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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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How to set up range for convolution distributions

I have little difficulty setting up the integration in the beginning, but have some serious difficulties in setting up the range. For example, where X~Uniform(0,1) Y~Uniform (0,1) Z=X+Y I can ...
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Distribution of the square of the sum of independent rayleigh variables

Suppose $\alpha_i$ is the $i$th independent Rayleigh distribution random variable following a Rayleigh probability density function (PDF) as \begin{equation} f_{\alpha_i}(r) = \frac{r}{\sigma_i^2} \...
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Filter that gives the max change in matrix value

I have a 3x3 matrix, I want to find the maximum change in value that any one element in the vector has with the centre element (given that this element has the opposite sign to the centre element). ...
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How to solve a Convolution Integral with one delta function.

I have a Convolution integral $$ \int_{t_0}^{t} \int_{t_0}^{\tau} C(t-t')C(\tau -t'') \delta(t''-t') dt'' dt'=\int_{t_0}^{t} C(t-t')C(\tau -t') dt'= ? $$ I do not know how to proceed any further, $\...
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Approximating Sobolev functions in $W^{1,p}(\mathbb{R}_+^n)$

Let $p \geq 1$. I know that there exists a continuous and linear extension operator $$ E : W^{1,p}(\mathbb{R}_+^n) \to W^{1,p}(\mathbb{R}^n) .$$ I read that from the existence of such an extension ...
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Is there a way to write the convolution form of a derivative?

In image filtering and computer simulations, derivatives can be applied to a image by performing a convolution of the image/data with an appropriate kernel matrix. For example, in 1D this could be $ ...
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How to minimize sum of matrix-convolutions?

Given $A$, what should be B so that $\lVert I \circledast A - I \circledast B \rVert _2$ is minimal for any $I$? $I \in \mathbb{R}^{20x20}, A \in \mathbb{R}^{5x5}, B \in \mathbb{R}^{3x3}. $ Note ...
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Convolutions: Let U~Unif(0,1) and X~Expo(1), independently. Find the PDF of U +X.

Let U~Unif(0,1) and X~Expo(1), independently. Find the PDF of U +X. Solution: $f_T(t) = \int_{-\infty}^{\infty}f_Y(t-x)*f_X(x)dx$ $= \int_{-\infty}^{\infty}1*\lambda e^{-\lambda x}dx$ Integrate ...
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existence of the convolution with a continuously differentiable function

It is well known (using Fubini's theorem) that the convolution of two $f,g\in L^1(\mathbb{R})$ functions is again in $L^1$, and thus $$ f \ast g(x)=\int_\mathbb{R}{f(x-t)g(t)dt}$$ converges for almost ...
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Does the 'continuous convolution relation' imply a discrete version?

In particular, I want to know if $L^p(\mathbb{R}^d)*L^q(\mathbb{R}^d)\subseteq L^r(\mathbb{R}^d)$ implies $\ell^{p}(\mathbb{Z}^d)*\ell^{q}(\mathbb{Z}^d)\subseteq \ell^{r}(\mathbb{Z}^d)$? My attempt is ...
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How to solve this Inverse Laplace Transform using Convolution Theorem? [closed]

I have the following problem: . I found difficulty in the inverse laplace of {s/((s^2 + 4)^3)}. So how can i solve that?
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1answer
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Convolution of a compactly supported function and an $L^1$ function.

