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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 notation.

If $s(\cdot,\cdot),\frac{\partial s}{\partial x}(\cdot,\cdot) \in C(\mathbb{R}\times\mathbb{R^{+}}), K(\cdot)\in C(\mathbb{R}) $ are integrable. The convolution between $\frac{\partial s}{\partial x}(\...
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constant function under convolution with 3 *

$p$ is prime Can someone show the intermediate steps, I don't understand the $1$st step even with the definition of a convolution in front of me. Thank you :)
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Show that convolution of two $L^1(\mathbb{R})$ functions is continuous

Suppose $f, g \in L^1(\mathbb{R})$. I want to show that their convolution is continuous. I can show continuity if one of the functions were in $L^\infty(\mathbb{R})$. I have tried to approximate ...
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Understanding/calculating the fourier coefficients of multiplied functions

I am hoping to get some clarifications/help on dealing with coefficients of a multi-dimensional Fourier series. First, I apologize for any mistakes or notations that may be off, I know just enough ...
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Log concavity of repeated convolution

Let $f\in C^\infty(\mathbb{R})$ be a smooth and positive function with support in $[-1,1]$, satisfying $\int f(t)\,dt = 1$. Define $g_1 = f$ and $g_{k+1} = f\ast g_k$. That is, $g_k$ is the $k$-fold ...
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Is integration by parts used in this equality?

The starting point is this convolution $$ \frac{\partial v_0}{\partial G}(t) = \int_0^t v_G(\xi) \Psi^{(0)}_G(t-\xi) \,d\xi. $$ Applying the product rule for differentiation \begin{align*} \frac{\...
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How does scaling in frequency domain affect real space?

I have a 3 dimensional array of real data corresponding to measurements in physical 3D space, and its corresponding data in spectral space. I want to scale certain specific frequencies in the ...
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Convolution of two step functions

Consider the probability distribution function $$ \Delta(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\geq \mu_j\} \hspace{1cm} \forall x \in \mathbb{R} $$ where $\lambda\equiv (\lambda_1,...,\...
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Showing that $f*\phi_t\to af$ in $\|\cdot\|_p$ as $t\to\infty$ with $f,\phi$ as given

From Folland's Real Analysis: Modern Techniques and Their Applications there is the following result: Folland, Theorem 8.14, (a): Suppose that $\phi\in L^1$ and $\int\phi(x)\text{d}x=a$. If $f\in L^...
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When X,Y are independent random variables we can use convolution to find the density of $X+Y$, can we do that for $X-Y$? [on hold]

When X,Y are independent random variables we can use convolution to find the density of $X+Y$, can we do that for $X-Y$? Is there an analogical case?
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Questions on the proof of $f*g\in C^\infty(\mathbb R)$ when $f\in L^2(\mathbb R)$ and $g\in C_c^\infty(\mathbb R)$

I am working through the proof that the convolution of a square integrable function with a compactly supported continuously differentiable function is itself continuously differentiable: "Let $f\in L^...
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Suggestion of article or book for convolution

I need a good book or article to learn about convolution.I have a course in Neural Networks and we have to make calculations by hand.
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Trying to find the CDF of $X+Y$ when $X\sim exp(\alpha)$ and $Y \sim exp(\beta)$ (independent) without convolution, but it doesn't seem to work

The textbook I am using, using convolution in order to find the CDF of the $X+Y$ when $X\sim exp(\alpha)$ $Y\sim exp(\beta)$, and X and Y are are independent. However, I have no background with ...
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Can we say anything about the first distributional derivatives of $g$, where $g$ is the solution to $-\Delta g =f\in L^p$ given by Riesz potential?

If $n\ge3$ and $\omega_n$ is the $n-1$-dimensional Hausdorff measure of the unit sphere in $\mathbb{R}^n$, define: $$K_n(x):=\frac{1}{(n-2)\omega_n} \frac{1}{|x|^{n-2}}.$$ Then $K_n$ is locally ...
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Recovering original image from its edges

Suppose we read an image $X$ with $1\times P$ dimensions (a single row and $P$ columns) and apply to it the simplest edge detector, that calculates the horizontal derivative say, $F = [1, 0, −1]$ to ...
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associative law for convolution of distribution with test functions

Let ${\mathcal D} = C^\infty_c$, the space of test functions (smooth functions $\Bbb R \to \Bbb C$ with compact support). Let $\mathcal D^*$ be the space of distributions (continuous linear ...
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Integro-differential equation with convolution

Given a $\mathcal{C^\infty}$ matrix-valued function $f$ from $\mathbb{R}^+$ to $\mathbb{R}^{n,n}$, I'd like to solve the following integro-differential equation: $$\ddot x(t) + \int_0^t f(\tau) \dot ...
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1answer
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Is Young's inequality useful here?

