Questions tagged [convolution]
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173 questions
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Special function: Pulse peak modified with a power term
PeakFit (Systat, v. 4.12) is a software for fitting experimental peaks obtained in physics or chemical experiments. Under the miscellenous peak functions, it shows the following equations with a name, ...
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Integral operator (compactness)
I am studying the compactness of some convolution operators. Let the convolution
$$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T(t-s)B(s)x\mathrm{d}s. $$
Here $T(\cdot)$ is a $C_0$-semigroup on some ...
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Apply gaussian blur to get original image [closed]
Suppose I have an image A. Is it possible to construct an image A' from A so I can get the ...
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Subtle distinction in "completeness"?
This is somewhat vague, but please bear with me. Complete metric spaces are supposed to take care of "gaps", they're understood as a natural extension of dense sets.
The convolution, defined ...
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When can a convolution be written as a change of variables?
Suppose $X$ is a random variable with a density $f(x)$ such that $f(x)$ is a convolution of some density $g$ with some other density $q$:
$$
f = g\ast q.
$$
Under what conditions does $X=h(Y)$, where $...
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Why does this convolution of the prime counting function $\pi$ look like a parabola?
In this previous question it is shown that the convolution of the prime counting function $\pi$ with itself, is related to the Goldbach conjecture:
$$\pi^*(n):=\sum_{k=0}^n \pi(k) \pi(n-k)$$
The ...
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Log-concavity of lattice-functions and convolution
I was looking at the definition of log-concavity:
A function $F:\mathbb{R}^n\rightarrow\mathbb{R}$ is said log-concave
iff $F(x)\geq 0\forall x\in\mathbb{R}^n$ and
$$F(x)^\lambda F(y)^{1-\lambda}\leq ...
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Deconvolution using the discrete Fourier transform
Summary: From discrete convolution theorem, it is understandable that we need 2N-1 point DFT of both sequences in order to avoid circular convolution. If we need to do deconvolution of a given ...
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What is the difference (if any) between "fourier transform" and "SO(3) fourier transform"?
What is the difference (if any) between "fourier transform" and "SO(3) fourier transform"?
I searched on Google but couldn't find a satisfiable answer.
Thanks in advance :)
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Convolution definition in an old educational article
I was reading an old article in IEEE Education magazine by Robbins and Fawcett titled "A Classroom Demonstration of Correlation, Convolution and the Superposition Integral" DOI: 10.1109/TE....
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2-Wasserstein metric on convolution of probability distributions
I have two related questions. Let $\mu$ and $\nu$ be two distinct probability measures on $\mathbb{R}^n$ with finite second moments, and $W_2(\cdot,\cdot)$ be the $2$-Wasserstein metric. The question ...
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Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?
Consider the following formula which defines a piece-wise function which I believe corresponds to a series representation for the Dirac delta function $\delta(x)$. The parameter $f$ is the evaluation ...
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Equivalent action of convolution of directional derivative
I have asked this question a while back on StackExchange but have not received any answer/comment. I received a suggestion to post the same question in here which is more research oriented.
Let $k*f(x)...
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How to solve the following $0= \int_{-\infty}^\infty e^{-\frac{(bt+\omega)^2}{2}} f(t+\omega) \frac{1}{i t} dt, \forall \omega \in \mathbb{R}$
Suppose that for a given $b\in \mathbb{R}$
\begin{align}
0= \int_{-\infty}^\infty e^{-\frac{(bt+\omega)^2}{2}} f(t+\omega) \frac{1}{i t} dt, \forall \omega \in \mathbb{R}
\end{align}
where $i =\sqrt{...
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Equivalent of a local limit theorem in the large deviation region and asymptotics of a convolution operator
Let $\{X_i \}_{i \in \mathbb{N}}$ be a sequence of i.i.d. random variables satisfying $\mathbb{E} X_1 = 0$ and $\mathbb{E} X_1 ^2 < \infty$. Assume that $\{S_n \}_{n \in \mathbb{N}}$ is a non-...
