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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0
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2answers
32 views

Evaluate convolution integral

Can someone tell me if I am calculating this integral correctly.
0
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1answer
144 views

Convolution integral involving two Heaviside functions

I am having trouble solving the following integral involving two Heaviside functions, obtained from a Laplace transform convolution: $\Large \int_0^t \frac{\tau}{\sqrt{\tau^2 - \alpha^2}} H(\tau - ...
1
vote
0answers
29 views

A small doubt regarding a previously asked limit of convolution

Previously, I asked this question to the forum. Pointwise limit of convolution Now, a question in this regard is coming to my mind. Suppose, we don't have the integral; i.e. we have the ...
5
votes
1answer
115 views

Max and sum of random variables

I have a set of independent random variables $\{A_1, A_2, B_1, B_2\}$. All of them have the same distribution function $F(x)$. I want to find distribution function of a variable $C$, where $C=max(A_1 ...
2
votes
1answer
46 views

Pointwise limit of convolution

Suppose $\omega$ is the standard mollifier in $\mathbb R$. Then, let $\omega_{\epsilon} (x):= \frac{1}{\epsilon} \omega \left(\frac{x}{\epsilon}\right)$. For $0 < t_{1} < t_{2}$ the following ...
0
votes
1answer
48 views

Combined arrival rate

Let us suppose a scenario with two clients, $a$ and $b$, each one generating load at rate $\lambda_a$ and $\lambda_b$, respectively. The server receives the requests from both clients. What will be ...
4
votes
0answers
219 views

Normal() convolved to Exp(polynomial)?

Is there a general exact solution for a Normal distribution convolved to $e^{y(x)}$, where $y(x)$ is a polynomial? $$e^{y(x)}*N(x,v)=\int_{-\infty}^{\infty}e^{y(z)}N(x-z,v)dz$$ ...
1
vote
1answer
48 views

If $\phi$ vanishes outside of $|x| < 2$, $\phi = 1$ where $|x| < 1$, prove/disprove $f * \phi$ is in $L^1$/$L^2$

Problem Statement: Let $\phi$ be a positive, smooth function. Suppose $\phi$ vanishes outside a compact subset of $\{x : |x| < 2\}$ and satisfies $\phi(x) = 1$ if $|x| < 1$. Let $f$ be a ...
1
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0answers
30 views

Convolution of independent but 'different' probability distributions

I have the following two probability distributions they relate to a particular ice-cream: ...
1
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0answers
35 views

Uniformly boundedness of convolutions

Assume $X$ is an absolutely continuous random variable with pdf $f:\mathbb{R}\to[0,\infty)$. Assume further there exists $M>0$ s.t. $|f(t)|\leq M \quad\forall t\in\mathbb{R}$. Let $X_1,\dots,X_n$ ...
2
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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. ...
0
votes
1answer
54 views

convolution of probability measures

What do we mean by convolution of measures? With example What is the difference between convolution of measures and convolution of functions? What is probability measure? Give an example of ...
0
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0answers
72 views

Derivative of mollification

This is in response to a claim made in the second line of the question here, namely: Given the standard mollifier $\eta$ and a locally integrable function $f:U \rightarrow \mathbb{R}^n$, by defining ...
2
votes
0answers
32 views

For which $F$ we have $F(f\ast g)= f\ast F(g)$ for all $f,g \in L^{1}$?

Young's inequality tells us that: $L^{1}\ast L^{p} \subset L^{p}, (1\leq p \leq \infty)$ My Question: What are examples of functions $F:L^{1}(\mathbb R)\to L^{p}(\mathbb R)$ with the property ...
0
votes
0answers
45 views

Contracting a sum of convoluted delta functions using plus/minus symbol?

So I'm currently reading a journal article (I must say, I'm not too savvy with math notation). However, I'm coming across the following relationship: http://i.stack.imgur.com/omwcN.png The equation ...
0
votes
0answers
23 views

Prooving that multiply by exponent in time domain yields a frequency shift in frequency domain using convolution.

im trying to proove that $F[x(t)e^{-jat}] = X(w-a)$ using convolution. using the convolution property i know i should get a convolution of $F(x(t))$ and $F(e^{-jat})$ So: $$ F[x(t)e^{-jat}]= 1/2\pi ...
3
votes
1answer
56 views

Discrete convolution of two sequences

Let $R$ be a commutative ring with unity. A finite sequence $x=\left< x_0,\dots,x_n\right>$ with elements in $R$ is called to be prime if there exists $a_0,\dots,a_n \in R$ such that ...
0
votes
0answers
37 views

Convolution and space-time Fourier transform

I have a general function $u(x,y,z,t)$. Then, (1) what would be the space-time Fourier transform of $$G \star \frac{\partial^n u}{ \partial t^n }$$ and (2) would the relation $$G \star ...
3
votes
1answer
58 views

Convolution can smooth an input function, is there an operation which bunches it up?