I have these related questions here that I could really use some help on. I believe there is a related question here although I don't think it is exactly the same... 1) Let $f\in L^1(\mathbb R)$ ...
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Find norm of image convolution

Suppose there is matrix A that is a black and white image. There's also matrix B - a filter. ...
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149 views
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Exercise about convolution of functions

I have found this excercise in theory of convolution (I started it the last week). I have been thinking about it for two days but I don't get solve it: Let be $1<p<2<q<\infty$ and $f:\...
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Convolution of probabilities on finite groups

I was reading a book on group and representation theory and came across the following which I don't understand, I'd appreciate any help. Suppose P and Q are probabilities on a finite group G. Thus $...
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Find the output of the filter with the given input

Suppose we have a filter $L$ defined by $(L\circ x)_n=(x\ast h)_n,$ where $$h_n=\begin{cases}\frac{\sin(\frac{\pi}{6}n)+\sin(\frac{\pi}{2}n)}{\pi n},&n\neq 0,\\ 2/3,&n=0.\end{cases}$$ Here $...
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Show $ \int_ {-\infty} ^{\infty} (\arctan(x+a)) {{1}\over {\sqrt{2 \pi T}}} e^{-x^2 /2T} dx$ can assume any value in $(-\pi/2, \pi/2)$

I would like to show $$ \int_ {-\infty} ^{\infty} (\arctan(x+a)) {{1}\over {\sqrt{2 \pi T}}} e^{-x^2 /2T} dx$$ can assume any value in $(-\pi/2, \pi/2)$ where $T>0$ is fixed and $a$ may be any ...
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Convolution of an unintegrable function and the convolution theorem

1) Does a convolution of an integrable function and an unintegrable function exist? 2) If the answer for the first question is possible, then: suppose a function $f(x)$ is integrable, and denote by $...
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Lebesgue Density $f$ with compact support, polynomial $p$ of degree $k$. Then convolution $f\star p$ is integrable and can be written as polynomial

Let $f(.)$ be a continuous Lebesgue-density-function with compact support $\mathrm{supp}f=\{x:f(x)\neq0\}$. Let $p(.)$ be a polynomial with degree $k$. Then the convolution $h:=f\star p$ is an ...
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A Fibonacci convolution

A Fibonacci convolution. Recall that $$F(x)=\sum_{n=0}^\infty F_n x^n =\frac{x}{1-x-x^2} =\frac{1}{\sqrt{5}} \left(\frac{1}{1-\Phi x} -\frac{1}{1-\bar{\Phi}x}\right).$$ (a) Prove that $\displaystyle ...
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Two dimensional Fourier Transform and Convolution of a periodic function

I want to calculate the Fourier transform of the following function $$ \frac{1}{T}e^{i\omega_n t}e^{-i\omega_m t^\prime}\Theta(t-t^\prime), $$ where $\omega_n=\frac{2\pi}{T}n$ ($n\in \mathbb{Z}$) and $...
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Fitting a spline: find coefficients using Fourier Transform?

I came up with a idea to estimate the coefficients of a B-spline fit by using the Fourier Transform but I don't know if it makes any sense to estimate them in this way. Given that $$s(x)=\sum_kc(k)\...
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1answer
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What does this notation mean? Double arrow with $z$ above

I am reading a paper about digital filtering (for the very first time) and I found this notation (double arrow with $z$ above) which I do not quite understand. Could you please give me some hint? ...
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1answer
31 views

Computing the PDF of $X + Y$ if $X, Y \sim \text{U}(0, 1)$ using a convolution

Theorem: If $X$ and $Y$ are independent continuous random variables with density functions $f_{X}$ and $f_{Y}$, the probability distribution $f_{X + Y}$ of $X + Y$ is given by $$f_{X + Y}(a) = ...
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Convolution exercise, is this really ok?

Let $f$ be a function such as, for every $\alpha > 0$, $$\omega (\alpha) = | \lbrace x \in \mathbb{R}^n : |f(x)| > \alpha \rbrace | \leq c (1 + \alpha)^{-p}$$ Prove that $f \in L^{r}(\mathbb{R}...
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Convolution of a gaussian and the derivative of the inverse error function

Problem Definition I need to calculate the convolution between a Gaussian function $g(x)$ and $d(x)$, the derivative of the inverse error function, $$d(x) * g(x) = \int_{-\infty}^{\infty} d(\tau) g(...
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Sequence of convolutions

I am trying to find $F_k(t)$ defined by $F_k(t) = \int\limits_0^\infty F_{k-1}(t-y)dF(y)$ and $F_1(t) = F(t)$ for the probability density function $f(t) = \begin{cases} \rho e^{-\rho (t- \delta)} \...
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28 views

Multiple Convolution closed form.

