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
votes
2answers
98 views

partially reconstruct information of function convoluted with boxcar kernel

the function (f) I want to reconstruct partially could look like this: The following properties are known: It consists only of alternating plateau (high/low). So the first derivation is zero ...
-2
votes
2answers
60 views

Lebesgue conduct integral

I. Suppose $f\in \mathcal{L^1}(R^n),g\in \mathcal{L^1}(R^n)$, then conduct integral $f*g$ is defined as $f*g(x)= \int_{R^n}f(x-y)g(y)dy$ for all $x$. My task is to prove following statements. (1) ...
2
votes
1answer
92 views

clarification asked for 'difference between convolution and crosscorrelation?'

I don't understand answer formulated in ways like this "Thus, $p\ast q$ is the distribution of $X+Y$. The cross-correlation $p\circ q$ is the distribution $c=(c_n)_n$ defined by ...
4
votes
0answers
301 views

Why matrix representation of convolution cannot explain the convolution theorem?

A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
2
votes
1answer
102 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 ...
0
votes
1answer
246 views

Is the convolution an invertible operation?

If I have a signal $f(x,y)$ (discrete) and I convolve this signal with a kernal $h(x,y)$: $y(x,y) = f(x,y) \star h(x,y)$ (where $\star$ is the convolution operator) Can I obtain $f(x,y)$ given only ...
0
votes
1answer
133 views

Integration function spherical coordinates, convolution

How can I calculate the following integral explicitly: $$\int_{R^3}\frac{f(x)}{|x-y|}dx$$ where $f$ is a function with spherical symmetry that is $f(x)=f(|x|)$? I tried to use polar coordinates at ...
2
votes
2answers
153 views

convolution computation involving $e^{-x^2}$

In working a problem involving convolution, I have arrived at the following integral, but do not know how to compute it: $$2\int_0^{\infty}e^{-a(x-y)^2-by^2}dy$$ I thought that this integrand did not ...
0
votes
1answer
144 views

Function as a convolution product of other two

I need help with this: I have to prove that a function $f\in L_{2}(T)$ can be expressed as $f=g*h$ (convolution product) for some functions $g,h\in L_{2}(T)$ if and only if $(\hat{f}(n))_{n}\in ...
1
vote
2answers
301 views

Differentiability of convolution

First let me say that I have used the search bar and looked through all the "differentiability of convolution" questions that I saw, but none of them seem to cover this case. (If one of them did and I ...
4
votes
0answers
511 views

Infinite self-convolution for a function

I have a mathematical problem that leads me to a particular necessity. I need to calculate the convolution of a function for itself for a certain amount of times. So consider a generic function $f : ...
0
votes
1answer
56 views

Need help with convolution problem.

I'm just learning about the convolution integral and am stuck on this example problem: Given $x_1(t) = \left \{\begin{array}{lr} 1 : 0 < x < 1 \\ 0: \text{elsewhere} \end{array} ...
1
vote
0answers
73 views

Fubini theorum for integrating 1 dimension of a 3d convolution

I have 3D volume that is convoluted with a 3D blur function. Both are positive and integrate to a finite value. I can see experimentally (meaning playing with matlab) that this is true: $\int_{-a}^{a ...
7
votes
1answer
229 views

A Mathematical way to represent a image kernel?

How to represent the calculation in this image mathematically? For example: With the discrete convolution and Fourier Transform. It tries to do a calculation on the original image (image A/input) ...
1
vote
1answer
338 views

What's the difference between convolution and crosscorrelation?

What's the difference between convolution and crosscorrelation? So why do you use '-' for convolution and '+' for crosscorrelation? Why do we need the "time reversal on one of the inputs" when doing ...
2
votes
2answers
516 views

Convolution: Laplace vs Fourier

Are there real world examples when it is better to use laplace instead of fourier to compute a convolution? And vice versa. Fourier can use negative numbers (as in 'integrates from minus infinity to ...
1
vote
1answer
645 views

Multiplying polynomial coefficients

Take: $u(x)$ and $v(x)$ to be integer polynomials, and then interpret them as sequences in the obvious way: i.e. you put the $i$th term to be the coefficient of $x^i$. Then you'll find that $u\ast ...
2
votes
2answers
194 views

