All Questions
Tagged with convolution integration
387
questions
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17
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Integration of the product of a compact supported convolution [closed]
I know that in general case we have $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} f(s)g(t-s) ds dt = \left( \int_{-\infty}^{+\infty} f(t) dt \right) \left( \int_{-\infty}^{+\infty}g(t) dt \right) ...
- 13
4
votes
1
answer
94
views
Convolution preserve the boundary condition
Here, I want to know if convolution will preserve the Neumann condition or not. Suppose $K$ is a continuous function on some interval $[-L,L]$, and $u$ is some 'good enouth' function on $[0,L]$ that ...
- 133
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20
views
Derivative of convolution of a continuous function with a continuously differentiable function.
Suppose $f \in C^1(\mathbb R)$ has compact support and $g \in
> C(\mathbb R)$ is bounded and $\lVert g \rVert _1 < \infty$. Prove that
the convolution $f * g$ is continuously differentiable and
$...
- 485
5
votes
2
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248
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Calculate $\sum_{n=1}^\infty (1-\alpha) \alpha^nP^{*n}(x)$
$$
\mbox{Let}\quad
P'(x)=\sum_{j=1}^n a_j\frac{a^jx^{j-1}}{(j-1)!}e^{-ax},x\geq 0
$$
be the density function of a mixture of Erlangs and let $\alpha\in(0,1)$:
Is is possible to determine an analytic ...
- 91
1
vote
1
answer
25
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A 3D integral (Hylleraas wave function)
In quantum mechanical context I would like to evaluate this function (Hylleraas type wave function for Helium atom ground state):
$$ I(\boldsymbol{r}) = \int_{R^3} d^3 r' e^{- a_1 r'} \Vert \...
- 191
2
votes
0
answers
48
views
Is there a good method to recover the original functions from their convolution?
Occasionally, I failed to detect the function which is just the convolution of two simple functions. For instance
$$
2a\sin{at}*\sin{at}=\sin{at}-at\cos{at}
$$
One possible way is to observe the ...
- 1,329
4
votes
1
answer
179
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defective renewal equation
I am reading this paper of Lin and Willmot. I dont understand how they come up with formula $2.7$ and why $\tilde{\phi}(s)=\frac{\tilde{H}(s)}{1+\beta-\tilde{g}(s)}$. Can someone help me?
So i want to ...
- 351
0
votes
0
answers
71
views
Fourier transform and convolution of product of Heaviside and indicator of a ball
Let $r > 0$ and $\mathbf{w}\in\mathbb{R}^n$, and denote by $rB_n$ the ball of radius $r$ in $n$ dimensions. Consider the functions: $f_1(\mathbf{x}) = \mathbf{1}(\langle\mathbf{w},\mathbf{x}\rangle ...
- 993
1
vote
0
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42
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Solving the double compounding integral
I have the following double integral which I would like to solve:
$$\int_{0}^{100} f_S(\sigma) \sigma \int_{-\infty}^{\infty} r e^{-\frac{(r-k\sigma)^2}{2\sigma^2}} \frac{1}{\sigma\sqrt{2\pi}} \, dr \,...
- 11
0
votes
1
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41
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Is it possible to demostrate, supported by the Fubini-Tonelli that integrate $h(x)(f*g)(x)$ is equivalent to integrate $f(x)(h*g)(x)$ [closed]
there is a lot of post about integral of the convolution of two fuction. Is it possible to demostrate that using variable changes as in the Fubini the following equivalence is valid:
\begin{align}\...
0
votes
0
answers
71
views
Seeking Probability Function Invariant under Normal Gaussian Convolution
I'm currently working on a problem where I need to find a probability function,
$P(x)$, that remains unchanged after a normal Gaussian convolution. Specifically, the function should satisfy the ...
- 770
0
votes
1
answer
32
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Numerical method to solve an 'almost' convolution integral
I'm trying to solve the following Volterra integral of the first kind for $w(x)$.
\begin{equation}
P(x)=\int_{-\infty}^{\infty}[d_1(x-x_0)+d_2(x)]w(x_0)dx_0
\end{equation}
My attempt so far: I'm aware ...
- 115
0
votes
0
answers
51
views
Convolution of two PDF
I am probably overlooking like, all the important details, but when trying to work out how to take the convolution of two pdfs I am going as follows:
according to https://en.wikipedia.org/wiki/...
- 1
1
vote
0
answers
135
views
Partial derivative of a convolution
I have some doubts regarding the following partial derivative:
\begin{equation}
\frac{\partial}{\partial x(t)} \int_{t_0}^{t} h(t-\tau) x(\tau)\, \mathrm{d}\tau.
\end{equation}
Assuming that $h(t)$ is ...
- 7
1
vote
1
answer
145
views
Sum of uniform and beta distribution
Suppose $X ∼ Beta(a = 3; b = 1; θ = 1)$ and $Y ∼ U (−2, 2)$ are independent. Derive an expression for the cumulative distribution function of $X + Y$.
I am trying to do this by a convolution but I am ...
- 13