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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75 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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1answer
62 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
58 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: $(f(t)\ast g(t))(\tau) = \int_{-\infty}^\infty{f(t)g(\tau-t)dt}$ ...
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1answer
394 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
44 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
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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
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1answer
82 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 ...
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2answers
63 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 ...
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2answers
43 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
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0answers
111 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 $$\mathcal{L}[f*g]=\mathcal{L}[f]\,\mathcal{L}[g],$$ ...
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3answers
69 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
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1answer
146 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 [ ...
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1answer
140 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 ...
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0answers
26 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 ...
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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
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1answer
54 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 ...
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0answers
37 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
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1answer
43 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.
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1answer
42 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!
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1answer
92 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} ...
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0answers
51 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 & ...
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0answers
57 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!
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0answers
30 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 $b_1$ ...
2
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1answer
49 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} ...
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0answers
29 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
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1answer
72 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 ...
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0answers
21 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 ...
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1answer
500 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) $?
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0answers
32 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.
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0answers
51 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 ...
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2answers
104 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 ...
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1answer
42 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 ...
1
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1answer
50 views

Solve initial value problem with unspecified right-hand side $g(t)$

Consider the initial value problem $$y''-6y'+9y=g(t),\quad y(0)=1,\ y'(0)=3.$$ 1) Use the Convolution Theorem to find the solution to the IVP for any piecewise continuous function $g(t)$ that is of ...
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1answer
92 views

Young's inequality for convolutions

Let's assume that the convolution $f * g$ is continuous with $\lim_{|x| \to \infty}(f*g)(x) = 0$ and that $f, g \in L^2$. Then the following inequality holds $$ \| f * g \|_{\infty} \leq \| f \|_2 ...
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0answers
33 views

Extend bivariate to multivariate convolution formula?

In reference to this post, the pdf for dependent random variables $X_1+X_2$ is given by: $$f_{X_1+X_2}(z) = \int_{-\infty}^{\infty} f_{X_1,X_2}(x,z-x) \mathrm dx$$ How does this formula extend to ...
2
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0answers
22 views

Convolution of two bernoulli distributions

Find the probability mass function of the sum of X ∼ Bernoulli(p) and an independent Y ∼ Bernoulli(q) variable. I started by letting Z=X+Y So $$P_z(Z)= \sum_{i=0}^{1}f_x(x) f_y(z-x) $$ $$ ...
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0answers
16 views

Approximation technique of common probability distributions that can be convoluted and integrated fast

I am looking for a approximation technique of functions with two conditions: It is possible to perform a fast approximate convolution with the approximate functions. It is possible to numerically ...
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1answer
44 views

Convolution with heaviside function, argument of the heaviside carry on to the dirac function?

So I have this equation to demonstrate: $$ x(t)*u(t)= \int_{-\infty}^t x(\tau)d\tau $$ , where $u(t)=\int_{-\infty}^t \delta(\tau)d\tau$ I opened the convolution as $ \int_{-\infty}^\infty ...
2
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1answer
217 views

Convolution of sine and unit step function

I started studying signal convolution recently and the first sample problem I got is to find convolution of sine and unit step function (Heaviside function). Here is what I have right now. ...
0
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1answer
80 views

Convolution CDF formula?

In reference to this post, the pdf of dependent random variables $A+B$ is given by: $$f_{A+B}(z) = \int_{-\infty}^{\infty} f_{A,B}(a,z-a) \mathrm da = \int_{-\infty}^{\infty} f_{A,B}(z-b,b) \mathrm ...
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1answer
83 views

How to obtain the convolution directly (not graphical) of the two functions $e^{-t}u(t)$ and $e^{-2t}u(t)$?

I'm having trouble solving this convolution integral graphically. I don't understand where I stop sliding my function $h(t-\lambda)$ since $x(t)$ doesn't have a boundary as lambda approaches infinity ...
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0answers
44 views

Equality condition for convolution's $L^p$ norm.

Suppose that $1< p< \infty$, $f\in L^1(R)$, and $g\in L^p(R)$ and that $\|f*g\|_p=\|f\|_1\|g\|_p$. Show that then either $f=0$ a.e or $g=0$ a.e I have solved for $g=0$ a.e. if $||f||_1>0$ ...
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0answers
48 views

convolve chirp with rect

I'm trying to evaluate $$g[x] = f[x] \ast f[x]$$ where * is the convolution operator and $$f[x] = RECT(\frac{x-2.5}{5}) \cdot exp (+i \pi x^2)$$ I assume the best approach to this equation is: ...
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0answers
30 views

estimate for derivative of convolution

Let $u\in L^\infty(\mathbb R\times (0,\infty))$ be a function such that $$u(x+z,t)-u(x,t)\leq c\left(1 + \frac 1 c\right)z\tag{$*$}$$ for some constant $c\in\mathbb R$ and almost all $x,z\in\mathbb ...
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1answer
31 views

what does support of convolution of functions says geometrically?

Let $f,g \in L^{1}(\mathbb R)$ we define $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy $ for all most all $x,$ and denote $\text{supp} (f)$ the support of $f.$ Fact: If $A$ is the closure of $\{x+y: ...
2
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1answer
67 views

convolution of non-zero functions

Let $f,g$ be two continuous functions with compact support. Show that if $f$ and $g$ are not identically $0$, then neither is $f\ast g$. This statement seems rather elementary, and I would prefer if ...
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0answers
70 views

Convolution of convex polygons and a Gaussian

I need to find the closest solutions for convolution of convex polygons/circles with a Gaussian function for computer graphics purposes. I was only able to find solutions for rectangles, like this ...
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3answers
379 views

Combining two convolution kernels

Is it possible to combine two convolution kernels (convolution in terms of image processing, so it's actually a correlation) into one, so that covnolving the image with the new kernel gives the same ...
0
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1answer
29 views

Deconvolution of two delta functions (solving $y(t) = A x(t-a) + B x(t-b)$)

I would like to calculate $x(t)$, when only $y(t)$ with $y(t) = A x(t-a) + B x(t-b)$ is known. Since this is a linear shift invariant operation (convolution), the inverse relation must be of the ...
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1answer
37 views

Using dirac delta functions to get $h(t)$ that satisfies $[u(t+1/2)-u(t-1/2)] \ast h(t) = [u(t+6)-u(t+2)]$ where $u(t)$ is unit-step function

Using dirac delta functions, how does one get $h(t)$ that satisfies $[u(t+1/2)-u(t-1/2)] \ast h(t) = [u(t+6)-u(t+2)]$ where $u(t)$ is unit(heaviside)-step function and $\ast$ is convolution?