedia, the free encyclopedia
Mathematics desk
< September 3 << Aug | September | Oct >> September 5 >
Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.


September 4

[edit]

(Moved to Science Desk: Wikipedia:Reference_desk/Science#Heliosphere_Moved_from_Math_Desk..)

Finding a relation between two expressions

[edit]

I have a function defined on a 2-sphere. Next I fix a certain angle . This angle is of course the angular distance from the North Pole. I need to consider only a slice of the function along one of the parallels. I need a Fourier transform expression for this function:

(1)

I also need to consider the numerical approximation of the integral above (Discrete Fourier Transform):

Forward Transform:

(2)

Inverse Transform:

(3)

I am mired in computations. This is a very small part of them, but this part affects other results. There are many FFT's and DFT's on the web and every time I compute individual members they are different from method to method but if I use them for the Inverse transform using corresponding methods, of course, I get a perfectly restored original function no matter which method I use.

In fact I need the result of the first -related expression but computing integral (1) is computationally prohibitive. I wonder if a coefficient could be found analytically connecting the complex numbers and ? It could look like this:

I use the Inverse transform for controls only and in the final variant I will not need it.

Thanks. --AboutFace 22 (talk) 17:32, 4 September 2017 (UTC)[reply]

I do not think that such an expression exists. Moreover any relation between and is likely to be non-linear and to depend on function itself. For example, if , then but . Ruslik_Zero 18:43, 4 September 2017 (UTC)[reply]
I use this simple discrete fourier transform formula
for
where is the complex conjugate of , and has nothing to do with the transcendental numbers and . Note that . This transform is its own inverse, because
for .
Bo Jacoby (talk) 18:54, 4 September 2017 (UTC).[reply]

Thank you @Ruslik0 and @Bo Jacopy. Intuitively I was ready for it. --AboutFace 22 (talk) 18:58, 4 September 2017 (UTC)[reply]