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
15 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
27 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
34 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
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0answers
149 views
+50

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
26 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
73 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 ...
5
votes
0answers
41 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
17 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
33 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
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0answers
18 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
94 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
23 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
13 views

A functional equation involving convolution

Let's say I have a smooth function $\varphi:\mathbb R \to \mathbb R$ which could be either: a convolution kernel or a difference of two convolution kernels: $\varphi=\varphi_1-\varphi_2$. I am ...
1
vote
1answer
35 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
38 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
19 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
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0answers
25 views

Almost everywhere convergence of convolution with mollifiers

I read that for $j\in L^1({\bf R}^n)$ with $\|f\|_1=1$ 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
19 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
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0answers
39 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
21 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
40 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
41 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
40 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
72 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
39 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
39 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
88 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
28 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
84 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
52 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
19 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
12 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
25 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
16 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
38 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
40 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
58 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
23 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 & ...
1
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0answers
44 views

What is a convolution kernel?

What is a convolution kernel? (in measure theory, probability theory) In which book can I read about kernels on measurable spaces and convolution kernels? Thank you!
1
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0answers
15 views

Convolution integral

I got unfortunally stuck by performing a (quite simple?) convolution integral. Given are those functions: $$f_1(t) = k_1\cdot e^{b_1\cdot t}$$ and $$f_2(t) = k_2\cdot t$$ where $k_1, k_2$ and $b_1$ ...
1
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0answers
159 views

A problem with kernels on measurable spaces

Let $(E, \mathcal{B}(E)), (F, \mathcal{B}(F))$ be two measurable spaces. A $kernel$ from $(E, \mathcal{B}(E))$ to $(F, \mathcal{B}(F))$ is an application $N : p \mathcal{B} (E) \rightarrow p ...
2
votes
1answer
24 views

Calculating PDF of $Z$ from $X,Y$ when $Z=X+Y$, given the PDFs of $X$ and $Y$

A Student is taking an exam which has two parts, X and Y, with each part given a score from 200 to 800. The students probability distribution for each part is given by $$ f_X(x)= \begin{cases} ...
1
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0answers
18 views

Inverse Laplace transformation correct?

I'm actually on the way to solve a little bit complicated differential-equation. Therefore I used the Laplace transformation. I've already solved it but I am actually not sure, whether my solution ...
0
votes
1answer
23 views

Exponential decay convolved with a gaussian

I need to convolve an exponential decay (defined as the exponential $Ae^{-\lambda t}$ from $0$ to $+\infty$) with a Gaussian of known standard deviation $\sigma$, in other words I need to compute the ...
0
votes
0answers
18 views

Is it true that for every signed probability distribution `f`, there are positive distributions `g` and `h` st. `fg=h`?

While reading the article Half of a Coin: Negative Probabilities, I came across the following theorem: For every generalized g.f. f (of a signed probability distribution) there exist two ...
0
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1answer
25 views

Convolution Theorem involving a constant.

Should one have f(x) and g(x), and wants $f(x) \ast g(x) $ from what i understand this can be quite difficult, however should $f(x)=\alpha$, a constant, what is $f(x) \ast g(x) $?
0
votes
0answers
29 views

Laplace transform involving two functions of t

I need to solve the following $$ \int_0^{\infty} f(t)g(t)e^{rt} dt$$ Where $$g(t)=t^n$$ Letting r=-s we have the definition of $$ \mathcal{L} [ f(t)g(t) ]$$ and am unsure how to continue.
1
vote
0answers
24 views

Discretization of a convolution integral for constrained optimization problem

I'm working on a constrained optimization problem in which an unknown forcing function, $u(\eta)$, is in the integrand of a convolution integral. To find an optimal shape for $u(\eta)$, the integral ...
0
votes
2answers
43 views

Finding convolution of exponential distribution

So this is a probability question, and I am asked to find $P(0.6 < Y <= 2.2)$ where $Y = X_1 + X_2$ X1~U(0,1) and X2~exp(2). Our professor worked it out, but I do not understand his ...
0
votes
1answer
19 views

Solving convolution $f(t)*g(t)$ where $f(t) = u(t) - u(t-2)$ and $g(t) = e^{-2t}u(t)$ where $u(t)$ is heaviside step function

How does one solve convolution $f(t)*g(t)$ where $f(t) = u(t) - u(t-2)$ and $g(t) = e^{-2t}u(t)$ where $u(t)$ is heaviside (unit) step function? I tried using Fourier transform of both functions to ...