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
votes
2answers
64 views

Sums of independent random variables (more than two) [closed]

I read that the convolution of two iid random variables is $$(f * g) (z) = \int f(z-y) g(y) dy$$ What is the general formula for more than two RVs? For example, for three RVs.
2
votes
1answer
2k views

About integrating product of two sinc function using Fourier transform

So the problem is which I think is pretty straight-foward by using Fourier transform and convolution property of two sinc functions and evaluating the convolution at 5. However, I got sinc(t) for ...
2
votes
1answer
52 views

Comparing Coefficients in Summations

Suppose I have the following equality: $$\sum_{k=0}^{n-a}\sum_{j=0}^{k}\binom{n}{k}\binom{k}{j}\frac{f(a,k)\cdot g(b,n-k)}{n!}=\sum_{k=0}^{n-a}\binom{n}{k}\binom{n-k}{a}\frac{z^k \cdot ...
2
votes
1answer
436 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 [ ...
2
votes
2answers
132 views

Integration of dirac function explanation

I have a problem that need your help. I have a gray image. We denotes $I(x)$ is gray level of a pixel in the image and $f(z)$ is a function of $z$(ie: histogram function...)-where $z$ is the set of ...
2
votes
2answers
630 views

Fourier transform as diagonalization of convolution

I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator $$ A_f(g) = \int f(\tau)g(t-\tau)d\tau $$ and apply it to $g(t)=e^{ikt}$. ...
2
votes
2answers
164 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
2
votes
2answers
87 views

How can this integral be rewritten with convolutions?

I've got $f:\mathbb{R}\rightarrow\mathbb{R}$ bounded and I'm trying to write `$\mathtt{f}$,' a discrete version of $f$, where each element in the domain takes on the average of the corresponding ...
2
votes
1answer
241 views

Convolution convergent in $L^\infty$

Suppose $f\in L^\infty(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_\epsilon(x)=\dfrac{1}{\epsilon}K\left(\dfrac{x}{\epsilon}\right)$$ Is it always true that ...
2
votes
1answer
103 views

Young's inequality

Let $U \in L^1(\mathbb{R}^d)$ and $\rho \in L^1(\mathbb{R}^d)$ such that $\rho \ge 0$ and the support of $\rho$ is included in $B(0,1)$ (the euclidean unit ball of $\mathbb{R}^d$). Is there a way to ...
2
votes
2answers
1k views

How to calculate a 1D convolution summation?

I hope I said that right. I'm trying to follow along with a convolution example but maybe I am in over my head. I don't understand how in this example they get the values on the right. For example, I ...
2
votes
1answer
86 views

Is it true that $f\in W^{-1,p}(\mathbb{R}^n)$, then $\Gamma\star f\in W^{1,p}(\mathbb{R}^n)$?

I am trying to understand the following paper. In page 1191, in the beggining of the proof of Theorem 2.9. the authors consider the convolution $$v=\Gamma\star f$$ They claim that $v\in ...
2
votes
1answer
83 views

Fourier analysis exercise

I need a hand with this question: If $f\in{L_1(\mathbb{R})}$ and $g\in{L_2(\mathbb{R})}$, then prove that $\widehat{f*g}=\hat{f}\cdot \hat{g}$ As a tip, i have been told to prove that: ...
2
votes
1answer
163 views

convolution-distributions

We denote by $E'(\mathbb{R})$ the set of distribution with compact support , and $\mathcal{D}(\mathbb{R})$ is the set of function $\mathcal{C}^{\infty}$ with a compact support. 1) I want to compute ...
2
votes
1answer
239 views

What is the distribution of empirical covariance between two independent normal distributions?

Suppose that we have two independent normal distributions $\mathcal{N}_{1}(0,s)$, $\mathcal{N}_{2}(0,t)$ what is the distribution of empirical covariance (or empirical correlation if this make my ...
2
votes
1answer
113 views

Fourier transform of product

I would like to know the fourier transform of the product of the Cauchy probability distribution $f(x)=\frac{1}{\pi (1+x^2)}, -\infty<x<\infty$ with itself. I know that the fourier transform of ...
2
votes
1answer
110 views

Evolution operator

We call a function that assigns a starting value of a time-dependent differentialfunction to a solution of a later timevalue as the evolution operator $E(t)$. Look at the thermal equation $$ ...
2
votes
2answers
554 views

Find derivative of convolution with gaussian

Let $A(\sigma)$, $\sigma > 0$ be an operator that acts on bounded continuous functions $f$ on $\mathbb{R}$ by the rule $$ (A(t)f)(x) = \int\limits_{\mathbb{R}} f(y)\frac{1}{\sqrt{2 \pi ...
2
votes
1answer
949 views

FFT with a real matrix - why storing just half the coefficients?