Understanding convolution

Take: $$ (u*v)(k) = \sum_{i=-\infty}^\infty u(i)v(k-i). $$ The $k$ is there, it's because you want to define $$ \ldots\ldots, (u*v)(-3), (u*v)(-2), (u*v)(-1), (u*v)(0), (u*v)(1), (u*v)(2), ...
1
vote
1answer
195 views

Sifting Property of Convolution

This is going to be a dumb question, but I can't figure it out, so here goes $ f(t)\quad \bigotimes \quad \delta \quad (t\quad -\quad { t }_{ o }) $ = $\int { f(\tau )\delta (t\quad -\quad { t }_{ ...
0
votes
1answer
159 views

difference between convolution of two densities and mixture density?

I am wondering about the difference of the convolution of two probability density functions and the mixture of those two. This is not the same right? But what is the difference and how can it be ...
3
votes
1answer
138 views

Convergence of convolution of $L^p$ function with a sequence of distributions

let $h_n\in C_c^\infty (\mathbb{R}^d)$ s.t. $\int h_n dm = 1$ and $\operatorname{supp}(h_n)\to {0}$. I've proven that $h_n\to\delta_0$ in $\mathcal{D}'(\mathbb{R}^d)$, now I'm trying to show that for ...
1
vote
2answers
115 views

Convolution with sign function

I am having some trouble calculating the convolution $ (f*g)(t) $ between these two functions: $$ f(t)=e^{-t}1(t) $$ where $1(t)$ is the unit step function, and $$ g(t)=\mathrm{sgn}(t) $$ Using ...
0
votes
1answer
120 views

Convolution of dependent discrete random variables

We have a set $X_1, X_2, \ldots, X_n$ of correlated discrete random variable with a given correlation matrix. How can one compute the sum $X_1 + X_2 + \cdots+ X_n$ knowing the probability mass ...
4
votes
2answers
149 views

How to make 3D object smooth?

I want to make the below picture into an egg with smooth surface. For the implementation in Mathematica, please, see this thread here. This thread considers mathematical methods to achieve the goal ...
1
vote
1answer
137 views

$\mathcal{L}^p$ spaces and convolution

Suppose that $f \in \mathcal{L}^p$ and $g \in \mathcal{L}^q$, and $p,q$ are conjugate exponents. Then prove that (a) $h(x) = \int_{-\infty}^{\infty} f(t) g(x+t) \, dt$ defines a bounded continuous ...
1
vote
0answers
42 views

Cancellation of summations

I am working on some stuff related to the convolution property of the discrete Fourier transform. If we consider: $$\sum_{p = 0}^{N-1}\hat{s}_{p}e^{ik_{p}x_{m}} = \sum_{p = ...
3
votes
0answers
171 views

The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ...
2
votes
2answers
629 views

Convolution of integer sequences

I read this somewhere: $U$ and $V$ are defined on the set $\mathbb{Z}$ of integers. The convolution of $u$ and $v$, noted as $(u*v) (k)\,\theta$ or $ (u \otimes v) (k)\theta$, is a new sequence whose ...
1
vote
1answer
314 views

How do I take the complex convolution of this impulse response and input?

I derived a an impulse response of $h[n] = (3/4) (-j3/4)^{n} u[n]$, where $u[n]$ is the unit step function. I have an input $x[n] = u[n-5]$. I can find a vector representation of the convolution of ...
3
votes
0answers
240 views

Convolution Exercise Homework

Put $\varphi(t)= 1- \cos \;t\;\;\;$ if $\;\;\;0 \leq t \leq 2 \pi$, $\varphi(t) = 0$ for all other real $t$. For $-\infty < x < \infty $, define $$ f(x)= 1,\;\;\;\;\;\;\;\;\;\;g(x) = ...
0
votes
1answer
1k 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
1answer
176 views

Lower bounds of laplace transform of characteristic functions

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
1
vote
1answer
256 views

Graphical understanding of the convolution of discrete distributions.

