Fourier transform of a function in spherical coordinates

In summary, the conversation discusses the relationship between Fourier conjugates in the spherical basis for two functions, where one is the Fourier transform of the other. The conversation also includes a coordinate transformation from Cartesian to spherical coordinates and addresses mistakes made in the transformation. Finally, it is mentioned that if a function's Fourier transform is independent of certain variables, then the function itself is also independent of those variables.
  • #1
redtree
285
13
TL;DR Summary
I am trying to understand the relationship between Fourier conjugate bases in spherical coordinates
I am trying to understand the relationship between Fourier conjugates in the spherical basis. Thus for two functions ##f(\vec{x}_3)## and ##\hat{f}(\vec{k}_3)##, where

\begin{equation}

\begin{split}

\hat{f}(\vec{k}_3) &= \int_{\mathbb{R}^3} e^{-2 \pi i \vec{k}_3 \cdot \vec{x}_3} f(\vec{x}_3 d\vec{k}_3

\end{split}

\end{equation}

where ##\vec{x}_3 = [x_1,x_2,x_3]## and ##\vec{k}_3 = [k_1,k_2,k_3]##
In spherical 3-space coordinates,

\begin{equation}

\begin{split}

\hat{f}(\varrho, \xi_1, \xi_2) &= \int_{0}^{\infty} \int_{0}^{1} \int_{0}^{1/2} e^{-2 \pi i (\varrho r + \xi_1 \theta_2 + \xi_2 \theta_2)} f(r,\theta_1,\theta_2) dr d\theta_1 d\theta_2

\end{split}

\end{equation}

where ##\vec{x}_3 = [r,\theta_1,\theta_2]## and ##\vec{k}_3 = [\varrho,\xi_1,\xi_2]##
Thus, for a function ##\hat{f}\left( \big(\vec{k}_3\big)^2 \right)##, where in spherical coordinates ##\big(\vec{k}_3\big)^2 = \big( \varrho \big)^2##,

\begin{equation}

\begin{split}

\hat{f}\left( \big(\vec{k}_3\big)^2 \right) &= \hat{f}\left(\big( \varrho \big)^2 \right)

\\

&= \int_{0}^{\infty} e^{-2 \pi i \varrho r} f(r^2) dr

\end{split}

\end{equation}

such that ##\hat{f}\left( \big(\vec{k}_3\big)^2 \right)## is independent of ##\theta_1## and ##\theta_2##. Is that correct? Am I missing something?
 
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  • #2
How are ##\theta_1## and ##\theta_2## defined?
 
  • #3
##0 \leq \theta_1 \leq 1/2 ##, such that ## 0 \leq 2 \pi \theta_1 \leq \pi##
## 0 \leq \theta_2 \leq 1##, such that ## 0 \leq 2 \pi \theta_2 \leq 2 \pi##
 
  • #4
I’m not sure I really understand, but in any case you definitely did your coordinate transformation wrong. Show us how you did it.
 
  • #5
\begin{equation}
\begin{split}
x_1 &= r \sin{2 \pi \theta_1} \cos{2 \pi \theta_2}
\\
x_2 &= r \sin{2 \pi \theta_1} \sin{2 \pi \theta_2}
\\
x_3 &= r \cos{2 \pi \theta_1}
\end{split}
\end{equation}
where ##r \geq 0##, ##0 \leq \theta_1, \leq \frac{1}{2}## and ##0 \leq \theta_2 \leq 1##
 
  • #6
Of course your first integral should be over physical space, not wave-vector space. That integral should include (using your notation I think) ##d\vec{x}_3## which has dimensions of volume. However, when you transformed the coordinates you somehow have ##dr d\theta_1 d\theta_2## which has dimensions of length. That should clue you in that it cannot possibly be correct. Have you transformed integrals from Cartesian to spherical coordinates before?
 
  • #7
redtree said:
\begin{equation}
\begin{split}
x_1 &= r \sin{2 \pi \theta_1} \cos{2 \pi \theta_2}
\\
x_2 &= r \sin{2 \pi \theta_1} \sin{2 \pi \theta_2}
\\
x_3 &= r \cos{2 \pi \theta_1}
\end{split}
\end{equation}
where ##r \geq 0##, ##0 \leq \theta_1, \leq \frac{1}{2}## and ##0 \leq \theta_2 \leq 1##
Where is the mistake in the coordinate transformation?
 