I know that when I perform a real to complex FFT half the frequency domain data is redundant due to symmetry. This is only the case in one axis of a 2D FFT though. I can think of a 2D FFT as two 1D ...
2
votes
1answer
705 views

Commutativity of Convolution in higher dimensions

I have a basic question about how to show that convolution in dimension $n$ is commutative - or maybe it is rather a question about change of variables .. So on $\mathbb{R}$ I know how to show ...
2
votes
1answer
150 views

Integrability of the function $f_1(y)f_2(x-y)$ for almost all $x$. (convolution)

What I'm trying to prove is that for $f_1,f_2 \in \mathcal{L}_1$ the function $y \mapsto f_1(y)f_2(x-y)$ is integrable for almost all $x$, or: $$ F(x) = \int f_1(y)f_2(x-y)\,dy < \infty \text{ ...
2
votes
1answer
208 views

Maximal ideals in a circular discrete convolution algebra

Let $G$ = $\mathbb{Z_n}$ and let $A$ = $\ell^1(G)$ with convolution, over $\mathbb{C}$. Let $A_{\mathbb{R}}$ denote the subring of real valued functions in $A$, so $A_{\mathbb{R}}$ is an algebra over ...
2
votes
1answer
43 views

Convolution - Hölder inequality

I wonder if you guys can help me out with a question(not homework). I have $\phi(x)=\int_\mathbb{R} |f(t)g(x-t)|dt$ where $f \in L^1(\mathbb{R}) $ and $g \in L^p(\mathbb{R})$ and p and p' are ...
2
votes
1answer
108 views

Convolution algebra $L^1(G)$ for non sigma-finite $G$

Let's assume that $G$ is locally compact and Hausdorff topological group, hence it carries a Haar measure, $\mu$. We can than consider space of integrable functions $L^1(G)$ (class of functions to be ...
2
votes
1answer
69 views

Is the convolution of two continuous functions continuous?

The title is the question: Is it true, that the convolution of two continuous functions is continuous again?
2
votes
1answer
62 views

Differential operator applied to convolution

Suppose that $g\in \mathcal{S}(\mathbb{R^n})$ (Schwartz space) and $f\in L^p(\mathbb{R^n}).$ The idea is to prove that the differential operator $D^\alpha$ does not follow the Leibniz rule when ...
2
votes
1answer
298 views

Find the PDF of X1 +X2 +X3.

The problem said: If X1,X2,X3 are independent random variables that are uniformly distributed on (0,1), find the PDF of X1 +X2 +X3. The theory I have said: Following the theory and the ...
2
votes
1answer
63 views

Does infinite repeated convolution with the same normal distribution converge?

According to Wikipedia, This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal. ...
2
votes
1answer
56 views

What is Convolution?

The definition of convolution is known as the integral of the product of two functions $$(f*g)(t)\int_{-\infty}^{\infty} f(t -\tau)g(\tau)\,\mathrm d\tau$$ But what does the product of the functions ...
2
votes
3answers
85 views

Adding two random variables with convolution

I am trying to understand the purpose of convolution of two probability functions. Also when it is appropriate to use the convolve function on two independent probability distributions. ...
2
votes
2answers
118 views

When is convolution associative?

Convolution is associative on e.g. integrable function on $\mathbb{R},$ but not on distributions. What about the convolution of measures on an unimodular group $G$?
2
votes
1answer
41 views

Convolution of various functions

There is asked in an example to do convolution $ h_1(t)*h_2(t) + h_3(t)*h_4(t) $ where $h_1(t) = e^{-2t}u(t)$ $h_2(t) = 2e^{-t}u(t) $ $h_3(t) = e^{-3t}u(t) $ $h_4(t) = 4\delta(t) $ and then the ...
2
votes
1answer
63 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 ...
2
votes
1answer
234 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 ...
2
votes
1answer
160 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 ...
2
votes
2answers
124 views

Convolution with delta function

I am merely looking for the result of the convolution of a function and a delta function. I know there is some sort of identity but I can't seem to find it. $\int_{-\infty}^{\infty} ...
2
votes
1answer
150 views

