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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15 views

Convolution of independent but 'different' probability distributions

I have the following two probability distributions they relate to a particular ice-cream: ...
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28 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 $X_1,\dots,X_n$ ...
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3answers
40 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. ...
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0answers
14 views

Uniform convergence of convolution using density argument [on hold]

An example in using $L^p$ norm inequality: Example1.24. If p=q=2, or more generally if q=p′,then $r=∞$. In this case, the result follows from the Cauchy-Schwartz inequality, since for all $x ∈ ...
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1answer
24 views

convolution of probablity 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 ...
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0answers
26 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 ...
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0answers
25 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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0answers
29 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 ...
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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 ...
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1answer
27 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 ...
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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= ...
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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: $$f(x) := e^{-\frac{1}{1-x^2}}1_{(-1,1)} $$ ...
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1answer
41 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 ...
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2answers
42 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
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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 ...
3
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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 ...
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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} ...
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0answers
26 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 ...
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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} * ...
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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 ...
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0answers
22 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
154 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. ...
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0answers
63 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 ...
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1answer
22 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 ...
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49 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} ...
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57 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 ...
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1answer
137 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 ...
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0answers
30 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} ...
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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 ...
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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] ...
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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)) ...
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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 ...
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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 ...
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1answer
47 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 ...
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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 ...
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0answers
38 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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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2answers
53 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 $G$?