Fourier series for a random function

In summary, the conversation is discussing a problem involving the representation of an uneven surface using Fourier series with random coefficients. The coefficients must meet certain conditions and the question is how to determine the P-coefficients, which may be related to probability density distributions or Monte-Carlo numerical methods. The coefficients have a mean of 0 and are uncorrelated, but there may be an error in the Russian text regarding the correlation for certain values of m and n.
  • #1
sukharef
54
0
Hello!

My problem consists of :
there is a representation of an uneven surface in terms of Fourier series with random coefficients:
ec431c6bff5d.jpg


The random coefficients are under several conditions:
359609e3105e.jpg


W - function is undefined.

Maybe you've confronted with such kind of expressions.
The question is : how to determine P-coefficients? Maybe they can be presented by means of some well-known probability density distributions or smth with a help of Monte-Carlo numerical methods?

Thank you in advenced.
 
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  • #2
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 
  • #3
All the conditions say the coefficients have mean 0 and are uncorrelated. Otherwise any distributions with the given second moments is possible.

There appears to be an error in the Russian (I can't read Russian). The correlation =0 for m≠m' OR n≠n' while the correlation is the second moment for m=m' AND n=n'.
 

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sinusoidal functions with different frequencies and amplitudes. It is used to decompose a function into its fundamental frequency components.

2. How is a Fourier series calculated?

A Fourier series is calculated by using a mathematical formula that involves an infinite sum of sine and cosine functions with different coefficients. The coefficients are determined by integrating the original function over a specific interval.

3. Can a Fourier series be applied to non-periodic functions?

No, a Fourier series is only applicable to periodic functions. However, it can be extended to represent non-periodic functions by using a technique called Fourier transform.

4. What is the significance of the coefficients in a Fourier series?

The coefficients in a Fourier series represent the amplitude and phase of the sinusoidal functions that make up the series. They determine the shape and frequency of the original function.

5. How is a Fourier series useful in real-world applications?

A Fourier series is used in various fields such as signal processing, image analysis, and data compression. It allows us to analyze and manipulate complex signals and extract important information from them.

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