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Suppose $f(x)$ and $g(x)$ are two functions. What is intuition or idea behind the convolution of $f$ and $g$? After taking the convolution we will get a new function. What is the geometric relation between $f$, $g$, and $f*g$ ?

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Have a look at this: jhu.edu/signals/convolve –  Giuseppe Negro Jan 23 '13 at 5:26
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The images at en.wikipedia.org/wiki/Convolution are very useful, as pointed out by @macydanim. Since convolution is linear in each function, you can try to get a feeling for the general situation by considering step functions using linear combinations of the first animation. –  user108903 Jan 23 '13 at 7:45
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I suspect this can be answered by a google search or a wikipedia search on the site. Usually people think $f*g$ as an "average" of $f$ with $g$, such that $f*g$ has at least as nice properties as $f$ and $g$. I am sure other people at here can give a much better answer, but hopefully the meaning of it will be clear if you encounter Fourier series or Fourier transform, since that's where they appear most naturally and frequently.

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