Partial derivative of convolution

I have a convolution: $$g(x,\alpha) = \int_D \phi(t)f(x-t,\alpha)dt,$$ where $D$ is compact. I need to calculate $\frac{\partial}{\partial \alpha}g(x,\alpha)$. Under what conditions: ...
2
votes
2answers
37 views

convolution problem given $h(x)=1/2$ for $0<x<2$ and $0$ otherwise

I have a convolution problem in the form $$g(x)= \int_{-\infty}^\infty h(y)h(x-y)\,dy$$ where they give me the function $h(x)=1/2$ for $0<x<2$ and $0$ otherwise. I have never done a ...
2
votes
1answer
99 views

Laplace transform of a product of two functions

I have read questions and answers about this topic and i am still confused, using this formula we can calculate the Laplace transform of a product of two functions: $$ L[a_{(t)} b_{(t)}]={{1}\over{2 ...
2
votes
1answer
48 views

Convolution computing

How we can compute the convolution product $$\Big(\sum_{n=0}^{+\infty} \delta_n^{(n)}\Big) \star \Big(\sum_{n=0}^{+\infty} \delta_n\Big)$$ where $\delta$ is Dirac distribution? Thank's for the help
2
votes
1answer
539 views

$g, f, \hat {f} \in L^{1}(\mathbb R)\cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R) \implies \widehat{(fg)}= \hat{f} \ast \hat{g} ? $

Let $f, g\in L^{1}(\mathbb R)$ and it Fourier transform of $f$, $\hat{f} (y) = \int _ {\mathbb R} f(x) e^{-2\pi i x \cdot y} dx, \ (y\in \mathbb R)$ and the convolution of $f $ and $g$; $f\ast g ...
2
votes
2answers
741 views

How to show that the difference of two Gumbel distributed random variables follows a Logistic distribution?

How can you show that when you have two random variables $X,Y\sim\text{Gumbel}[0,1]$ , then $X-Y\sim\text{Logistic}[0,1]$ . I tried to use the convolution formula ...
2
votes
1answer
68 views

What is the easiest way to find the inverse Laplace of F(s)?

$$ F(s)= \frac{1}{(s-1)^2(1-1/s^2)} $$ Do I have to multiply by $s^2/s^2$ and then use partial fractions or is there a way to use the convolution theorem?
2
votes
1answer
291 views

Convolution converging uniformly on real line

I'm working on this question and stuck with the following part: Suppose $f\in L^\infty(\mathbb{R})$ and $K,K_1,K_2,\ldots\in L^1(\mathbb{R})$ with $K_n\rightarrow K$ in $L^1$. Why is it true that ...
2
votes
1answer
1k views

Convolution is uniformly continuous and bounded

Suppose $f\in L^\infty(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Show that the convolution $f\ast K$ is a uniformly continuous and bounded function. The definition of ...
2
votes
1answer
258 views

Convolution and uniform continuity

If $f\in L^{\infty}(\mathbb{R}^n)$ and $f$ is continuous at $x$, then $$\lim_{k\to\infty}(f*\phi_k)(x)=cf(x)$$ If $f\in L^{\infty}(\mathbb{R}^n)$ and is uniformly continuous, then $f*\phi_k\to cf$ ...
2
votes
1answer
99 views

Does negative distributive property of convolution over cross correlation holds?

Let $\star$ denote convolution binary operation and $\otimes$ denote cross correlation binary operation between two functions. Let $f,g,h$ be functions. Does this negative distribution property ...
2
votes
1answer
328 views

Convoluted Lorentzian and Fourier Transformation

To describe a measurement, I have to calculate the convolution of three functions $f,g,h$: $f(x)=\frac{1}{(W/2)^2+x^2} \, , W>0$ $g(x)=e^{\beta x} \frac{(\beta x-2)e^{\beta x}+\beta ...
2
votes
1answer
93 views

A continuous random walk of length 1

Suppose one starts at origo in in the plane and takes $N$ steps of length $1/N$ in a random direction, what is the distribution of the resulting distance from origo as $N$ approaches infinity? For one ...
2
votes
1answer
582 views

Using FFT in matlab

I am not completely sure if this is where a MatLab question belongs, so if not, please direct me where I should ask. But onto my question. I am working on trying to deconvolution a signal with ...