Intuitive understanding of convolution?

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SUMMARY

The discussion centers on the need for an intuitive understanding of convolution, particularly in the context of solving the heat equation and its relation to Fourier transforms. The user expresses difficulty grasping the theoretical aspects of convolution despite familiarity with its mechanics. Recommendations include exploring resources that explain convolution in depth, particularly those that connect it to Fourier transforms, which were historically significant in its development.

PREREQUISITES
  • Basic understanding of Ordinary Differential Equations (ODEs)
  • Familiarity with Partial Differential Equations (PDEs)
  • Knowledge of Fourier transforms
  • Experience with mathematical concepts related to kernels and functions
NEXT STEPS
  • Study the concept of convolution in detail through textbooks on Fourier transforms
  • Watch introductory lectures on convolution from Stanford's YouTube channel, starting from the first lecture
  • Explore online resources that provide visual explanations of convolution mechanics
  • Investigate the historical context of convolution's development in relation to Fourier analysis
USEFUL FOR

Students and professionals in mathematics, particularly those studying differential equations, as well as anyone seeking to deepen their understanding of convolution and its applications in analysis and signal processing.

filter54321
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I had a terrible adjunct professor in ODEs and got little or no theory. I'm not in PDEs and my much better professor just (re)introduced convolutions while generalizing the heat equation to Rn - unfortunately it was not a reintroduction for me.

Later chapters in the book deal with transforms, which are, I think "special" convolutions where you mix the subject function with a specific kernel function. Apparently I need to understand this concept.

Any resources for getting an intuitive understanding of what a convolution *is* and why one would want to do such a thing? I played around on Youtube and Wikipedia and I see the mechanics but not the theory.

Thanks
 
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Convolution was first derived somewhere along the progress of when mathematicians were trying to solve the problems associated with Fourier transform.

It would be best if you could find a book on Fourier Transform. You could also find the explanation on Stanford's Youtube channel but I think you would need to start from lecture 1 to understand it.
 

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