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
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.