# Fourier transforms of power of a function

• sumesh.pt
In summary, the conversation discusses an equation involving the Fourier transform and differentiation. The equation appears to be incorrect and unclear, as the summation on the right-hand side is unspecified. It is suggested that if the equation is rewritten using Parseval's theorem and a differentiation property, it may hold true. The speaker requests clarification on this explanation.
sumesh.pt
I am not able to comprehend this :
/int dx (df(x)/dx)^2 = \sum (q^2 F(q)^2 where F(q) is the Fourier transform of f(x).

Can some one throw light?
thanks.

I'm typesetting your attempt at LaTeX properly so that I can read it:

sumesh.pt said:
I am not able to comprehend this :
$$\int\, dx \left(\frac{df(x)}{dx}\right)^2 = \sum q^2 F(q)^2$$

where F(q) is the Fourier transform of f(x).

Can some one throw light?
thanks.

I don't think that the equation, as it stands, is true. It's also not very clear because you haven't specified what range you are summing over on the right-hand side. If you told me that the Fourier transform of what was on the left-hand side was equal to the right-hand side, I might believe it. Because you mentioned the power of signal, I suspect that this might be a combination of Parseval's relation for Fourier transforms (or series?) and the differentiation property of said transform (or series).

Thanks for the reply.
In RHS summation is over all possible q. Could you please clarify the last part of your explanation where you say that it would be true if it is written as FT(LHS) and use a combination of Parsevals theorem and some differentiation property. How do I see that?
Thanks.

## 1. What is a Fourier transform of a power of a function?

A Fourier transform of a power of a function is a mathematical operation that decomposes a function into its individual frequency components. It is used to analyze the frequency content of a signal or function.

## 2. How is a Fourier transform of a power of a function different from a regular Fourier transform?

A regular Fourier transform only considers the amplitude of a function at each frequency, while a Fourier transform of a power of a function also takes into account the power or energy at each frequency.

## 3. What is the practical application of Fourier transforms of power of a function?

Fourier transforms of power of a function have various practical applications, such as in signal processing, image processing, and data analysis. They are also used in fields like physics, engineering, and neuroscience to analyze and understand complex systems.

## 4. Can a Fourier transform of a power of a function be inverted?

Yes, a Fourier transform of a power of a function can be inverted using the inverse Fourier transform. This allows us to reconstruct the original function from its frequency components.

## 5. Are there any limitations to using Fourier transforms of power of a function?

One limitation is that the function must be continuous and have a finite energy for the Fourier transform to exist. Additionally, the Fourier transform assumes that the function is periodic, which may not always be the case in real-world applications.

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