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

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80 views

Sobolev norm of a convolution

Let $\eta$ be a rapidly decaying function such that it is radial and $(\mathscr{F}\eta)(\xi)=1$ for $\vert\xi\vert\leq 1$. (Here $\mathscr{F}$ is the Fourier transform). Let's put ...
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0answers
20 views

Proof of Convolution Properties

Let u,v, w $\in l_1(Z)$. I need to prove that the following are true: their convolution $u*v$ is also in $l_1(Z)$ if u and v are two probability vectors, then their convolution is also a ...
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0answers
59 views

Hessian Of Convolution's Quadratic Form

For the discrete inputs $\mathbf{x} \in \mathbb{C}^{M}$ and $\mathbf{y} \in \mathbb{C}^{N}$, I want to find the Hessian of $\Vert x \ast y \Vert_2^2$, where $\ast$ is the discrete convolution ...
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1answer
48 views

Calculate the convolution of the product of two simple functions. (5.6-12)

Synopsis: I cannot duplicate the answer given in a very reputable online symbolic integral calculator as shown in this link ($x$ is $\tau$) although my answer does appear very similar. This tells me ...
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1answer
43 views

Calculate the inverse Laplace transform by convolution. (5.6-26)

Synopsis: I cannot duplicate the answer in my text although I do get somewhat close. This tells me that my method is correct but I am making another kind of error -- perhaps in my integration? The ...
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1answer
12 views

Calculate the inverse Laplace transform by convolution. (5.6-25)

Synopsis: I cannot duplicate the answer in my text although I do get somewhat close. This tells me that my method is correct but I am making another kind of error -- perhaps in my integration? The ...
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1answer
33 views

A general theory of convolution product

in my childhood, I learned about convolution products for function over $\mathbb R$ (1). For quite a while now, I have played with polynomial rings, where also, the product is sometime called a ...
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1answer
63 views

Calculate the convolution of the product of a unit step function and t. (5.6-14)

Request Please check my work. I am not certain how to calculate the convolution of the unit step function. Given: Find the convolution of $f(t)=t$ and $g(t)=u(t-1)$. $$h(t)=(f*g)(t)=\int_0^t ...
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1answer
18 views

Calculate the convolution of the product of two sine functions. (5.6-13)

Request: I cannot duplicate the answer in the book although I do get very close. This tells me that my method is correct but I am making another kind of error -- perhaps in my integration? Given: ...
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1answer
31 views

Calculate the convolution of the product of two identical sine functions. (5.6-7)

Request I am very new to this so please bear with me. I cannot duplicate the answer in the book although I do get very close. This tells me that my method is correct but I am making another kind of ...
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3answers
32 views

Convolution Product $\sin t *\sin t$ by complex replacement.

I want to compute $$\sin t * \sin t=\int_0^t \sin u \sin (t-u) \, d u.$$ I have already tried complex replacement: $$\int_0^t \sin u \sin (t-u) \, \mathrm d u = Im{\int_0^t e^{iu} e^{i(t-u)}\,d u ...
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1answer
19 views

Energy preservation in convolution

Is the energy preserved in convolution? I convolve two functions: $$g(t) = f_1(t) \cdot f_2(t)$$ provided that the integrals of $f_1$ and $f_2$ remain unchanged, is the integral of $g(t)$ always ...
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0answers
110 views

Solving a non-linear ODE in which one term is a convolution

Does anyone have any clue of how to solve the following ODE: \begin{align} \frac{F'(t)}{1-F(t)} &= p + (q-w)F(t) + w [F'(u) * e^{-\nu u}](t) \\ \frac{F'(t)}{1-F(t)} &= p + (q-w)F(t) + w ...
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0answers
14 views

Convolution, clarification about exercise

I'm studying convolution, and I have found this exercise, but I haven't understood a step. Let's consider this function: $$f(t)=\begin {cases}1, t\in [-1,1] \\ 0, t\not \in [-1,1]\end{cases} ...
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1answer
39 views

