Can someone explain this step in the proof of the convolution theorem?

Therefore, the last step is correct.In summary, the conversation discusses a step in the proof of the convolution theorem, specifically the last step where the integral is written as a product of two separate integrals. The person is questioning the validity of this step, as they believe it is only allowed when y is independent of x. However, it is explained that after the substitution of y = z - x, y becomes a dummy variable and is still independent of x in the integral over all space. This makes the last step correct.
  • #1
codiloo
2
0
I fail to understand a step made in this proof:
http://en.wikipedia.org/wiki/Convolution_theorem"

more specifically the last step where the integral is written as a product of 2 separate integrals (each equal to a Fourier transform):
from:
30aa5ce6a4881f46515121f4ccfee81d.png

to:
97cc52195eb954a68ef235c3969e3f02.png

I'm quite rusty on my integration, but as far I can remember this operation is only allowed when y is independent of x. (since y is taken out of an integral over x). But since we substituted y = z − x this is not the case. Can somebody explain me why this step is correct? (and why I'm wrong)
 
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  • #2
After the substitution, y is just a dummy variable and consequently independent of x. If it was a definite integral, the substitution would have moved the dependency on x to the integral limits but because the integral is over all space, they are still independent of x after the substitution.
 

FAQ: Can someone explain this step in the proof of the convolution theorem?

What is the convolution theorem?

The convolution theorem is a mathematical concept that relates the convolution operation in the time domain to the multiplication operation in the frequency domain. It states that the Fourier transform of the convolution of two functions is equal to the product of their individual Fourier transforms.

Why is the convolution theorem important?

The convolution theorem is important because it allows us to simplify complex mathematical operations involving convolutions by transforming them into simpler multiplication operations in the frequency domain. This is particularly useful in signal processing and image processing applications.

What are the steps involved in proving the convolution theorem?

The proof of the convolution theorem involves several steps, including applying the Fourier transform to the convolution of two functions, using the convolution theorem for the Fourier transform, and then taking the inverse Fourier transform to obtain the desired result.

Can you explain the step involving the convolution theorem for the Fourier transform?

The convolution theorem for the Fourier transform states that the Fourier transform of the convolution of two functions is equal to the product of their individual Fourier transforms. This allows us to simplify the convolution operation by transforming it into a multiplication operation, making it easier to solve.

How is the convolution theorem used in practical applications?

The convolution theorem is used in a wide range of practical applications, including signal processing, image processing, and digital filtering. It allows for efficient and accurate calculations of convolutions, making it a valuable tool in various fields of science and engineering.

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