I want to prove that for a given $0<\alpha<N$ and for all $0<\varepsilon< N-\alpha$ there exists $C>0$ s.t. $$ \Vert\vert x\vert^{-\alpha}\ast\vert u \vert^2\Vert_{L^\infty(\mathbb{R}^N)...
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1answer
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confusion on characteristic wrt convolution and product measure

I have recently come across characteristic functions. Let $X,Y$ be random variables on $(\Omega, \mathcal{F}, P)$ Let $\widehat{P_{X}}$ and $\widehat{P_{Y}}$ denote the respective characteristic ...
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Fourier Transform of Dyson's Equation

I'm attempting to show that the Fourier transform of Dyson's equation for a constant potential V, \begin{equation} G(\mathbf{r},t,\mathbf{r}_0,t_0) = G^0(\mathbf{r},t,\...
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Partial Differential Equations: Find an explicit solution of IVP for diffusion equation

Find an explicit solution of IVP for the diffusion equation: $u_{t}=u_{xx}, x \in\ {R}, t>0$ $u(0,x)=x, x\in R$ So, I used the normalization integral, differentiated it in t and then ...
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Expression for density of sum of two jointly distributed random variables

I read on here that given two jointly distributed random variables $X$ and $Y$, the density of their sum (let $Z=X+Y$) can expressed as $$f_Z(z) = \int_{-\infty}^{\infty}\int_{-\infty}^{z-x} f_{X,Y}(...
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Convolution of molifiers with measure

I would like your opinion on a computation i found in a statistical mechanics paper : Let $\nu$ a prob measure on $\mathbb{R}^{d}$, $V:\mathbb{R}^{d} \rightarrow \mathbb{R}$ continuous, belongs to $L^...
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Two dimensional Laplace transform of convolutions

On a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, let us consider the joint law of two random variables $X$ and $Y$ $$\nu(B_1\times B_2):=\mathbb{P}(X\in B_1, Y \in B_2) $$ where $B_i\in\...
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Convolution of identical density functions

Let $X,Y$ be random variables each with density $f(x)=2xI_{x ∈[0,1]}$ , where $I_{A}$ is the indicator function on the event $A$. What is the density of the random variable $U:=X+Y$? I start by using ...
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67 views

Convolution of two probability distributions

I would like your help to find the correct definition of convolution of two probability distributions. I found several references on that, but very complicated, involving random variables, and more. ...
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Convolution of $\chi_{[0,1]}$ with itself [duplicate]

The characteristic function of a set $E$ is defined as follows: $\chi_{E}(x) :=1 \space \text{if} \space x\in E, \space \text{and} \space \chi_{E}(x) := 0 \space \text{if} \space x \notin E.$ Find a ...
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How to solve this DE involving a convolution $\int_0^t ds \ e^{-\kappa s} \cos(\omega s) f(t-s)$?

I've got a DE of the form $$ \frac{df}{dt} = A - B f(t) - C \int_0^t ds\ e^{-\kappa s} \cos(\omega s) f(t-s) $$ which I want to solve given an initial condition $f(0^{+})=f_0 \in \mathbb{R}$. All the ...
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Convolution with compact support function

I have a basic question which I am confused about. If I have two functions $f(x)$, and $g(x)$, where $g(x)$ has a compact support say $[-M,M]$. Can I always say that I can just consider the integral ...
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1answer
36 views

Convolution of a Binomial and Uniform Distribution

I am given that $X$ is a random variable with a Binomial distribution with parameters $(n,p)$ and that $Y$ is a random variable with a Uniform distribution on $(0,1)$. We assume independence. I want ...
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Is there a term for the additivity of parameters in distributions closed under convolution?

Consider a collection of independent random variables $X_i$, each with probability density function $p(x;\theta_i)$. Suppose the variable $Z=\sum_iX_i$ has as density function $p(x;\sum_i\theta_i)$, ...
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Linearity in Proof of Young's Convolution Inequality

Convolution Inequality. Let $f \in \mathcal{L}^p(\lambda^d), \ g \in \mathcal{L}^1(\lambda^d)$ and $1 \leq p < \infty$. Then $f * g$ is defined almost everywhere with $f*g \in \mathcal{L}^p(\lambda^...
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Why is Convolution Well-Defined (Simple Example)

I am having trouble understanding why convolution is well-defined. Let's take a simple example: $(\Omega, \mathcal{F}, P)$ probability space and $X_{1}, X_{2}$ two real random variables where $P(X_{...
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What is a Spectral Graph Convolution?

I'm reading about Graph Neural Networks and I would like to understand more about first-order and second-order approximations of spectral graph convolution. What is a Spectral Graph Convolution? ...
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Is there a preference if one of the functions in convolution of Mellin transform is divergent?