Calculate the convolution of two constants. (5.6-1)

Request I am very new to this so please bear with me. I cannot duplicate the answer in the book. I believe I may be making a methodical error. Please correct it for me. Given: Find the convolution ...
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1answer
56 views

Multiple self-convolution of rectangular function - integral evaluation

I am trying to find an $n$-multiple convolution of a rectangular function with itself. I have a function $f(x) = 1$ for $0<x<1$, 0 otherwise. I define $$ g_2 (y) = \int_{-\infty}^{\infty} ...
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0answers
36 views

Proving $\mu\ast K_n\to\mu$

Let $\{K_n\}$ be approximating unit and $\mu\in M(\mathbb{T})$. Show that $\mu\ast K_n\to \mu$ weakly means $$\int f(t)d(\mu\ast K_n)(t)\to\int f(t)d\mu(t)$$ Suppose $\mu\in L^1(\mathbb{T})$, I ...
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1answer
25 views

What is $f(t) * g(-t)$ (convolution)?

I know that the definition of convolution is the following: $$ f(t) * g(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) \mathrm d \tau $$ Then, which is the correct one between the two: $$ f(t) * ...
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0answers
108 views

Integral of the product of two Normal distribution CDF (erf)

How do I solve the following? $$ \lim_{x \rightarrow \infty} \int_0^{x} \left[ 1 + \text{erf} \left( \frac{\epsilon - a}{b} \right) \right] \left[ 1 + \text{erf} \left( \frac{\epsilon - c}{d} \right) ...
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0answers
21 views

Expectation of a convolution between a WSS random process and an LTI system

Let $X(t)$ be a wide sense stationary random process―i.e., its expectation is a constant and its autocorrelaton function is a function only of time differences―and let $Y(t) = X(t) * h(t)$ where ...
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1answer
14 views

Find the convolution of $x(t)*h(t)$

I am studying for an exam and have the following question: $$x(t) = u(t)$$ $$h(t) = [e^{-t}-e^{-2t}]u(t)$$ where u(t) is a unit-step function. I need to find the ...
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1answer
20 views

Calculate Convolution of a funcion

Given $$f(t)=\mathbb{1}_{[-\frac a 2,\frac a 2]}(t)=\cases{1\qquad t\in[-\frac a 2,\frac a 2]\\0\qquad\text{otherwise}}\quad(0<a<\pi)$$, Calculate $f\ast f$, the convolution on $\mathbb{T}$ ...
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0answers
45 views

Trouble writing Double Summation

I have the following: ...
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0answers
35 views

Using Substitution for Convolution

Suppose I have the following product: $$\sum_{k=0}^{a}\alpha_kx^k\sum_{k=0}^{b}\beta_kx^k\sum_{k=0}^{c}\gamma_kx^k$$ Note that the bounds are finite and not equal; $a\neq b \neq c$. I'm looking to ...
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6answers
132 views

Solve $\int_{0}^{t}\cos x\sin (t-x)dx$

I want to solve the following integral $$\int_{0}^{t}\cos x\sin (t-x)dx$$ What I tried to do is integration by parts twice, but it just brought me straight back to $$\int_{0}^{t}\cos x\sin ...
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2answers
51 views

Convolution and absolute value

I have some problems to do the convolution between $f(x)=e^{-|x|}$ and $g(x)=e^{-|x|}$. $$f*g=\int_{-\infty}^{+ \infty}f(y)g(x-y)\, dy=\int_{-\infty}^{+ \infty}e^{-|y|}e^{-|x-y|}\, dy$$ I have ...
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1answer
37 views

Convolution Properties

I have a quick question about certain algebraic properties of convolution. If I have 3 functions $f(x)$, $g(x)$ and $h(x)$, is the following true? $\Big[ f(x) . g(x)\Big] \circ h(x) = \Big[f(x) \circ ...
2
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0answers
50 views

