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Questions tagged [convolution]

Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schwartz's sense) or measures.

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Integration of the product of a compact supported convolution

I know that in general case we have $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} f(s)g(t-s) ds dt = \left( \int_{-\infty}^{+\infty} f(t) dt \right) \left( \int_{-\infty}^{+\infty}g(t) dt \right) ...
Cantor's user avatar
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Solving 1st order PDE including convolution

I'm studying Van Kampen's "Stochastic processes in physics and chemistry" and stuck to some exercise (p.78): That is, solving \begin{equation} \frac{\partial P(y, t)}{\partial t}=\int_{-\...
Patche's user avatar
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Is the Hadamard Product of two laplacian operators allowed to get some kind of biharmonic operator?

I'm currently working on my masters thesis in computer science and from this point I'm not that into this subject. Right know I try to understand the steps the authors of this paper did to get the ...
dontoronto's user avatar
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Is there an exact expression for the full width half maximum of a sech^2 curve convolved on itself?

As some simple math can show, a Gaussian convolved onto itself is also a Gaussian. Importantly, the FWHM of the original gaussian compared to that of its convolved counterpart is different by a factor ...
ChemGuy's user avatar
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Matrix multiplication expressed as convolutions? [closed]

I know that the 2D discrete convolution operation can be expressed as a sparse matrix multiplication, but can the reverse be done easily? Does anyone know if there is a way to express any matrix ...
Run Zhou Ye's user avatar
2 votes
1 answer
74 views

Proving a distribution is not infinitely divisible

I'm trying to show the following: Show that the distribution on $\mathbb R$ with density $f(x) = \frac{1-\cos(x)}{\pi x^2}$ is not infinitely divisible. The characteristic function of this ...
D Ford's user avatar
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4 votes
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Evaluating the convolution of $e^{-at^2}$ and $e^{-bt^2}$ via Fourier transforms

Problem Statement: Use the convolution theorem on the function $ f(t) = e^{-at^2} $ and $ f(t) = e^{-bt^2} $, $ a, b \in \mathbb{R} $. Calculate $ (f \ast g)(t) $. I got a hint that I should first ...
math123's user avatar
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Convolution between $L^1$ function and a singular integral kernel

I meet a problem when reading Modern Fourier Analysis(3rd. Edition) written by L.Grafakos. On pg.82 he writes: Fix $L\in\mathbb{Z}^+$. Suppose that $\{K_j(x)\}_{j=1}^L$ is a family of functions ...
ununhappy's user avatar
9 votes
1 answer
272 views

How to prove $(F\ast\sin)(x)=-\sin(x)$, where $F(x)=\frac{1}{2}|x|$?

Wikipedia states in this article about fundamental solutions that if $F\left( x \right) = \tfrac{1}{2} \left| x \right|$, then $$\left( F \ast \sin \right)\left( x \right) := \int\limits_{-\infty}^{\...
Alucard-o Ming's user avatar
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convolution of the fundamental solution with the homogeneous solution

I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero? Let $U$ and $E$ ...
Alucard-o Ming's user avatar
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Analycity of $f*g$ with $f$ and $g$ smooth on $\mathbb{R}$ and analytic on $\mathbb{R}^*$

Posted also on MO with a bounty Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ analytic on $\mathbb{R}\backslash\{0\}$ but non-...
NancyBoy's user avatar
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1 answer
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Algorithm to compute a convolution recursively

Let $$ f(t) = \int_0^t k(t-s)g(s) \, ds. $$ Assume that $g$ is only given in a grid $t_j = j\delta_t$, and that we wish to compute similarly $f$ on the same grid. What's an efficient algorithm to ...
G. Gare's user avatar
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Clarification Needed on 1D Convolution and Kernel Purpose

I am confused about the definition of 1D convolution. Given $ a = [-\frac{1}{2}, \frac{1}{2}] $ and $b = [1, 1, 1, -1, -1, -1] $, what will be the result of the convolution $( a * b )$? From my ...
Daniel's user avatar
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Convolution of slightly multivariate Gaussians slightly modified

Starting with $ p(a) = \int p(a|b) p(b) db$ replace $p(b)$ with $\tilde{p}(b) = \mathcal{N}(b; \mu_b, \Sigma_b + \tilde{D})$ where $\tilde{D}$ is an additive diagonal covariance. Assuming ...
scleronomic's user avatar
4 votes
1 answer
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Convolution preserve the boundary condition

Here, I want to know if convolution will preserve the Neumann condition or not. Suppose $K$ is a continuous function on some interval $[-L,L]$, and $u$ is some 'good enouth' function on $[0,L]$ that ...
SaltedFishKing's user avatar
10 votes
2 answers
571 views

What's the necessary and sufficient condition for a real sequence to be written as the self-convolution of another real sequence?