The convolution of Mellin transform is $$ \sigma \left( x \right) = \int _x^1 f \left( \epsilon \right) h \left( \frac{x}{\epsilon} \right) \frac{1}{\epsilon} \mathrm{d} \epsilon , $$ if both $f \...
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Effect of a convolution with a Bernoulli distribution on Rényi divergence

Let $P$ and $Q$ be two probability distributions on $\mathbb{Z}$. Let $D_\alpha(P\|Q)$ be the Rényi divergence of order $\alpha$ of $P$ and $Q$: $$ D_\alpha(P\|Q)=\frac{1}{\alpha-1}\sum_i\frac{P(i)^\...
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Why does $\int_{-\infty}^\infty f(x-y)g(y)dy$ exists when $\int_{-\infty}^\infty f(x)dx<\infty$ and $\int_{-\infty}^\infty g(x)dx<\infty$

Why does $\int_{-\infty}^\infty f(x-y)g(y)dy$ exists when $\int_{-\infty}^\infty f(x)dx< \infty$ $\int_{-\infty}^\infty g(x)dx< \infty$ where $f,g$ are maps from $\mathbb{R}$ to $\mathbb{R}_+$ ...
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Finding an unknown function out of an DE (involving an integral and an inverse Laplacetransform)

Well, I've the following problem for an unkown function $x(t)$: $$x'(t)\cdot\text{a}+\text{b}\cdot\frac{x'(t)}{x(t)+\text{c}}+\frac{\partial}{\partial t}\left\{\int_0^tx(\tau)\cdot\mathcal{L}_\text{s}...
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Convolution theorem for a function that doesn't vanish at infinity

I'm studying a paper of Rapp and Kassal (1968), in which a second order differential function that doesn't vanish at $\pm \infty$ is solved. I will put here the parts of the paper that matter: $\...
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Convolution with unusual limits

I would like to find the Fourier transform of the following function: $$ \mathcal{F}(x) = \int_{\vert x\vert}^\infty \! {f}^\ast\left((r-x)/2\right){f}\left({(r+x)}/{2}\right)\operatorname{d}\!r $$ ...
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find the probability about sum of random variables

Let $X_1, X_2, X_3, Y_1, Y_2, Y_3, Z_1, Z_2, Z_3$ be random variables which have uniform distribution between 0 and 1. It means, the average of $X_1 = 0.5$ Let: $X=X_1 + X_2 + X_3,$ $Y=Y_1 + Y_2 + ...
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Given an input image of 128x128, and output 29x29. What is the size of the area of the input that is middle pix in output

Given an input image of 128x128, after putting it through a convolutional neural network the output is 29x29. What is the size of the area (in pixels) of the input that correspond to the middle pixel ...
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Young's convolution inequality: Equivalent representations

According to Wikipedia Young's inequality for convolutions states that For functions $f \in L^p$ and $g \in L^q$ one has $|| f*g ||_r \leq ||f||_p ||g||_q$ $\hspace{6.75cm}$ (Eq. 1) with $...
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Fourier transform of the convolution of a Dirac comb with the product of a complex exponential function and a rect function

Straight to the math question: How can I calculate the following 1-dimensional spatial Fourier transform? $\frac{1}{2\pi}\int_{-\infty}^{\infty}\left(e^{i(n-1)k_0\frac{x^2}{2R}}\mathrm{rect}(x/w)*\...
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Solving for moments of $u(x,t)$ given the equation $u_t=xu+f\ast u$

Given the evolution equation $$ \frac{\partial u}{\partial t}(x,t)=xu(x,t)+\int_{-\infty}^\infty f(x-y)u(y,t)dy $$ with initial condition $u(x,t)=\delta(x-x_0)$, is there a way to solve for either or ...
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1answer
38 views

Integro-differential equation including a convolution of the first derivative.

I am having difficulty finding the right approach to solving the following differential equation, $$ y''(t)+\int_t^Tg(s-t)y'(s)\,ds=f(t), $$ with the boundary conditions, $$y(0)=y_0\,,\quad y(T)=0.$$ ...
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1answer
62 views

Approximate Identity

Let ${({\varphi}_{n}})_{n=1}^{\infty}$ an Approximate Identity in Schwartz Space. Let $\alpha \in \mathbb{Z}^+$. Is it true or not the following statement? \begin{equation} \lim _{ n\...
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1answer
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Commutative Convolution. Problem 26 Royden 2 ed.

Let $f$ and $g$ be functions in $L^1(—\infty,\infty)$, and define $f\ast g$ to be the function $h$ defined by $h(y) = \int f(y — x)g(x) dx$. Why $f\ast g=g\ast f$? I have this: If $y-x=z$ then $\int ...
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2answers
52 views

Convolution is well defined in $L^1$

I know that this question has been "answered" before, but I am struggling to understand the solutions previously given. Let $f$ and $g$ be in $L^1(R, L, m)$ a) Show that $f*g$ is well defined for a....
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1answer
37 views

Show that $g_1(\mu_1,\sigma_1)\ast g_2(\mu_2,\sigma_2)=g\left(\mu_1+\mu_2,\sqrt{\sigma_1^2+\sigma_2^2}\right)$

The definition (convention) I have been using for the Fourier transform is $$\mathscr{F}[f(t)]=g(\omega)=\frac{1}{\sqrt{2 \pi}}\int_{t=-\infty}^{\infty}f(t)e^{i\omega t}dt\tag{1}$$ and the inverse as ...