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Recovering a function from its Gaussian convolution
Let $\varphi(x)=\frac{1}{\sqrt{2\pi}}\exp(-x^2/2)$ be the Gaussian density and
$f:\mathbb{R}\to\mathbb{R}$ another measurable function.
Under what conditions can $f$ be recovered from its convolution ...
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Vector convolution?
I am working on a research problem which leads to the following optimization problem:
\begin{equation}
\hat{M} = \operatorname*{arg\,max}_M \Bigl\lVert\sum_{k=0}^{M-1} {\mathbf y}_k \exp\left(-j 2\pi ...
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The square-integrability of $p$ and $\nabla u$
We consider the stationary Stokes problem in $\mathbb{R}^n$
$$\DeclareMathOperator{\Dvg}{\nabla\cdot}
\begin{cases} \Delta u + \nabla p = f & \text{ in $\mathbb{R}^n$} \\
\Dvg u =0.
\end{cases}
$...
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Existence of solutions to first-order PDE involving convolution
Let $f(x,\alpha)$ be a smooth function of compact support in $x$. Now, let its $\alpha$-dependence be determined by the following first-order equation,
\begin{align}
\frac{\partial}{\partial \alpha} ...
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Does convolution by a Schwartz function preserve symbol classes?
I am working on a problem involving pseudodifferential operators, and I need a property of the operator "convolution by a Schwartz function". I apologize in advance if the question is ...
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Problem regarding vanishing set of convolution
Let $f$ vanishes on an open set containing 0. So there exists $l>0$ such that $f$ vanishes on $B(0,2l).$ So we can choose $g\in C_c^\infty (\mathbb{R}^n)$ (supported on $B(0,l)$) such that $f*g$ ...
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Optimization with convolution in the objective function
I would like to minimize the following objective function
$$
\| H \ast A - (H \cdot I) \ast B \|_F^2
$$
w.r.t. $H$, where
$H$, $I$, $A$, and $B$ are all square matrices of the same size ($I$ is a ...
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Why do convoluted convolved Fibonacci numbers pop up from this triangle?
Start with this triangle (OEIS A118981). This triangle is simple to generate with the following recurrence relation (though $T(0,0)$ ends up different from the OEIS version):
$$
T(0,0) = 2;T(1,0) = 1;...
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The derivative of a filter with respect to a output signal [closed]
I have two signals, $d(t)$ and $p(t)$, respectively the input and the output of the matching filter $w(t)$, i.e.
$$
d(t)*w(t)=p(t)
$$
where $*$ denotes convolution.The impulse response $w(t)$ may be ...
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Is $g(v)=\mathbb{E}[f(v+W)]$ a differentiable function of $v$ when $f$ is continuous and $W$ is multivariate normal?
Suppose $f$ is a continuous function on $\mathbb{R}^n$, and $W$ has a multivariate normal distribution on $\mathbb{R}^n$. If the expectation
$$g(v)=\mathbb{E}[f(v+W)]$$
is defined for all $v \in \...
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What is convolution intuitively?
If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
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Anticommutation of convolution products on trace class operators of quantum groups
This question was originally posted to MathStackExchange.
Let $\mathbb{G}$ be a locally compact quantum group and let $W$ and $V$ be the left and right fundamental unitaries, i.e., they implement the ...
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Fast computation of convolution integral of a gaussian function
Given a convolution integral
$$
g(y) =\int_a^b\varphi(y-x)f(x)dx=\int_{-\infty}^{+\infty}\varphi(y-x)f(x)\mathbb{I}_{[a,b]}(x)dx
$$
where
$\varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\...
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When is $\lVert f*g\rVert_\infty=\lVert f\rVert_1\lVert g\rVert_\infty$?
If $1\leq p<\infty$, it is easy to find nice necessary and sufficient equality conditions for the convolution inequality $$\lVert f*g\rVert_p\leq\lVert f\rVert_1\lVert g\rVert_p\qquad (f\in L^1(\...
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Can we show that the characteristic function of an infinitely divisible probability measure has no zeros
Let $E$ be a normed $\mathbb R$-vector space, $\mu$ be a probability measure on $\mathcal B(E)$ and $\varphi_\mu$ denote the characteristic function$^1$ of $\mu$.