An easy to remember description of what the convolution of two functions is, is to say that one is a weight function and the result is a weighted average of the other function. The canonical example ...
2
votes
0answers
46 views

Clarification on Wolfram Mathworld's explanation of the connection between Gelfand Transform and Fourier Transform

http://mathworld.wolfram.com/GelfandTransform.html In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can ...
1
vote
1answer
53 views

Convolution vanishes on an interval

Fix a "test" function $f(x)=x\exp(-x^2)$, which is nonzero except $x=0$. Suppose that $g$ is a function with some necessary regularity. Consider the convolution. $$ (f\ast g ...
-1
votes
1answer
51 views

Convolution of two piecewise functions using Laplace transform [closed]

I'm practicing Laplace transforms and I stumbled upon one question which I am not exactly sure how to tackle. The question is: Using Laplace transforms (or otherwise) calculate the convolution of ...
2
votes
1answer
86 views

solving integral

How can I solve this integral to get the result as follow: $${\sqrt{\alpha} \over 2\pi} \int_{0}^{t} {1\over \sqrt{r^{3}(t-r)}}[\sin({\alpha\over2r})+\cos({\alpha\over2r})] \mathrm{d}r= ...
0
votes
0answers
35 views

Compute the convolution of two compactly supported functions

I'm looking for a concrete example to understand the computation procedure for convolution: Let $f, g \in C_{0}^{\infty}(\mathbb R)$ be defined as follows: $$f(x) := e^{-\frac{1}{1-x^2}}1_{(-1,1)} $$ ...
0
votes
1answer
58 views

Is the convolution pointwise bounded?

A problem from an old exam: Prove or disprove: if $p,q \in [1,\infty)$ such that $p^{-1}+q^{-1}=1$ and $f\in L^p, g\in L^q$, then the convolution $f*g$ is pointwise bounded. First of all: what ...
2
votes
2answers
75 views

How to take derivatives of a convolution when the kernel's derivative is in the distribution sense?

I came need to take the derivative of the following convolution: $$ \int_{-\infty}^\infty \operatorname{sgn}(x-y)e^{-|x-y|}f(y) \, dy $$ However, the derivative of the kernel only exists in the sense ...
3
votes
1answer
45 views

On a problem of weak convergence for a particular convolution of pr. measures

Assume that $\{P_n: n \in N\} $ and $\{Q_n: n \in N\} $ are sequences of probability measures. Assume that $P_n \stackrel{w}{\to} P. $ Also, assume that $Q_n = \delta_{b_n}, $ the Dirac measure and ...
3
votes
1answer
62 views

Convolution of probabilities

It is a well known fact that for a random variable $Z=Y_1+Y_2+...+Y_n$ where $Y_i$ are independently distributed then the probability density function of $Z$ is the convolution of the density ...
1
vote
0answers
55 views

Convolution of a gabor function and gaussian noise?

I am convolving the same image with a 2D Gabor over different gaussian noise masks that are generated in every trial. The convolution naturally takes time, is there any way to speed up the process by ...
1
vote
0answers
52 views

A doubt regarding derivative of convolution!!

In the following calculation: $\int_{\mathbb R^{d}} u_{o \epsilon} div (\phi) dx = \int_{\mathbb R^{d}} (u_{o} * \psi_{\epsilon}) div(\phi) dx = \sum_{i=1}^{d} \int_{\mathbb R^{d}} ( u_{o} * ...
1
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0answers
35 views

Can convolution be used to measure the difference between two sequences?

Say I have an infinite sequence $S_1$ and another finite sequence $S_2$. If I calculate $$ E = S_1 ∗ S_2 $$ does it somehow reflect whether $S_2$ appears somewhere in $S_1$? What if an approximate ...
1
vote
1answer
370 views

Non-linear Systems, Impulse Responses, and Convolution

In linear system theory, it is easy to find the particular solution to differential equation by means of convolving the system's impulse response with the forcing function. My question is why can we ...
0
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0answers
30 views

Sobel method on data points

From what I've seen of the Sobel method, one takes an source image $A$, and applies the matrices $G_x = \begin{pmatrix} -1 && -2 && -1 \\ 0 && 0 && 0 \\ 1 && 2 ...
9
votes
0answers
144 views

What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?

