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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3
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33 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
votes
0answers
28 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} ...
1
vote
0answers
24 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
45 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
25 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
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0answers
17 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
20 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 ...
7
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0answers
80 views
+50

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
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0answers
83 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
61 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
116 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
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0answers
57 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 ...
1
vote
1answer
17 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 ...
3
votes
0answers
47 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} ...
1
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0answers
54 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
101 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
28 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
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 ...
2
votes
1answer
52 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
10 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
20 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
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0answers
16 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
46 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
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3answers
65 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
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0answers
37 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 ...
2
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0answers
28 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 ...
1
vote
1answer
32 views

Convolution with one of the variables is mixed and the other continuous

Suppose $X$ and $Y$ are independent random variables with CDF $F$ and $G$ and nonnegative support. If $X$ has a point mass $p$ at $0$ and otherwise some "density" $f$ (that is, ...
0
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0answers
23 views

Integration of convolution

I'm trying to solve the following equation $$\int\limits_{-\infty}^t \,(f\ast g)(t')dt'.$$ $f$ could be a kind of $\delta$-function: $f(t) = \delta(t)$ but should not be limited to be one. $g$ is ...
0
votes
1answer
28 views

Simple case of Young's inequality

I have a question concerning Young's inequality stated as follows: $||a∗b||_{ℓ_q}≤||a||_{\ell_1}||b||_{ℓ_q},~~~~ 1≤q≤∞$. Here you can find something on $\ell_q\big(\mathbb{Z}\big)$: Young's ...
2
votes
2answers
45 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 $G$?
0
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0answers
23 views

A convolution equation with two unknowns

I consider the following convolution-type equation with two unknowns $f_1$, $f_2$: $$ a_1 * f_1 + a_2 * f_2 = 0, $$ where $a_1$, $a_2 \in L^1(\mathbb R)$ and $*$ is the ordinary convolution. This ...
5
votes
1answer
67 views

Reversing results for sums of independent variables

Please let me use a specific example to illustrate the general title above. (1) It is well known that if $X$ and $Y$ are independent and $X,Y\sim N(0,1)$ then $$ Z\equiv X^2+Y^2\sim\chi_2^2 $$ where ...
1
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0answers
27 views

On the convolution of $f(x)=\sin x/x$ and $g(x)=1-|x|$

I am having trouble with computing the convolution of $f(x)=\sin x/x$ and: \begin{equation} g(x)=\begin{cases} 1-|x|,& -1 \leq x \leq 1 \\ 0, & x \notin [-1,1] \end{cases} \end{equation} I ...
0
votes
1answer
19 views

1D FFT on rotated image column by column

I am facing a problem: performing 1D FFT on a rotated column by column on a rotated image, described as following: Original Image: Rotated Image: What I have: original image convolution ...
3
votes
2answers
41 views

Steinhaus-like problem

I know there are similar problems on here, but I believe this is not a duplicate. Let $E \subset \mathbb{R}$ be a measurable set of positive finite measure. Define $f:[0,\infty) \rightarrow ...
0
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0answers
16 views

Concatenated model for 2D?

I am looking for a way to perform 2D convolution through a concatenated model. I am not looking for a faster way of doing 2D convolution. My objective is to find a scheme where I can perform it in ...
0
votes
0answers
9 views

Prove that $\int_{\mathbb R^n} u_\epsilon \, divT=\int_F div(T*\rho_\epsilon). $

Let $F\subseteq \mathbb R^n$ be a set and let $\chi_F$ its indicator function, regularized with functions $\rho_\epsilon\in C^\infty_c(\mathbb R^n)$ such that $u_\epsilon:=\chi_F* \rho_\epsilon\to 0$ ...
2
votes
1answer
34 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
67 views

a Bound for functions in $L^p$ after convolution with a $G_\lambda$ almost a heat Kernel

The following questiion comes from the article of Stroock & Varadhan (Diffusion processes with continuous coefficients I - 1969 - pg 378 ) We consider the operator $G_\lambda$ $$G_\lambda ...
0
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1answer
40 views

Convolution in Matlab with different “sampling”

I am trying to figure out how to "normalize" the convolution that Matlab does (using the "conv" operator). If I have a rect function with spacing T and I do the convolution of that function with ...
4
votes
2answers
95 views

Estimate of a convolution from a paper by Michael Christ

I don't understand Lemma 2 of the paper Hilbert transforms along curves, II: A flat case by Michael Christ. The situation is as follows. I slightly simplified it from the exact context in the source. ...
0
votes
1answer
19 views

Does the operand in a convolution have a particular name?

In a convolution: $$(f*g*h)(t) = \int f(x)g(y)h(z) \delta(t-x-y-z) dxdydz$$ do the operands $f,g,h$ have a specific name, besides the general "operand"?
2
votes
1answer
47 views

How can I find the limit?

Let $f\in$$L^1(\mathbb{R})$and $g\in$$L^1(\mathbb{R})$$\cap$$L^\infty(\mathbb{R})$. My question is how can I find the value $\lim_{x \to \infty}f\ast g (x)$ ? I know that $f\ast g $ is continuous and ...
0
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0answers
34 views

Transformation of function of three random variables

I have been doing exercises in basic probabilities, and there is a type of questions which I am never quite sure of the answer, so I wanted to get some opinion about the way I do it. The question is ...
0
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0answers
19 views

What groups act transitively on the unit disk

After long computations to find a formula for the solution of the Dirichlet problem on the strip, my Complex Analysis professor observed that the solution on the strip and on the disk is written in ...
3
votes
1answer
35 views

Convolution product, Lp spaces

I wonder how to prove the following statement, Let p,q be real numbers s.t $1\leq p \leq\infty$, $1\leq q \leq\infty$ and $ \frac{1}{p}+ \frac{1}{q}=1$ Let $f \in L^p(\mathbb R^n)$ and $g \in ...
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0answers
37 views

Why is polynomial convolution equivalent to multiplication in F[x]/(xn−1)?

Why is polynomial convolution equivalent to multiplication in $F[x]/(x^n-1)$? From this, I still can not understand how to get this $$ \begin{align} &f*g ...
0
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0answers
51 views

Normal Exponential Convolution proof

I am seeing a scientific paper where they explain background correction modelled as two random independent variables one with exponential distribution and the other with normal distribution. ...
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0answers
22 views

Shifting a function for convolution

We have $f(t)$, $h(t)$ and we want to compute the convolution of these two functions. So we will have a dummy variable τ for the product: f(τ)h(t-τ) as we know... Why we are not permitted to ...
0
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0answers
19 views

Graphical transformation : reflect and shift

I know that x[-n] will be reflection of x[n] along y-axis and x[n+k] will shift x[n] to left by k points. Now if I take x[n] 1. x'[n]=x[-n] should reflect along y axis 2. x'[n+k]=x[k-n] should shilf ...