Fourier transform of $e^{-\frac{x^2}{2}}\frac{1}{\mathsf{sinc}( x )} $

I have to find inverse Fourier transform of \begin{align*} F(\omega)=e^{-\frac{\omega^2}{2}}\frac{1}{\mathsf{sinc}( \omega )} \end{align*} I was wondering whether it exists maybe in a sense of ...
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0answers
22 views

Basic questions on convolution

I am new to convolution. Below is some derivation related to convolution I saw in a paper. Hope to get some help here. (The paper is "Comparing nonparametric and parametric regresssion fit" published ...
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2answers
46 views

Convolution of 2 uniform random variables

I really do not know how to do this. Let $X$ have a uniform distribution on $(0,2)$ and let $Y$ be independent of $X$ with a uniform distribution over $(0,3)$. Determine the cumulative distribution ...
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2answers
37 views

Simple convolution problem

Let $X$ be continuous uniform over $[0,2]$ and $Y$ be continuous uniform over $[3,4]$. Find and sketch the PDF of $Z = X + Y$, using convolutions. So I have: $$f(x) = 1/2, 0 \leq x \leq 2 $$ $$f(y) ...
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0answers
197 views

Solve integral(convolution) equation

I have been trying to find a solution to the following convolution equation: \begin{align*} e^{-ax^2/2}*\ln \left(f(x)*e^{-ax^2/2}\right)+e^{-bx^2/2}*\ln \left(f(x)*e^{-bx^2/2}\right)=cx^2 ...
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0answers
27 views

Ratio of convolution expressed as a convolution $ \frac{(f_1*f_2)(x)}{(f_3*f_4)(x)}=(g*h)(x)$

Suppose we have four non-negative valued function $f_1,f_2,f_3,f_4 \in L^1$. Can we express ration of convolutions $ \frac{(f_1*f_2)(x)}{(f_3*f_4)(x)}$ as a convolution i.e. \begin{align*} ...
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2answers
81 views

Simplify ratio of integrals $\frac{\int f(x-t) t e^{-t^2/2} dt}{\int f(x-t)e^{-t^2/2} dt}$

I am trying to simplify the following expression: \begin{align*} \frac{\int_{-\infty}^\infty f(x-t) t e^{-t^2/2} dt}{\int_{-\infty}^\infty f(x-t)e^{-t^2/2} dt} \end{align*} by getting it in terms of ...
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1answer
100 views

Convolution of two indicator functions can't be constant

Let $A,B \subset S^1$ be measurable sets (considering $S^1$ with say the lebesgue measure). I'm trying to prove that if the convolution $1_A*1_B$ is constant then one of $A$ or $B$ is a full measure ...
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1answer
30 views

why does the integral of convolution equal to the product of their integral separately?

$(f*g)(x)$ is called convolution and is the integral of $f(x-y)g(y)$ with respect to $y$ on $\mathbb{R}^n$. But why the integral of $f*g$ is equal to product of integral of $f$ and $g$. Wiki says it ...
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1answer
35 views

Convolution with $\theta_t$(x) = $\frac{1}{t} \theta\bigl(\frac{x}{t}\bigr)$ for $\theta(x)$ with certain conditions

Let $\theta:\mathbb{R}\to\mathbb{R}$ be a measurable bounded function with bounded support such that $\int_\mathbb{R} \theta(x)dx = 1$ and $\theta\ge0$. Also let $f:\mathbb{R}\to\mathbb{R}$ be a ...
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0answers
19 views

How to prove these two equations

How to prove: $$x(t)*\delta^{(n)}(t) = \frac{d^n}{dt^n}x(t)$$ and $$x(t)*u(t) = \int_{-\infty}^tx(s)ds$$ To the first one, I think I could use the following formula: $$ ...
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0answers
101 views

Using the Kuramoto-Sivashinsky operator applied on the Korteweg–de Vries Soliton as a filter for image processing

When the Kuramoto-Sivashinsky operator (Kuramoto-Si) is applied to the Korteweg–de Vries Soliton (Soliton) we obtain a very interesting filter which is able to process an image via convolution. An ...
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1answer
45 views

Is the solution of a PDE always the convolution of the Green function?