Define the convultion of two functions f and g as: $$(f*g)(x)=\int_{-\infty}^{\infty}f(\tau)g(x-\tau)d\tau$$ Is there any closed form for a multiple convultion: $(f_1*f_2*...*f_p)(x)$?
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$u(x,t)=\left(f*\mathcal{H}_{t}^{(n)}\right)(x)$ is solution to the n-dimensional heat equation

Consider the time-dependent heat equation in $\mathbb{R^{n}}$: $$\displaystyle\frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial x_{1}^{2}}+\cdots+\frac{\partial^{2}u}{\partial x_{n}^{2}},\...
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Does $\int_{-\infty}^\infty f(x) dx < \infty$ where $f \ge 0$ implies $\sup_{x \in \mathbb R} f(x)<M $?

Question. Does $\,\int_{-\infty}^\infty\, f(x)\, dx < \infty,\,$ where $\,f \ge 0,\,$ imply that $\,\,\mathrm{ess}\sup_{x \in \mathbb R}\, f(x)<\infty\,? $ My attempt Since $f \ge 0$ and $\...
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Fourier Transform of B-splines linear combination, any application to spectral analysis of non stationary series?

In a framework of non-linear regression one could opt for using B-splines in order to approximate the values $y$ assumed by an unknown function $f(x)$. If we denote with $B_j(x,p)$ the value of the ...
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1answer
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Young's Inequality for Convolutions; when $r = \infty$

I've been trying to prove what seems to be a little generalized version of the Young's Inequality for Convolutions. Here's the statement of the Theorem: Let $1\leq p, q \leq \infty$, $\frac{1}{p}+\...
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1answer
57 views

If $\,f$ and $g$ are continuous with compact support then $f*g$ (convolution ) is also continuous with compact support [closed]

I do not know how to find a compact that satisfies $f$ and $g$ support. Could someone explain how find this new compact ? How $f$ and $g$ are continuous in a compact then are bounded in this ...
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60 views

Convolution with Gaussian

Let $f, g\in \mathcal{S}(\mathbb R)$ (Schwartz class function), $\delta_0$ (dirac delta distribution). Consider distribution as follows: $$G(x, y)= f(x)g(x)\delta_0(y)-f(y)g(y)\delta_0(x), \ (x, y\...
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Conditional distribution from the sum of uniform distributions

I am trying to find the conditional probability distribution function of $Y$ $$F(Y\mid X_1,X_2)$$ given that $Y$ is distributed uniformly on $[0,1]$, $$X_1=Y+Z_1$$ and $$X_2=Y+Z_2$$ where $Z_1$ and $...
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1answer
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Convolution Notation: The difference between (f*g)(x) and f(x)*g(x)

What is the difference between (f*g)(x) and f(x)*g(x) [1] for convolutions? Are they the same? I ask this because I have been asked to prove the Reflection of Convolution property for my course in the ...
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Double integrals - how are the boundaries chosen?

I was looking at the proof of the theorem stating that for two independent rv's $X,Y$ with density functions $f,g$ the density function of the rv $X+Y$ is the convolution of $f$ and $g$. After ...
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The solution of heat equation satisfies $u(x,t)\rightarrow 0$ as $|x|+t\rightarrow \infty$

Suppose $u$ solution of the heat equation $u_{t}=u_{xx}$, where $f\in S(\mathbb{R})$ belongs to Schwartz space, $u(x,t)=\left(f*\mathcal{H}_{t}\right)(x)$ setting $u(x,0)=f(x)$. I need to prove that $...
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“Unit sample response” relation to “step reponse” $(u[n] \to \delta[n])$

Book says: $$\delta[n]=u[n] - u[n-1]$$ Therefore, the unit sample reponse $h[n]$ is related to unit step reponse, $s[n]$, as follows: $$h[n]=s[n] - s[n-1]$$ My question is, how to prove this ...
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39 views

$f$ continuous, moderate decrease and $\int_{-\infty}^{\infty}f(y)e^{-y^{2}}e^{2xy}dy=0$ implies $f=0$.

