Questions on the (continuous or discrete) convolution of two functions.
9
votes
3answers
71 views
Why convolution regularize functions?
There is a tool in mathematics that I have used a lot of times and I'm still not confortable with. In fact I can't figure out (by this I mean that I cannot understand it geometrically) why does ...
1
vote
0answers
28 views
What's the exact definition for convolution?
I tried to solve the problem in Stein's Real analysis, 1ed, P94, Ex 21 (c), which asked to show that for any two measurable functions $f,g$ on $R^d$, the convolution of $f$ and $g$, $$(f\ast ...
1
vote
0answers
24 views
General approach to prove the smoothness of convolution
Consider $\mathbb{R}^d$ and $D_i = \partial /\partial x_i$, in many cases
$$D_i(f*g) = D_if*g = f*D_i g,$$
given one of $f,g$ is smooth and the other is $L^p$ integrable.
I am wondering if there is ...
0
votes
1answer
71 views
+50
What will be the support of the convolution of two test functions.
If $g\in C^{\infty}_c$ defined on $\Bbb R^n$ and K is the support of function $g$. I want to find the support of $g_\epsilon$. Where $g_\epsilon$ is regularization of $g$.
Regularization of $g$ is ...
0
votes
0answers
7 views
Closeness of a family of function under convolution.
I'm interested in functions defined over the non-negative integers that are a product of an exponential function and a polynomial. So a standard term of such a function is something like
$$
f(k) = ...
2
votes
0answers
22 views
changing the parameters of a function
Lets say we have
$h[n] = ((1/2)^n )(u[n])$
now if we are ask, find h[k-n], then isn't it we should just swapped every 'n' with 'k-n'. So it turns out
$h[k-n] = ((1/2)^{k-n})(u[k-n])$
But why here ...
0
votes
0answers
22 views
Convolution of logistic function and gaussian distribution
I am trying to solve the folowing problem:
$$\int \exp\left(-\frac{(x-u)^2}{2\sigma^2}\right) \log(1+\exp(ax + b)) \,dx$$
which I think is very complicated and there is no closed form solution(?)
...
2
votes
1answer
59 views
Fourier analysis exercise
I need a hand with this question:
If $f\in{L_1(\mathbb{R})}$ and $g\in{L_2(\mathbb{R})}$, then prove that $\widehat{f*g}=\hat{f}\cdot \hat{g}$
As a tip, i have been told to prove that:
...
0
votes
0answers
24 views
Convolution of Inverse Gaussian distribution
Im having problems with showing that the sum of two inverse gaussian distributed random variables are stable under convolution, i.e.
let $f_{T_a}(t) = \frac{a}{\sqrt{2 \pi t^3}} e^{-\frac{(a- \nu ...
0
votes
1answer
22 views
Function with $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$
I am looking for a function $f:\mathbb{R}\to \mathbb{R}$ and $\epsilon>0$ such that there is no $\delta>0$, for him any $x\in\mathbb{R}$: $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$
...
0
votes
0answers
28 views
Inclusion regarding the support of the convolution of two functions
Let $u \in L^1(\mathbb R^n)$ and $v \in L^p(\mathbb R^n)$ where $1 \le p \le \infty$. Show that $$\textrm{spt}(u \ast v) \subseteq \overline{\mathrm{spt}(u) + \mathrm{spt}(v)}$$ where the addition ...
-1
votes
1answer
30 views
Prove that L[f' ' ](s)$ = $sL[f](s)
Can anyone prove this question ?
Let $f$:$\mathbb{R}$$→$$\mathbb{C}$ be continuous function such that $f$$(0)$ $=$ $0$ and that $f'$ be a piecewise continuous function and absolutely integrable on ...
0
votes
1answer
57 views
Using Convolution Theorem to find the Laplace transform
In previous questions I have used Laplace transform to find the inverse Laplace transform.
I have worked through this work booklet ...
4
votes
2answers
30 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
44 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
0answers
76 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 ...
2
votes
0answers
55 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 ...
-1
votes
0answers
15 views
Uniform Convergence of Convolutions
$\displaystyle Q_n(t) = ne^{-nt}$
$Qn(t)*f(x) = \int_0^{\infty} Q_n(t)*f(x-t) dt$
How do I prove that this converges uniformly? It seems to be similar to the proof of the
Weierstrass ...
2
votes
1answer
46 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
59 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
21 views
Integration function spherical coordinates, convolution
How can I calculate the following integral explicitly:
$$\int_{R^3}\frac{f(x)}{|x-y|}dx$$
where $f$ is a function with spherical symmetry that is $f(x)=f(|x|)$?
I tried to use polar coordinates at ...
0
votes
0answers
35 views
Discrete convolution: where do I go from there?
I took this from the book Signals and Systems by Haykin.
I have the following discrete system: $y[n] = u[n]*u[n-3]$, where $*$ is the discrete convolution and $u[n]$ is the unit step function, and ...
0
votes
0answers
31 views
Strange convolution equation
In an article ( https://www.dropbox.com/s/3012v4s1ngpimvg/gridding_Schomberg_Trimmer.pdf ) about implementation of Gridding method for parallel-beam tomography there's an equation(#47 in the article):
...
1
vote
2answers
48 views
convolution computation involving $e^{-x^2}$
In working a problem involving convolution, I have arrived at the following integral, but do not know how to compute it:
$$2\int_0^{\infty}e^{-a(x-y)^2-by^2}dy$$
I thought that this integrand did not ...
0
votes
1answer
33 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
1answer
41 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 ...
3
votes
0answers
88 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
27 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
42 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 ...
6
votes
1answer
133 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
176 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 ...
1
vote
2answers
85 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
51 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 ...
1
vote
2answers
74 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
40 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 }_{ ...
1
vote
0answers
42 views
convolution: Fourier transform to convolution matrix [closed]
Do you know convolution very well? Can you explain the similarities and the differences between convolution, its relation to the discrete-time Fourier transform, the fast fourier transform, the ...
0
votes
1answer
33 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
51 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
35 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
51 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 ...
2
votes
1answer
64 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
78 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
28 views
Cancellation of summations
I am working on some stuff related to the convolution property of the discrete Fourier transform. If we consider:
$$\sum_{p = 0}^{N-1}\hat{s}_{p}e^{ik_{p}x_{m}} = \sum_{p = ...
3
votes
0answers
74 views
The norm of an operator
Let $\rho(x)$ be a weight function in a unit sphere, such that
\begin{equation}
\begin{array}{l}
\displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\
\displaystyle 2. \rho(x)\in ...
2
votes
2answers
154 views
Convolution of integer sequences
I read this somewhere:
$U$ and $V$ are defined on the set $\mathbb{Z}$ of integers. The convolution of $u$ and $v$, noted as $(u*v) (k)\,\theta$ or $ (u \otimes v) (k)\theta$, is a new sequence whose ...
1
vote
1answer
72 views
How do I take the complex convolution of this impulse response and input?
I derived a an impulse response of $h[n] = (3/4) (-j3/4)^{n} u[n]$, where $u[n]$ is the unit step function. I have an input $x[n] = u[n-5]$. I can find a vector representation of the convolution of ...
3
votes
0answers
70 views
Convolution Exercise Homework
Put $\varphi(t)= 1- \cos \;t\;\;\;$ if $\;\;\;0 \leq t \leq 2 \pi$, $\varphi(t) = 0$ for all other real $t$. For $-\infty < x < \infty $, define
$$
f(x)= 1,\;\;\;\;\;\;\;\;\;\;g(x) = ...
-1
votes
1answer
119 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
65 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
80 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 ...
