A quesiton about multidimensional Fourier transform

In summary, the conversation discusses the Fourier transform and the use of a 4-dimensional analog to spherical coordinates to eliminate the integration about angular variables. The speaker also mentions that the integral over the 4-dimensional unit sphere includes the factor cos(u).
  • #1
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my question is the following

let be the Fourier transform [tex] \int_{-\infty}^{\infty}d^{4}p \frac{exp( ip*k)}{p^{2}+a^{2}} [/tex]

here [tex] p^{2}= p_{0}^{2}+p_{1}^{2}+p_{2}^{2}+p_{3}^{2} [/tex]

is the modulus of vector 'p' , here * means scalar product

for the scalar product i can use the definition [tex] p*k= |p|.|k|.cos(u) [/tex]

so ony the modulus appear, my question is if there is a way to set cos(u) =1 to get rid of the integration about angular variables, thanks
 
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  • #2
There is (I don't know the form offhand) a 4 dimensional analog to spherical coordinates. Let p be the magnitude of (p1,p2,p3,p4). The differential will look like p3dpdA where A is the 4-d analog of surface area of the unit sphere. The A integral is over the 4-d unit sphere - includes the cos(u) factor. The p integral is from 0 to ∞.
 

1. What is a multidimensional Fourier transform?

A multidimensional Fourier transform is a mathematical operation that decomposes a function or signal into its constituent frequencies in multiple dimensions. It is an extension of the traditional Fourier transform, which operates on one-dimensional functions.

2. What is the purpose of using a multidimensional Fourier transform?

The purpose of using a multidimensional Fourier transform is to analyze complex signals or functions that vary in multiple dimensions. It allows for the separation of different frequencies and their respective contributions to the overall signal or function.

3. How is a multidimensional Fourier transform different from a traditional Fourier transform?

A multidimensional Fourier transform differs from a traditional Fourier transform in that it operates on functions or signals that vary in multiple dimensions, as opposed to just one dimension. It also produces a multidimensional spectrum, rather than a one-dimensional spectrum.

4. What are some practical applications of a multidimensional Fourier transform?

A multidimensional Fourier transform is commonly used in fields such as image and signal processing, medical imaging, and quantum mechanics. It is also used in the study of complex phenomena, such as weather patterns and fluid dynamics.

5. What are the limitations of a multidimensional Fourier transform?

One limitation of a multidimensional Fourier transform is that it assumes the function or signal being analyzed is periodic, meaning it repeats itself infinitely. It also requires the function or signal to be square-integrable, meaning its energy is finite. In addition, it is not always suitable for analyzing non-linear or time-varying signals.

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