Definition For a sequence $a_0,a_1,\cdots,a_n$, the corresponding self-convolution is another sequence $\displaystyle b_m=\sum\limits_{i+j=m}a_ia_j$ where $0\leq m\leq 2n$. Calculating the self-...
grj040803's user avatar
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Associativity of Convolutions

In Folland's real analysis textbook, there are the following propositions: Assuming that all integrals in question exist, we have $$ (f*g)*h=f*(g*h) $$ The proof is based on the Fubini's theorem.But ...
12345's user avatar
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Help with understanding combination of probability distributions

I have two probability mass functions (PMFs) across the surface of a sphere. They are localised Gaussians (a few degrees in expanse), whose centres have arbitrary positions though they are quite close ...
Jacob Ayre's user avatar
-1 votes
1 answer
65 views

Inverse Fourier Transform - convolution of exponential and rectangular window

I'm trying to get the response in the time domain of the convolution between the exponential $u(t)e^{-at}$ and the rectangular window ($u(t+1)-u(t-1)$). I had already obtained its result by ...
nickalicas's user avatar
0 votes
1 answer
53 views

Is convolution theorem on $l^2(\mathbb{Z})$ valid?

I have a doubt about Fourier transform $F:L^2([0,2\pi])\to l^2(\mathbb{Z})$. If $f,g\in l^2(\mathbb{Z})$ then $f*g\in l^2(\mathbb{Z})$, then, $\mathcal{F}^{-1}(f*g)\in L^2([0,2\pi])$. Question $\...
eraldcoil's user avatar
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The Fourier transform of product of derivatives

I have the task to compute the Fourier transform of the product in matlab: $$ \left( \frac{\partial u(t, x)}{\partial x} \right)^2 \left( \frac{\partial^2 u(t, x)}{\partial x^2 } \right)$$ I was ...
unknown's user avatar
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Cross-correlation of a function with itself

I came up with the following question while writing on my thesis. We assume $f:\mathbb{R}\rightarrow\mathbb{R}$ to be a real valued $\mathcal{L}^1$-function. Then the cross-correlation of $f$ with ...
Christoph Richter's user avatar
1 vote
1 answer
145 views

A hard optimization problem

Consider the following function, for $1\leq j \leq N$ $$\tag{1} y_j=\sum_{k=0}^{M} \frac{e^{-\sum_{|i|\leq k}(k-|i|)x_{j+i}/v}-e^{-\sum_{|i|\leq k}(k+1-|i|)x_{j+i}/v}}{\sum_{|i|\leq k} x_{j+i}} $$ for ...
sam wolfe's user avatar
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19 views

Weighted inequality on torus

In the Torus (circle). Let $[0,2\pi]\to\mathbb ]0,\infty[\colon \theta\mapsto w(\theta)$ a weight function, i.e. nonnegative and integrable on $[0,2\pi]$. If $\mathbb{Z}\to\mathbb{R}\colon k\mapsto m(...
eraldcoil's user avatar
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3 votes
1 answer
73 views

What is wrong with this proof that a linear, bounded, time invariant operator on $L_p$ must be a convolution?

I'm trying to understand if this is true and how to prove it, "If $T$ is a bounded, time invariant operator on $L_p(\mathbb{R})$, then $T$ is a convolution operator.'' Here's an attempt at a ...
travelingbones's user avatar
2 votes
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49 views

General solution for linear Volterra-like integral equation?

