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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2
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0answers
37 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 ...
1
vote
1answer
27 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) * ...
0
votes
0answers
121 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) ...
0
votes
0answers
29 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 ...
1
vote
1answer
18 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 ...
0
votes
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 $\mathbb{T}$ ...
2
votes
0answers
46 views

Trouble writing Double Summation

I have the following: ...
2
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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 ...
0
votes
6answers
134 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 ...
1
vote
2answers
60 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 ...
0
votes
1answer
47 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
votes
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 ...
2
votes
0answers
29 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 ...
-1
votes
2answers
54 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 ...
1
vote
2answers
42 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) ...
2
votes
0answers
206 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 ...
0
votes
0answers
29 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*} ...
0
votes
2answers
85 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 ...
3
votes
1answer
103 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 ...
0
votes
1answer
36 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 ...
0
votes
1answer
37 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 ...
0
votes
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: $$ ...
1
vote
0answers
104 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 ...
1
vote
1answer
53 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 ...
0
votes
0answers
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 ...
1
vote
1answer
45 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 $v\in L^1_{loc}(\mathbb{R}^n)$. ...
0
votes
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: ...
1
vote
1answer
24 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 ...
1
vote
0answers
44 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 ...
0
votes
0answers
55 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 ...
0
votes
0answers
46 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 ...
0
votes
1answer
34 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
votes
1answer
174 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
votes
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 ...
4
votes
1answer
45 views

Prove that $f\ast g$ is continuous if $f\in C(\mathbb{T})$ and $g\in R(\mathbb{T})$

Prove that $f\ast g$ is continuous if $f\in C(\mathbb{T})$ and $g\in R(\mathbb{T})$ (Meaning $f$ is continuous and periodic and $g$ is Riemann integrable and periodic). So basically, if we define ...
2
votes
1answer
79 views

Is this a convolution?

I have the following integral \begin{align*} \int_{-\infty}^\infty f(t) q(t+ax) dt \end{align*} where a is some constant. This integral look a lot like convolution (or correlation). My question is ...
3
votes
2answers
53 views

Correct definition of convolution of distributions?

Wikipedia states, that the definition of convolution of function $f$ with a distribution $T$ is $$\langle T\ast f,\varphi\rangle=\langle T,\tilde{f}\ast\varphi\rangle$$ where $\langle ...
0
votes
2answers
42 views

What is the convolution here?

I'm reading Knuth/Graham/Patashnik's: Concrete Mathematics. In here, it's not clear to me what is the convolution, is it the act of writing as this? Is this convolution somehow ...
2
votes
0answers
104 views

Changing the order of integration in the proof that Laplace maps convolution to multiplication

I was reading the proof that Laplace transform maps the convolution of two functions to the multiplication of their transforms. Or mathematically $$\mathcal{L}[f*g]=\mathcal{L}[f]\,\mathcal{L}[g],$$ ...
0
votes
3answers
52 views

convolution integral limits

There are 2 kinds of convolution: The limit of the integral is from minus infinity to plus infinity The limit is from zero to t. When we use the first and when we use the second? $$\int ...
2
votes
1answer
114 views

Integral of the convolution of two functions: $\int_{-\infty}^{\infty} (f*g)(x)dx$

There is this proof for the integral of convolution between two functions: $$\begin{align}\int_{-\infty}^{\infty} (f*g)(x)dx&=\int_{-\infty}^{\infty}\left [ ...
0
votes
1answer
111 views

Convolution with dirac delta - proof

I have dirac delta defined as $\delta(f)=f(0)$, where $f(x)$ is an arbitrary function. I have defined convolution of distribution and function as $T\ast f=T(\tilde{f}\ast\varphi)$, where ...
1
vote
0answers
23 views

How to calculate convolution of function defining a measure

Given the function $F(t)=2-2e^{-t}$ defining a measure on $(\mathbb{R}_+,\mathfrak{B}(\mathbb{R}_+))$ and I want to calculate the convolution of this function with itself. I tried to do that by using ...
0
votes
0answers
15 views

norm bounded Convolution in matrix space

There is a stable matrix $A$ with eigen values in unit circle,for discrete time system : $x(k+1)=Ax(k)+f(k)$ can we prove: $||\Sigma_{j=0,..,k} A^{k-j}f(j)||_2<= ||f(k)||_2/{(1-A_{max})} $ where ...
2
votes
1answer
47 views

Sum of uniformly distributed random variables in a given range

I am trying to find the sum of n uniformly distributed i.i.d random variables in the range [0-W]. I am aware that if the variables are distributed in the interval (0,1) then their convolution is given ...
1
vote
0answers
25 views

Calculate FFT of 1/r green's function

I am trying to write the Poisson equation solver in C, using FFTW library. For given density of charge I need to calculate potential assuming periodic boundaries. My idea is to use convolution, simply ...
2
votes
1answer
40 views

How to find solution of the integral equation?

$$y(t) + t \int_0^t y(v)dv = 1 + \int_0^t vy(v)dv$$ I found the answer to be $y(t) = \cos{t}$. I have no idea how they go this answer. I would appreciate any suggestions how to solve this.
1
vote
1answer
41 views

Does convolution preserve strict log-concavity?

Suppose $f, g$ are strictly log-concave functions. Then the convolution $f * g$ will also be log-concave. However, will it also be strictly log-concave? Thanks!
0
votes
1answer
82 views

Convolution of discrete uniform distributions

For two independent, discrete, uniformly distributed random variables $X$ and $Y$, I wish to obtain the distribution of the sum $Z=X+Y$. I have the densities: $$f_X(x)=\left\{\begin{matrix} ...
0
votes
0answers
42 views

Product of two random variables - Resulting Distribution and Correlation?

Let $X \sim \mathcal{N}(0,1)$ and let $Z$ be a random variable independent of $X$ such that \begin{align*} P(Z=z) = \begin{cases}\frac{1}{2} & z=-1\\ \frac{1}{2} & z = 1\\ 0 & ...