I am studying convolution and trying to get a visual sense of the process. From the wikipedia page I understand what 2 continuous gaussian or 2 uniform distributions will produced when convolved, but ...
1
vote
2answers
263 views

Convolution of continuous function with $\mathcal{C}^{1} $ function

I'm having difficulty with the following problem: Let $f:\mathbb{R}\to\mathbb{R}$ be continuous on $\mathbb{R}$ such that $\mbox{Supp}\left(f\right)$ is compact and let $g:\mathbb{R}\to\mathbb{R}$ ...
2
votes
0answers
88 views

Convolution and Smoothness Conditions

Suppose $f(x),g(x)\in L_1(\mathbb{R})$, with both $|f(x)| \leq 1$, $|g(x)| \leq 1$ and $|f(x)| \rightarrow 0$, $|g(x)| \rightarrow 0$ for $|x| \rightarrow \infty$. Given that we have two other ...
2
votes
1answer
170 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 ...
1
vote
2answers
353 views

How can you do convolution graphically?

I don't exactly remember whether I should get the common area under the curves of the functions being convolved or I should multiply them and get the area under the resulting curve.
1
vote
1answer
353 views

convolution square root of uniform distribution

I need to find a probability distribution function $f(x)$ such that the convolution $f * f$ is the uniform distribution (between $x=0$ and $x=1$). I would like to generate pairs of numbers with ...
0
votes
1answer
220 views

what's the distribution of the inverse of a random variable that follows a negative binomial distribution?

I was studying the method of moments estimation of parameters, and I encountered the following problem. I have a geometric distribution as following: $P(X=k) = p(1-p)^{k-1}$, and a sample size of n, ...
3
votes
4answers
1k views

Repeated convolution of probability distributions

Question Let $$S_k=\sum_{i=1}^k X_i$$ be the sum of $k$ independent random variables. I am interested in closed-form expressions of the pdf of $S_k$. In general, the pdf is given by the $k$-fold ...
7
votes
0answers
84 views

Properties of a continued fraction convolution operation

Usually the partial numerators of a continued fraction are all 1s. Has anyone considered the operation where you convolve 1 continued fraction with another, in other words, make a new continued ...
0
votes
1answer
1k views

Finding limits of integration in convolution

I am struggling to fully get how to choose proper limits of integration when calculating convolutions. Right now I am stuck on a problem where I have to show that when taking the Fourier transform of ...
2
votes
1answer
65 views

Convolution of $L_1$ functions is $L_1$

Let $f,g\in L_1(\mathbb R, m)$ where m is the Lebesgue measure. Prove that: a) $f(x-t)g(t) \in L_1(\mathbb R, m)$ as a fuction of $t$ almost all $x$ b) $h\in L_1(\mathbb R, m)$ where $h=(f \ast ...
2
votes
2answers
334 views

the fourier transform of a “double convolution”

Suppose I have a function $$ m(x) = f(x)\int_{-\infty}^{\infty} h(w)g(w-x)dw = f(x)h*g(x) $$ I want to find the Fourier transform of m(x) in terms of the Fourier transforms of $f,h,g$ but for the ...
1
vote
1answer
51 views

A problem about Convolution

If $\phi\in C_0^{\infty}(\mathbb{R}^N)$ and $\psi\in L_{loc}^1(\mathbb{R}^N)$ is defined by $\psi(x)=|x|^{2-N}$, $N\geq 3$ , does $\phi\star\psi$ is in the Schwartz space? Note: $\star$ stands for ...
1
vote
1answer
179 views

PDE - (homogeneous) Heat equation - Solution?

today I have a question in PDE. It concerns the heat equation: Formulate the (homogeneous) heat equation for functions $f:(0,\infty)\times\mathbb{R^n} \longrightarrow\mathbb{C}$. Derive an equation ...
1
vote
1answer
323 views

Intuition behind the convolution of two functions

Suppose $f(x)$ and $g(x)$ are two functions. What is intuition or idea behind the convolution of $f$ and $g$? After taking the convolution we will get a new function. What is the geometric relation ...
2
votes
1answer
82 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 ...
0
votes
1answer
91 views

Filter signal through convolution

I am a little bit unsure if I've set up the following problem correctly: Consider the signal $$f(t) = e^{-t}(\sin(5t) + \sin(3t) + \sin(t) + \sin(40t)) \quad 0 \leq t \leq \pi$$ Filter this signal ...
2
votes
0answers
64 views

Show compactness of an evolution operator

Consider the heat equation $$ u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$ with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$ and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$. 1.) ...