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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2
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3answers
247 views

Does the Convolution of Two Series Require Absolute Convergence?

Let $A=(a_n)_{n=0}^\infty$ and $B=(b_n)_{n=0}^\infty$ be sequences of real numbers. Let $C = (c_n)_{n=0}^\infty$ be the sequence such that $$c_n = {a_0}{b_n} + {a_1}{b_{n-1}} +\cdots+ {a_n}{b_0}.$$ ...
2
votes
1answer
244 views

Using FFT in matlab

I am not completely sure if this is where a MatLab question belongs, so if not, please direct me where I should ask. But onto my question. I am working on trying to deconvolution a signal with ...
0
votes
1answer
36 views

Convolution and integrating over G(t)

I'm struggling with the following expression in a statistics script: $$H(x) = \int_{-\infty}^\infty F(x-t) dG(t)$$ What does the dG(t) mean exactly? I've never seen that notation before. Background: ...
2
votes
0answers
300 views

How to express multiplication of two spherical harmonics expansions in terms of their coefficients?

Consider a spherical harmonics expansion/series like this: $$f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)$$ Presumably if we take two functions on ...
2
votes
1answer
58 views

Asymptotics at the origin of the convolution with an approximation to the identity.

In short, I am trying to find sufficient conditions for an approximation to the identity function $K_h$ so that, for $h$ small enough and fixed, the asymptotics at the origin of an $L^1 \cap L^2$ ...
1
vote
1answer
327 views

Convolution of distributions.

We are given with distributions $f,g \in D'(\Bbb R)$. If $suppf\subset (-\infty,a)$ and $supp(g)\subset(b,\infty)$ then prove that $f*g$ is well defined distribution. where $a$ and $b$ are real ...
3
votes
2answers
156 views

How to prove that operator is not compact in $L_2 (\mathbb{R})$

I have the operator $(Af)(x) = \int _{\mathbb{R}} e^{{-(x-t)^2}/2} f(t) dt$. It seems to me that it isn't compact and I'm looking for some general <=> criterion for integral operators to be ...
2
votes
1answer
122 views

The differentiability of convolutions

Yes, again, this type of question. Similar ones this and this. I come with another variant. Let $f\in\mathcal{S}$, i.e. Schwartz function, and $g\in L^{p}(\mathbb{R}^d),p\in[1,\infty]$. The following ...
3
votes
2answers
106 views

Problem of convolution.

If we are given with a polynomial $\mathcal P$ and a compactly supported distribution $g$. Can we prove that their convolution will be a polynomial again?
2
votes
2answers
155 views

What if the cauchy product of two series in $\mathbf{Z}$ is null

I have a problem I do not find a solution. Given two series $\left(a_n\right)_{n \in \mathbf{Z}}$ and $\left(b_n\right)_{n \in \mathbf{Z}}$ which have a cauchy product $\left(c_n\right)_{n \in ...
1
vote
0answers
136 views

Fourier transform of convolution in a finite range

Can anyone help me evaluate the Fourier transform of of the following function, $t \in \mathbb{R}$, $\lambda \in \mathbb{C}$, $g:\mathbb{R} \rightarrow \mathbb{R}$, $f(t) = \int_{t_0}^t ...
3
votes
2answers
2k views

convolution of exponential distribution and uniform distribution

Given $X$ an exponentially distributed random variable with parameter $\lambda$ and $Y$ a uniformly distributed random variable between $-C$ and $C$. $X$ and $Y$ are independent. I'm supposed to ...
1
vote
1answer
59 views

expanding convoluted integrand

I have a function on the form $$g(y) = \int_{-\infty}^{\infty}{e^{-v^2}f(y-v)dv}$$ I know that $g(y)$ is linear around $0$, $g(y\approx 0)\approx yG$, and I am interested in finding this gradient $G$. ...
2
votes
1answer
128 views

On using fourier transforms to solve the root of a convolution

In continuation of Lower bounds of laplace transform of characteristic functions. My question is: Can anyone point out where i'm going wrong in the derivation below. It's been a while ...
0
votes
1answer
65 views

Proving an identity regarding the Cauchy problem (using convolutions)

Given $u_0 \in C_c(\mathbb{R}^n)$, consider the solution of the Cauchy problem $$u(x,t) = \int_{\mathbb{R}^n} \Gamma (x - y,t)u_0(y) dy \qquad x \in \mathbb{R}^n,t>0\, \, .$$ Given $0<s<t$ , ...
2
votes
2answers
61 views

Evaluate $\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$

I'm trying to evaluate the integral $$\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$$ where $\chi_{[0,1]}(x)=1$ is the characteristic function, i.e. equals $1$ for $x \in ...
16
votes
3answers
1k 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
205 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 ...
2
votes
0answers
106 views

General approach to prove the smoothness of convolution

Consider $\mathbb{R}^d$ and $D_i = \partial /\partial x_i$, in many cases $$D_i(f*g) = D_if*g,$$ given one of $f,g$ is smooth and the other is $L^p$ integrable. I am wondering if there is a general ...
2
votes
1answer
593 views

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 ...
3
votes
0answers
47 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 ...
2
votes
1answer
78 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
1answer
40 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 $x\in\mathbb{R}$: $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$ ...
-2
votes
1answer
105 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
299 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 ...
5
votes
2answers
101 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
329 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
103 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
261 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
137 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
158 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
147 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
316 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
538 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
74 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
234 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
342 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
534 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
667 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
199 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
213 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
164 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
144 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
126 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
125 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
152 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
140 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 ...