If we are given Poisson's equation in a space: $$\nabla ^2 u=F$$ The solutions (those who admit Fourier transform) are given by: $$u(x)=\int_\mathbb{R^n} G(x,y)F(y)dy$$ Where $G$ is the Green ...
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30 views

A functional equation involving convolution

Let's say I have a smooth function $\varphi:\mathbb R \to \mathbb R$ which is a convolution kernel. I am interested in the following equation: $0=A+B~(\varphi*v)+C~v+D~v~(\varphi*v)$, where ...
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1answer
41 views

Derivative of convolution is the convolution with a derivative

I am trying to solve this exercise: Let $\alpha$ be a multi-index. Show that $\partial^{\alpha}(u * v)=(\partial^{\alpha}u)*v$, where $u\in C_0^{k}(\mathbb{R}^n)$ and $v\in L^1_{loc}(\mathbb{R}^n)$. ...
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1answer
39 views

Is there a convolution mistake in my method?

I have the input signal $x(t)$  And impulse response $h(t)=20 e^{-1000t} u(t)$ in which u(t) is the unit step function. When I try a convolution, I thought the solutions would be something like: ...
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1answer
21 views

Distributions convolution with heaviside

How to solve this? $$ g(t) = t \Theta (t) $$ $$ g * g(t)$$ I had hope to be able to use the $\delta $ function in some way to get eaiser calculations, but I can't see how. Is there any way to ...
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0answers
40 views

Almost everywhere convergence of convolution with mollifiers

I read that for $j\in L^1({\bf R}^n)$ with $\|j\|_1=1$ and $f\in L^1_{\rm loc}({\bf R}^n)$ the mollifiers $j_\epsilon(x):=\epsilon^{-n}j(x/\epsilon)$ exhibit $j_\epsilon\ast f\in L^1({\bf R}^n)$ and ...
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0answers
48 views

convolution integral involving modified Bessel functions of the first kind

I'm stuck with this convolution integral... \begin{equation} f_{Z}(z)=\int^{\infty}_{0}f_{1}(x)f_{2}(z-x)dx = \mbox{ } ??? \end{equation} which represents the pdf of the sum $Z = X_1 + X_2$ of two ...
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0answers
44 views

Convolution of a pdf $f$ with a Gaussian $g$: distance between $g\ast f$ and $g$?

I have been looking for references on the following matter: let $f$ be the pdf of any real-value random variable ($f$ is not necessarily continuous wrt Lebesgue measure), and $g=g_{\mu,\sigma}$ be a ...
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1answer
29 views

Relation between Correlation and Convolution

We have two functions of time $f(t)$ and $g(t)$, for which convolution and correlation are defined as following: Convolution: $(f(t)\ast g(t))(\tau) = \int_{-\infty}^\infty{f(t)g(\tau-t)dt}$ ...
0
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1answer
124 views

Convolution between impulse response

I read a paper, and am confused about the following: Suppose $W$ is an operator with impulse response (IR) $w$. And suppose $w^n$ is the IR of $W^n$. My question is the following: ...
2
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1answer
42 views

Prove that $C^\infty(\mathbb{R}^n)$ is dense in $L^2(\mathbb{R}^n, (1 + |\xi|^2)^s d\xi$

I would like to show that $C^\infty(\mathbb{R}^n)$ is dense in the space $L^2(\mathbb{R}^n, (1 + |\xi|^2)^s d \xi)$ (here, $s$ is an arbitrary element of $\mathbb{R}$). I am familiar with the ...