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Why do we define the convolution?

Why is convolution useful?

What is the purpose of the geometry of convolution of two functions in plane? Can we draw the convolution of two functions without compute the integration?

What is idea of definition of convolution?

What is relation between convolution and probibility?

I have already read https://en.wikipedia.org/wiki/Convolution


marked as duplicate by Rahul, Math1000, user223391, Leucippus, levap Dec 10 '16 at 2:55

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    $\begingroup$ You may find this post to be useful $\endgroup$ – Omnomnomnom Dec 9 '16 at 17:25
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    $\begingroup$ Was there something in the Wikipedia article that you didn't understand? What would you like clarified? $\endgroup$ – Thomas Andrews Dec 9 '16 at 17:28

In one interpretation of the convolution of two functions, one of them is interpreted as the input signal to a system, and the other the impulse response of that system. Therefore, the convolution gives the system's response to the input. See Section 17.4.1 in this book: https://books.google.com/books?id=q5KRCwAAQBAJ&pg=PA109&dq=zorich+mathematical+analysis+vol+I&hl=en&sa=X&ved=0ahUKEwilx9yf0OfQAhXrrVQKHdbFBdkQ6AEIJzAC#v=onepage&q=convolution&f=false

As a special case, the convolution of a function $f(t)$ with the Dirac delta function gives $f(0)$ (the response of the system at $t=0$).

Because of the above, the input signal is thought of as "smearing" the impulse response on the time axis. Mathematically, it means that convolution of a given function $f$ with a smooth function (a "smooth smearer") results in something smoother than $f$. This type of smoothing is extremely useful when you want to use differentials.

It also turns out that, on mapping from the time domain to the frequency domain (using a Fourier or Laplace transform), convolution of time-dependent functions maps to the product of frequency-dependent functions.

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    $\begingroup$ That is one view of convolution. There is a more symmetric view that comes up in probability. $\endgroup$ – Thomas Andrews Dec 9 '16 at 17:29
  • $\begingroup$ @ThomasAndrews. I agree. I just wanted to start by tying it to something tangibly empirical, like physical devices and measurements. $\endgroup$ – avs Dec 9 '16 at 18:22
  • $\begingroup$ The problem is the absolute statement "one of them is interpreted..." It should be "in one interpretation of ... one of them is interpreted...." You start with a strong statement which can be misleading. $\endgroup$ – Thomas Andrews Dec 9 '16 at 18:27
  • $\begingroup$ @ThomasAndrews, I agree again. Edited accordingly, and thanks. $\endgroup$ – avs Dec 9 '16 at 19:18

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