# Questions tagged [convolution]

Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schwartz's sense) or measures.

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### Convolution Integral with Same Direction in Integrand

I am working on the convolution below, however I have gotten stuck. I am not sure how to think about changing the bounds of the integrals to give me an answer. Here is the problem, where $u(t)$ is the ...
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### Fourier Transform Method for Solving Fredholm Translation-Invariant Covariance-Kernel Integral Equations

I'm examining a problem involving the Fredholm integral equations of the second kind and trying to apply Fourier transform techniques. Particularly, the challenge arises when seeking to express the ...
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### Do there exist mathematical transforms other than the Fourier Transform for which there exists some sort of a fast convolution theorem?

One nice property of the Fourier transform is it's famous convolution theorem : $$f*g = \mathcal{F}^{-1} \left\{ \mathcal{F}\left\{ f \right\} \cdot \mathcal{F}\left\{g\right\} \right\}$$ If we want ...
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### Convolution of two uniform probability densities (two square waves)

Assuming $X$ and $Y$ are i.i.d. random variables let $Z = X + Y$ $$f_X(x) = f_Y(y) = \begin{cases} 1/2 & -1 \le x \le 1 \\ 0 & \text{else} \end{cases}$$ Find the density ...
1 vote
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### Is the convolution of a tempered distribution and a Schwartz function also a function?

Let $T \in \mathscr{S}'(\mathbb{R}^n)$ and $f \in \mathscr{S}(\mathbb{R}^n)$. We may define their convolution as $$(T * f)(\varphi) = T(\tilde{f}*\varphi)$$ where $\tilde{f}(x) = f(-x)$. The above ...
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### Calculate the limit of $\langle \partial_x f,\rho_\varepsilon \ast \chi_B \rangle$

Let $f(x_1,x_2)=x_2^2\chi_E$ where $E=\{(x_1,x_2)\in \mathbb{R}^2:x_1\geq 0 \}$ and let $B$ be the unit ball in $\mathbb{R}^2$. Since $f$ is locally integrable, it can be interpreted as a distribution ...
1 vote
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### Equation for Gaussian kernel's effect on a frequency's amplitude

Applying a Gaussian blur/kernel with a sigma of $\sigma_{gau}$ to a sine/cosine wave of frequency $f_{sin}$ will cause what percent reduction in the amplitude $p_{amp}$ (not power) of the sine wave? ...
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### A nested double sum(to do with e?)

I’ve come across this nested double sum while doing an investigation but cannot seem to find a closed form for it. $$1+\sum_{i=1}^\infty{\frac{1}{i!} \sum_{j=0}^i}{\frac{1}{j!}}$$ This is about the ...
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### Is the integral $\int_{-\infty}^\infty dx \int_{-\infty}^\infty dy \frac{e^{i a x}}{\sqrt{x^2+y^2}}$ convergent?

Is the integral $$I = \int_{-\infty}^\infty dx \int_{-\infty}^\infty dy \frac{e^{i a x}}{\sqrt{x^2+y^2}}$$ convergent for real $a$? I have an idea to calculate it but I am not sure if it is correct: ...
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### Exact Successor State Distribution for a Pendulum

I want to solve the following problem. Suppose we have a simple pendulum, which follows the differential equation \begin{equation} \dot{x} = f(x) = [x_2, -\sin(x_1)]^T, \text{with } x=[x_1, x_2]^T. \...
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### Convolution of Double Coset Indicator Function in Hecke Algebra of Locally Profinite Group

Let $G$ be a locally profinite group (i.e. a topological group that is locally compact Hausdorff and totally disconnected or, equivalently, a Hausdorff topological group s.t. $1 \in G$ has a ...
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### Extension of single zero-crossing property

Let $f\in\mathscr{C}^2(\mathbb{R},\mathbb{R})$ a strictly increasing function, striclty convex on $(-\infty,0)$, strictly concave on $(0,\infty)$ and let $\sigma_1>\sigma_2>0$ be two real ...
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### Zero-crossing for convolution

Suppose that you have a function $f\in\mathcal{C}^2(\mathbb{R},\mathbb{R})$. Can one show that if the function $g$ defined by: $$g(x):=\int_\mathbb{R}f(s)e^{-(s-x)^2/2}ds$$ has to zeros $x_1$ and $x_2$...
1 vote
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### What is this operation?

Let $F[n]$ and $G[n]$ be arrays of length $N$. At first, $G=F$. After initialisation, $G$ is calculated by the relation ...
1 vote
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