  • #8
These equations

redtree said:
\begin{equation}
\begin{split}
x_1 &= r \sin{2 \pi \theta_1} \cos{2 \pi \theta_2}
\\
x_2 &= r \sin{2 \pi \theta_1} \sin{2 \pi \theta_2}
\\
x_3 &= r \cos{2 \pi \theta_1}
\end{split}
\end{equation}
where ##r \geq 0##, ##0 \leq \theta_1, \leq \frac{1}{2}## and ##0 \leq \theta_2 \leq 1##
Are fine, but when you used them to change the variables of integration you did most of it wrong. For example ##d\vec{x}_3## should transform (if I did the math right) to ##4\pi^2 \, r^2 \, \sin(2\pi\theta_1) \, dr \, d\theta_1 \, d\theta_2##. Also
$$
\vec{k}_3\cdot \vec{x}_3 = k_1\, r\, \sin{2 \pi \theta_1} \cos{2 \pi \theta_2} + k_2\, r\, \sin{2 \pi \theta_1} \sin{2 \pi \theta_2} + k_3\, r\, \cos{2 \pi \theta_1}
$$

This is standard stuff for changing coordinates in multiple integrals, as learned in a standard calculus sequence. Have you learned multivariable calculus?

jason
 
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  • #9
jasonRF said:
These equationsAre fine, but when you used them to change the variables of integration you did most of it wrong. For example ##d\vec{x}_3## should transform (if I did the math right) to ##4\pi^2 \, r^2 \, \sin(2\pi\theta_1) \, dr \, d\theta_1 \, d\theta_2##. Also
$$
\vec{k}_3\cdot \vec{x}_3 = k_1\, r\, \sin{2 \pi \theta_1} \cos{2 \pi \theta_2} + k_2\, r\, \sin{2 \pi \theta_1} \sin{2 \pi \theta_2} + k_3\, r\, \cos{2 \pi \theta_1}
$$

This is standard stuff for changing coordinates in multiple integrals, as learned in a standard calculus sequence. Have you learned multivariable calculus?

jason
Got it. Thanks!
 
  • #10
such that,
\begin{equation}
\begin{split}
\hat{f}(\varrho,\xi_1,\xi_2) &= \int_{0}^{1} \int_{0}^{1/2} \int_{0}^{\infty} \text{Exp}\left[-2 \pi i \varrho r \big(\cos{2 \pi \theta_1 } \cos{2 \pi \xi_1 } + \cos{2 \pi (\theta_2 - \xi_2) } \sin{2 \pi \theta_1 } \sin{2 \pi \xi_1 }\big) \right]
\\
&4\pi^2 \, r^2 \, \sin(2\pi\theta_1) \, dr \, d\theta_1 \, d\theta_2
\end{split}
\end{equation}
 
  • #11
Does it remain true that if ##f\left( (\vec{x}_3)^2 \right) = r^2##, then ##f(\vec{x}_3) = f(r)##, where ##f(r) = \mathscr{F}^{-1}[\hat{f}(\varrho)]##?
 

1. What is a Fourier transform?

A Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies. It is commonly used in signal processing and image analysis to analyze and manipulate data in the frequency domain.

2. What are spherical coordinates?

Spherical coordinates are a system of coordinates used to describe points in three-dimensional space. They consist of a radial distance, an azimuthal angle, and a polar angle, and are commonly used in physics and engineering to describe the location of objects in spherical or cylindrical shapes.

3. How is a Fourier transform performed in spherical coordinates?

In spherical coordinates, the Fourier transform of a function is calculated by integrating the function over all possible values of the azimuthal and polar angles. The resulting transform is a function of the radial distance and represents the frequencies present in the original function.

4. What are the applications of a Fourier transform in spherical coordinates?

A Fourier transform in spherical coordinates has several applications, including signal processing, image analysis, and solving differential equations in spherical geometries. It is also used in quantum mechanics and electromagnetic theory to solve problems involving spherical symmetry.

5. Are there any limitations to using a Fourier transform in spherical coordinates?

One limitation of using a Fourier transform in spherical coordinates is that it assumes the function being transformed is periodic in all three dimensions. This may not always be the case in real-world applications, and alternative methods may need to be used. Additionally, the computational complexity of performing a Fourier transform in spherical coordinates can be high, making it challenging to use for large datasets.

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