In traditional Fourier theory, the Dirac delta plays the role of an "identity" for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group ...
2
votes
0answers
110 views

About sparse polynomial squares

Given $p\in\mathbb{Q}[x]$, we define the weigth of $p$ as: $$ W(p) = \#\{n\in\mathbb{N}: [x^n]\,p(x)\neq 0\} $$ i.e. as the number of non-zero terms. By playing a bit with the Taylor series of ...
1
vote
2answers
68 views

Integrability of Maximal Convolution Operator

Let $f\in L^{\infty}(\mathbb{R}^{n})$ be supported in the unit ball and have mean zero. Let $\phi\in L^{1}(\mathbb{R}^{n})\cap C^{\alpha}(\mathbb{R}^{n})$ be a Holder continuous function with exponent ...
8
votes
2answers
662 views

Reconciling two intuitions about convolution

There are two intuitive things convolution does. In the time domain, it represents the distribution of the sum of two independent random variables. In the frequency domain, it's just multiplication. ...
1
vote
1answer
31 views

Translations AND dilations of infinite series

Sometimes, when working with infinite series, it's useful to add "dilated" or "translated" versions of the infinite series, term by term, back to the original. There are ways of making this rigorous ...
4
votes
0answers
59 views

Linear independence of primitive Dirichlet characters and convolution

This is not an exercise but merely a question I have. Fix $N \in \mathbb{N}$ and suppose there exist some values $a_k \in \mathbb{C}$, for $k \in \mathbb{Z}_N$, such that $$ \sum_{k \in \mathbb{Z}_N} ...
3
votes
0answers
95 views

Derivation of back-propagation equation $\frac{\partial E(\theta)}{\partial W^k}=x*\delta h^k+\tilde{h}^k*\delta y$ for convolutional autoencoders

I was reading the following paper on convolution stacked auto-encoders and they had the following convolution neural network (for auto-encoders, notice I didn't write the offset term [to avoid ...
4
votes
1answer
602 views

Is the $L_\infty$ norm of the $\mathcal{l}_2$ norm of this sequence of functions finite?

I am interested in proving or disproving the following claim and am stuck. We define a series of functions with the following properties. For each $i\in \mathbb{N}$ let $f_i\colon \mathbb{R}^+ \to ...
0
votes
0answers
54 views

2D Convolution notation confusion?

We can express 2D convolution between $f(m,n)$ and $h(m,n)$ as following \begin{align} g(m,n) &= \displaystyle \sum_{k_1=-\infty}^{\infty} \sum_{k_2=-\infty}^{\infty} ...
2
votes
1answer
61 views

An identity involving Gauss sums and convolution

For a Dirichlet character $\chi$ modulo $N$, the Gauss sum attached to $\chi$ is given by $$G_\chi(m) = \sum_{k \in \mathbb{Z}_N} \chi(k) e^{2\pi i mk/N}.$$ Suppose one has an $N$-periodic function ...
2
votes
1answer
59 views

What does it mean to convolve matrices of finite dimension?

If one is given two matrices $I$ and $K$ what does the notation: $$ I * K $$ mean rigorously/precisely? I do know the definition of convolution: $$ s[i, j] = (I * K)[i, j] = \sum_m \sum_n I[m,n] ...
0
votes
0answers
17 views

how to express separable 2D convolution using simple properties?

I have two 2D sequence $x(n_1,n_2)=f(n_1)g(n_2)$ and $y(n_1,n_2)$. How can I express this convolution \begin{equation} x(n_1,n_2)*y(n_1,n_2)=(f(n_1)g(n_2))*y(n_1,n_2)=(f(n_1)*y(n_1,n_2))*(g(n_2)) ...
1
vote
1answer
27 views

$\{f\in L^{1} : \hat{f} \in L^{p} \}$ closed under convolution?

Let $f, g\in L^{1}(\mathbb R),$ we may define the convolution of $f$ and $g$ as follows: $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy, (x\in \mathbb R).$ We note that $L^{1}(\mathbb R) \ast ...
0
votes
0answers
31 views

Understanding the convolution as a weighted average to eliminate noise $ s(t) = \int x(a) w(t-a) da$

I was reading Yoshua's Bengio [book][1] on convolutional neural networks and it has small section that described/explains the convolution in the context of estimating the location of a spaceship with ...
0
votes
1answer
51 views

convolution and integral limits

Let $\xi$ be an increasing function , and $f$ be a continuous function on the interval $[0,1]$. Take $\phi$ a smooth function such that $\int_0^1 \phi(s)\, ds= 1 $ and consider an approximation of ...
1
vote
3answers
83 views

History of convolution

Let $f, g\in L^{1}(\mathbb R),$ we may define the convolution of $f$ and $g$ as follows: $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy, (x\in \mathbb R).$ It is well known that it can be defined on ...
0
votes
0answers
42 views

Help with Fourier transform of product

I was reading this article in wikipedia, and I supposed $f,g \in L^1(\mathbb{R^n})$ such that their product $f \cdot g$ are in $L^1(\mathbb{R^n})$ too. So let $h=f \cdot g$, and ...