My final question is: what is the intuition behind convolution? what is its relation with the inner product? I would appreciate it if you include the examples I gave above and correct me if I am wrong.
3 The definition of convolution is known as the integral of the product of two functions $$ (f*g) (t)\int_ {-\infty}^ {\infty} f (t -\tau)g (\tau)\,\mathrm d\tau$$ But what does the product of the functions give? Why are is it being integrated on negative infinity to infinity? What is the physical significance of the convolution?
I am currently learning about the concept of convolution between two functions in my university course. The course notes are vague about what convolution is, so I was wondering if anyone could giv...
Since the Fourier Transform of the product of two functions is the same as the convolution of their Fourier Transforms, and the Fourier Transform is an isometry on $L^2$, all we need find is an $L^2$ function that when squared is no longer an $L^2$ function.
I am merely looking for the result of the convolution of a function and a delta function. I know there is some sort of identity but I can't seem to find it. $\int_ {-\infty}^ {\infty} f (u-x)\delta...
It the operation convolution (I think) in analysis (perhaps, in other branch of mathematics as well) is like one of the most useful operation (perhaps after the four fundamental operations addition, subtraction, multiplication, division) MY Question: How old the operation convolution is? In other words, the idea of convolution goes back to whom?
Here is something I've sometimes wondered about. If f g f g are both nonnegative proving commutativity of convolution can be done without a tedious change of variable. Indeed, let X X be a random variable with density f f and let Y Y be a random variable with density g g. Its easy to see that f f convolved with g g is the density of X Y X Y (or in your case X Y mod π X Y m o d 2 π). By ...
You should end up with a new gaussian : take the Fourier tranform of the convolution to get the product of two new gaussians (as the Fourier transform of a gaussian is still a gaussian), then take the inverse Fourier transform to get another gaussian.
I am aware that such "series" would never converge (in the traditional sense) unless they were countably supported, but oddly enough this helps me understand the definition for (continuous) convolution. Thank you. (It also makes me wonder if the algebraic theory for such objects, defined for instance for complex functions on groups with some nice measure, is nearly as rich as the theory of ...
My knowledge of convolution mainly comes from this document. If the answer is neither, then I wish someone can explain to me what convolution with Heaviside can be simplified into, especially if the integral's limit is from 0 to some finite t.