A linear Volterra integral equation looks like this (see the wiki) \begin{align} x(t) = f(t) + \int_0^t K(t, s)x(s)~\mathrm{d}s. \end{align} If the Kernel function $K$ is of the form $K(t, s) = K(...
Lyle's user avatar
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Convolution of $\mathcal{C}^\infty$ is analytic

Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$. Is the convolution (assumed to be well defined) defined as: $$(f*g)(x) = \int_\mathbb{...
NancyBoy's user avatar
  • 506
1 vote
3 answers
100 views

Try to give the solution of PDE with initial boundary

The equation is \begin{align} \partial_{t}\!\operatorname{u}\!\left(x,t\right) & = x^{2}\,\partial_{x}^{2}\operatorname{u}(x,t) + x\,\partial_{x}\operatorname{u}\left(x,t\right),\quad\quad(t,x)\in\...
George Lin's user avatar
0 votes
2 answers
66 views

Differentiating Entropy with respect to Convolution Parameters

First, some formula reminders for the sake of completion: $H(X) = -\sum_{i} p(x_i) \log p(x_i)$ is the entropy of a sequence $x_i$, where $p(x)$ is the discrete probability of x. A discrete ...
2 False's user avatar
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1 answer
74 views

Seeking Algorithm to Solve a Convolution Integral or Directly Convolve Two CDFs

Hi everyone, I'm working on a project where I need to find a way to directly convolve two cumulative distribution functions (CDFs) given in polynomial form, and solve the following convolution ...
guttf's user avatar
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Is the convolution between two CDF always well defined?

Given the integral convolution: $$(F_X * G_X)(x)=\int_{-\infty}^x F_X(t)G_X(x-t)dt $$ and also that the CDF of random variables are bounded between the interval $(0,1)$ and the flipping of one of them ...
Daniel Muñoz's user avatar
1 vote
1 answer
65 views

Circular shift of a function.

Consider a function $f$ that maps real numbers to real numbers with domain $[-a,a]$. I would like to describe the circular shift of this function by an amount $\delta$ such that, if I shift the ...
NicNic8's user avatar
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1 vote
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Prove that the convolution of the signals and its time reversal is an odd signal.

Suppose signal $g(t)$ is obtained by time reversal of signal $f(t)$ for all times $t$. Prove that the convolution of the signals $f$ and $g$ is an odd signal. My attempt at proof Given: $g(t)=f(-t)\...
Awe Kumar Jha's user avatar
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Derivative of convolution of a continuous function with a continuously differentiable function.

Suppose $f \in C^1(\mathbb R)$ has compact support and $g \in > C(\mathbb R)$ is bounded and $\lVert g \rVert _1 < \infty$. Prove that the convolution $f * g$ is continuously differentiable and $...
RatherAmusing's user avatar
1 vote
1 answer
47 views

Identifying Waveforms that Satisfy Specific Convolution Constraints

I am attempting to find a set of waveforms, denoted as $y_1$, $y_2$, and $y_3$, that satisfy the following convolution constraints, where $*$ is used to denote a convolution: $$ y_1,y_2,y_3 \text{ are ...
Tiny Tim's user avatar
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1 vote
1 answer
35 views

The operator norm regarding to the difference between a mollified function and the function itself

Let $\rho_\epsilon$ be a mollifier that has support in $B(0,\epsilon)$, define the operator $$T_\epsilon (f)=\rho_\epsilon*f-f,$$ for every $f\in L^2(\mathbb R^d)$, can we prove or disprove that the ...
Euler's little helper's user avatar
1 vote
0 answers
25 views

Classifying bounded linear functions satisfying convolution identity

Find all bounded linear functionals $T$ on $Y= \{f \in W^{1,1}[0,1]: f(0)=0\}$ such that there exists $K>0$ such that $$\int_0^1|Tf(\cdot-t)|\ dt\le K \|f\|_1\qquad \forall\ f\in Y\tag{1}$$ My ...
modeltheory's user avatar
1 vote
1 answer
76 views

Proof review Tempered Distributions: $(T*\varphi)^\wedge = (2\pi)^{n/2} \hat{\varphi}\hat{T}$ and computing $(x^\alpha)^\wedge$

I am learning about tempered distributions on schwartz space. But there are 2 questions in this text where Im unsatisfied with my solution... I ask kindly for your review of my solutions below, and if ...
NazimJ's user avatar
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1 vote
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Derivative of a convolution integral

