1
vote
0answers
50 views

Calculating convolutions of probability density functions

I have a PDE: $$\frac{\partial N (x,u)}{\partial x}=\int _0^uN(x,u)f(u-u')du'$$ $$N(0,u) = \delta (u)$$ Here $f(u)$ is a probability density function for $0 \le u \le u_{max}$, $\int _0 ^ {u_{max}} ...
0
votes
1answer
99 views

Function as a convolution product of other two

I need help with this: I have to prove that a function $f\in L_{2}(T)$ can be expressed as $f=g*h$ (convolution product) for some functions $g,h\in L_{2}(T)$ if and only if $(\hat{f}(n))_{n}\in ...
2
votes
2answers
252 views

the fourier transform of a “double convolution”

Suppose I have a function $$ m(x) = f(x)\int_{-\infty}^{\infty} h(w)g(w-x)dw = f(x)h*g(x) $$ I want to find the Fourier transform of m(x) in terms of the Fourier transforms of $f,h,g$ but for the ...
1
vote
1answer
276 views

Intuition behind the convolution of two functions

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 ...