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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5
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2answers
554 views

Convolution of irreducible characters of a finite group

If $\chi^{\lambda}$ and $\chi^{\mu}$ are the characters of two irreducible representations $V^{\lambda}$ and $V^{\mu}$ of a finite group $G$, is there a simple way of proving that : $$ \chi^{\lambda} ...
0
votes
2answers
25 views

Evaluate convolution integral

Can someone tell me if I am calculating this integral correctly.
0
votes
1answer
12 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 - ...
4
votes
0answers
72 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 ...
1
vote
0answers
23 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 ...
2
votes
1answer
38 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
32 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 ...
0
votes
0answers
16 views

Normal convolved to Exp(polynomial)?

Is there an analytic solution for a Normal (normalized Gaussian) distribution of variance $v$ convolved to $e^{y(x)}$, where $y(x)$ is an $m$-th order polynomial? Assume that $m$ is even and the ...
0
votes
2answers
852 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 ...
1
vote
1answer
38 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 ...
0
votes
1answer
29 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
votes
2answers
1k views

Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...
5
votes
0answers
129 views

How much does convolution with a C^m kernel increase the order of continuity.

EDIT: I rewrote the question, see the edit history for my previous attempt. My questions are: Is the following proof/reasoning essentially correct? How do I make it more precise, concise, correct? ...
0
votes
1answer
2k views

How to determine the step response using convolution of the signal's impulse response?

The step response can be determined by recalling that the response of an LTI to any input signal is found by computing the convolution of that signal with the impulse response of the system. ...
1
vote
0answers
23 views

Convolution of independent but 'different' probability distributions

I have the following two probability distributions they relate to a particular ice-cream: ...
2
votes
3answers
44 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. ...
1
vote
0answers
30 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 $X_1,\dots,X_n$ ...
0
votes
0answers
27 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
28 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
11 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 ...
2
votes
1answer
38 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
17 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 ...
0
votes
1answer
324 views

Matlab: Impulse response of linear time invariable (LTI) sine-signal

I'm preparing for a lab in a Signals and Systems course in my university, 5th semester. I've found old exercise material from the class and since I know some Matlab and have dealt with LTI systems ...
0
votes
0answers
23 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 ...
0
votes
1answer
514 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) $?
3
votes
1answer
22 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 ...
3
votes
1answer
37 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 ...
2
votes
2answers
45 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 ...
1
vote
1answer
48 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
vote
0answers
30 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
votes
1answer
30 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
84 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
13 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
42 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 ...
3
votes
1answer
40 views

Convolution of tempered distributions where one has compact support.

For $u\in\mathcal E'(\mathbb R^n)$ and $v\in\mathcal S'(\mathbb R^n)$, we defined $u\ast v$ by $\langle u\ast v, \phi\rangle = \langle v, \check u \ast \phi \rangle$ for all $\phi\in\mathcal S(\mathbb ...
3
votes
1answer
52 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 ...
0
votes
0answers
34 views

An inverse Laplace transform I

While viewing the problem "Find the inverse Laplace transform" the solution given by Amir Alizadeh can be reformulated into the form \begin{align} \mathcal{L}^{-1}\left\{ \frac{s \, (a - f(s))}{s-b} ...
8
votes
0answers
117 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 ...
1
vote
0answers
27 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 ...
8
votes
2answers
158 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
0answers
47 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
vote
0answers
27 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
0answers
25 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 ...
1
vote
2answers
63 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 ...
0
votes
0answers
23 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 ...
2
votes
0answers
90 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
0answers
58 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 ...
2
votes
1answer
52 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 ...
4
votes
1answer
138 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 ...
1
vote
0answers
64 views

Trellis Diagram - Viterbi decoding

I have the trellis diagram below which is used as Viterbi decoder. The coded message is the sequence of bits at the bottom of the picture. My question is this. t=0:The decoder starts from state 00 ...