I'm reading this material Topics in inverse problems and I'm having difficulty in understanding how the derivative of a convolution integral was obtained. In equation 4.5 (page 91), the function $y(t)$...
zooond's user avatar
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1 answer
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Suddenly applied discrete complex exponential inputs and convolution

I am reading the textbook Discrete-Time Signal Processing by Oppenheim & Schafer. I am confused about how to get $y[n]$. By discrete-time convolution, we have $$...
sleeve chen's user avatar
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3 votes
1 answer
136 views

Properties of fourier series on $SO(3)$

With standard fourier series we can use some identities like convolution theorem and Parseval's theorem: (convolution theorem) Fourier series of the convolution of $f$ and $g$ is the point-wise ...
cnikbesku's user avatar
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Dependent random variables whose convolution adds up

I want to find two dependent random variables $X$ and $Y$ with values in $\mathbb{Z}$ such that $P_X\star P_Y=P_{X+Y}$, where $\star$ is the convolution. What I've tried: I was "tickling" ...
Christoph Mark's user avatar
0 votes
1 answer
30 views

Derivative of Convolution with Respect to Input Image

Given a discrete convolution operation for image processing like this: $$ H(x, y) = \sum_{i=-a}^{a} \sum_{j=-b}^{b} K(i, j) \cdot I(x - i, y - j) $$ It is common for CNNs to take the derivative with ...
James Li's user avatar
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0 answers
17 views

Prove that the derivative of the mollification approaches the strong $L^p$ derivative

Let $g \in C_c(\mathbb{R})$ be such that $\int_\mathbb{R} g dx = 1$. Suppose that $f \in L^p(\mathbb{R}$ has a strong $L^p$ derivative, i.e. there is some $h$ such that $\displaystyle\lim_{y \to 0} \...
Squirrel-Power's user avatar
5 votes
2 answers
247 views

Calculate $\sum_{n=1}^\infty (1-\alpha) \alpha^nP^{*n}(x)$

$$ \mbox{Let}\quad P'(x)=\sum_{j=1}^n a_j\frac{a^jx^{j-1}}{(j-1)!}e^{-ax},x\geq 0 $$ be the density function of a mixture of Erlangs and let $\alpha\in(0,1)$: Is is possible to determine an analytic ...
Leon's user avatar
  • 91
2 votes
2 answers
88 views

Does convolution preserve local integrability in the sense that if $f \in L^p_{\text{loc}}$ and $g \in L^1$, then $f \ast g \in L^p_{\text{loc}}$?

Consider the Euclidian space $\mathbb R^n$, where $n \in \mathbb N$ is a fixed integer, equipped with the Lebesgue measure. Let $L^p_{\text{loc}}(\mathbb R^n)$($1 \leqslant p < \infty$) denote the ...
xyz's user avatar
  • 1,141
2 votes
1 answer
50 views

Density of similiar Sobolev space

Consider the space of functions defined as, $$ D = \{f \in L^p(0,\infty): f \in AC_{loc}(0,\infty) \text{ and } xf'(x) \in L^p(0,\infty)\}, $$ where $AC_{loc}(0,\infty)$ is the set of locally ...
Scottish Questions's user avatar
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33 views

Find a sequence polynomials that converges to $e^{x-y}$ on $S^1$ but diverge anywhere else

I'm trying to do Exercise 3.3.1 in scv.pdf Let $z=x+i y$ as usual in $\mathbb{C}$. Find a sequence of polynomials in $x$ and $y$ that converge uniformly to $e^{x-y}$ on $S^1$, but diverge everywhere ...
hbghlyj's user avatar
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34 views

Fourier transform with and without convolution theorem not equivalent

This is a problem involving Fourier transforming an integral relevant to the computation of Feynman diagrams, which is of the form: $S(r_1,r_2)=\int d^3 r_3 \space v(r_1,r_3)f(r_3,r_2),$ where $v(r_1,...
user2188518's user avatar
0 votes
0 answers
35 views

Preservation of strict log-concavity under convolution

I have spent an embarrassing amount of time trying to prove or disprove any of this. I am aware that a similar question was posted in 2014, but since I couldn't make anything out of the two sources ...
Jacob's user avatar
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