Let $f$ a continuous function, moderate decrease and satisfying \begin{equation} \int_{-\infty}^{\infty}f(y)e^{-y^{2}}e^{2xy}dy=0 \end{equation} for all $x\in\mathbb{R}$ I need to prove that $f=0$. ...
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No convolution Identity element in $L^1_{per} $ using Fourier series

We have to show that there is no identity element for the ring $ L^1_{per}( ]0,2\pi [) $ specifically using the Fourier coeffcients. Suppose that : $ \exists e , \: \: e * f = f \: \: \: \: \forall f ...
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Discrete convolution of two $\ell_p(\mathbb{Z})$ functions for $p > 2$

Given $f, g \in \ell_p(\mathbb{Z})$, we define the convolution of $f$ and $g$ as follows : $$(f\ast g)(x) :=~ \sum\limits_{y = -\infty}^\infty f(y)g(x-y).$$ It is easy to prove that : If $p = 1$ ...
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Continuous Tikhonov Regularization for Deconvolution

I am trying to solve the following deconvolution problem where $g(s)$ is a known real valued function and has finite energy: $$g(s) = \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}f(t)e^{-(t-s)^2/2}...
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Help with convolution

I'm learning about convolution operations right now, and I have to find the discrete time convolution between x[n] = 2^nδ[n − 1] and h[n] = 0.4^nu[n]. As I think I understand, the convolution ...
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Using convolution with two independent geometric random variables

Consider two independent geometric random variables $$X\sim \text{Geometric}(1/2) \ \ \ \ \text{and} \ \ \ \ Y\sim \text{Geometric}(3/4).$$ My goal is to find $$\mathbb{P}(X-Y=2).$$ Using the ...
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Is there a name for an integral of the form $\int d \vec{r} \int d \vec{r}\,'\, f(\vec{r})\, K(\vec{r} - \vec{r}\,')\, f(\vec{r}\,')$?

Is there a special name for an integral of the form $$\int d \vec{r} \int d \vec{r}\,'\, f(\vec{r})\, K(\vec{r} - \vec{r}\,')\, f(\vec{r}\,')\; ?$$ Here $\vec{r}, \vec{r}\,' \in R^d$, and the ...
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30 views

Stochastic Independence of $\tan(U_1)$ and $\tan(U_1+U_2)$ for uniform independent $U_1,U_2$

I found the following statement in Stoyanov's book on counterexamples (Section 7.2) quite interesting: Let $U_1$ and $U_2$ be independent and uniformly distributed on $(0,\pi)$. Then $\tan(U_1)$ and $\...
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32 views

Difficult integral and joint probability (calculating something like $Pr(X \geq b-a, X \geq Y-a)$) where $X$ and $Y$ are random variables

I'm trying to have some practice with joint probabilities, and I came up with a question that I'm struggling to answer. Suppose that we have two random variables $X$ and $Y$, where $X$ is ...
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1answer
30 views

Inequality involving convolutions

We have well known inequality for convolution: $\|g\ast f\|_{X} \leq \|g\|_{Y} \|f\|_{Z}$ where $Y, X, Z$ are suitable Lebesgue spaces, e.g., see Young's inequality. Let $f:\mathbb R^{2}\to \mathbb ...
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Convolution in Bialgebras

On the wikipedia page on convolution, there is a section on convolution in bialgebras. It's completely mysterious to me. If it has something to do with the regular concept of convolution, can some one ...