Prove Equivalence of Two Functions - Convolution Identity

In summary, the convolution identity is a mathematical theorem that states the convolution of two functions is equal to the product of their Fourier transforms. It is important because it simplifies complex expressions and makes problems more manageable. To prove equivalence using this identity, the steps involve taking the Fourier transforms, multiplying them, and then comparing the result to the convolution obtained using the original functions. This identity can be applied to any two functions with well-defined Fourier transforms and has real-world applications in signal and image processing, as well as in solving differential equations in physics and engineering.
  • #1
muzialis
166
1
Hi there,
working on a physical problem I found two functions that should be equivalent, and indeed they seem to be after a numerical check.

The functions are shown in the attached PDF. I can not figure a way to prove their equivalence analytically (the double integral especially gives me grief when trying to "unravel" it, to get to a single integral expression)
Any help would be so appreciated, thank you very much
 

Attachments

  • Convolution Question.pdf
    39.3 KB · Views: 248
Mathematics news on Phys.org
  • #2
G(2t-τ1-τ2) = G(t-τ1) G(t-τ2)/c
The double integral then factors into two separate integrals which are identical in effect.
 

1. What is the convolution identity?

The convolution identity is a mathematical theorem that states that the convolution of two functions is equivalent to the product of their Fourier transforms. In other words, the convolution of two functions in the time domain is equal to the product of their frequency domain representations.

2. Why is it important to prove equivalence of two functions using the convolution identity?

The convolution identity is important because it allows us to simplify complex mathematical expressions involving convolutions. By proving equivalence using this identity, we can reduce the amount of calculations required and make the problem more manageable.

3. What are the steps involved in proving equivalence using the convolution identity?

The first step is to take the Fourier transform of both functions. Then, multiply the two Fourier transforms together. Next, take the inverse Fourier transform of the product to get the convolution of the two functions. Finally, compare the result to the convolution obtained using the original functions to prove equivalence.

4. Can the convolution identity be applied to any two functions?

Yes, the convolution identity can be applied to any two functions as long as they have well-defined Fourier transforms. It is a general theorem that holds for all functions that meet this requirement.

5. Are there any real-world applications of the convolution identity?

Yes, the convolution identity has many real-world applications in fields such as signal processing, image processing, and physics. It is used to analyze and manipulate signals and images, and to solve differential equations in physics and engineering.

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