Assume $\mu$ is infinitely divisible, ...
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Is there a name for this type of matrix?
For my thesis in neural networks, I was trying to find a way to generalize a Sobel operator. I quickly thought of this:
$$
\begin{bmatrix}
a&b&c\\
d&0&-d\\
-c&-b&-a
\end{...
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Convolution of an Airy function with a Gaussian
I wonder if the convolution
\begin{equation}
f(y)=\int_{-\infty}^{+\infty} \mathrm{Airy}(a\cdot x)\cdot e^{-b(y-x)^2} dx
\end{equation}
can be solved analytically. Or in case not, if there is an ...
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Vanishing convolution between density and compactly supported function
Find a pair of functions $f,g:\mathbb{R}\to\mathbb{R}$ such that:
$f$ is smooth and compactly supported (say, on $[0,1]$ but this isn't crucial),
$g(x)>0$ for all $x\in\mathbb{R}$, $\int g(x)\,dx=...
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Norm of convolution operator
By Young's inequality for any $f\in L^p(\mathbf{R})$ the map $T_f:g\mapsto f\star g$ is a continuous operator from $L^q(\mathbf{R})$ to $L^r(\mathbf{R})$ where $1\leq p,q,r\leq \infty$ satisfy $1+\...
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Young's convolution inequality for weighted norms
Young's convolution inequality states that, for $1/p+1/q=1/r+1$ ($1\leq p,\, q, r\leq \infty$), we have $$\lVert f * g \rVert_r \leq \lVert f\rVert_p \lVert g\rVert_q.$$
It is implicit here that the ...
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Algebraic relation amongst an elliptic function and its convolution
NOTE: I edited this question, following the comments of Alexander Eremenko and Paul Garrett.
I have a question concerning elliptic functions that maybe you can help me shed light on. I am a ...
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What is the computational complexity of the calculation of $ \Psi(x) $?
What is the computational complexity of the calculation of $ \Psi(x) $ described below:
Let $\left\{ f_i : \{0,1,\dots,m\} \to \mathbb{R} \right\}_{i=1}^n$. For each $x \in \{0,1,\dots,m\}$ we ...
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Hopf "algebroid" structure of a groupoid convolution algebra?
This question is already posted in math.stackexchange, but didn't receive any answer. I'm not sure if this question fits in here, but surely someone in here can guide me to the correct answer.
To make ...
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Can it be represented by convolution and multiplication
I have functions $A, B, F, S$ that are zero on $(-\infty, 0)$.
And I have successfully represented the below equation as convolution and multiplication:
$\int_0^t {dt_1} \int_0^t {dt_2} B(t - t_2)F(...
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Relation between Cox-deBoor recursion and Convolution (b-spline basis)
Consider the Cox-deBoor recursion formula for producing b-spline basis functions given a knot vector:
$N_{i,0}(u)=1 $ if $u_i\leq u < u_{i+1}$
otherwise, $=0$
$N_{i,p}(u)=\frac{u-u_{i}}{u_{...
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Uniqueness of deconvolution after convolution?
I have the following question and I'd greatly appreciate any help!
Basically, I have an arbitrary probability distribution with pdf $f(x)$, we can assume it's continuous with support on $[0,\infty]$
...
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How can we show this estimate for the convolution of two probability measures?
Let $(\delta_k)_{k\in\mathbb N}\subseteq(0,\infty)$ be nonincreasing with $\delta_k\xrightarrow{k\to\infty}0$ and $(\varepsilon_k)_{k\in\mathbb N}\subseteq(0,\infty)$ with $\sum_{k\in\mathbb N}\...
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If $(\exp(\mu_n))_{n\in\mathbb N}$ is weakly convergent, is the normalized sequence convergent as well?
Let $E$ be a metric space and $\mathcal M(E)$ denoote the space of finite signed measures on $\mathcal B(E)$ equipped with the total variation norm $\left\|\;\cdot\;\right\|$.
I would like to know ...
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Convolution of ball measures
It is well known that convolution of two ball measures (i.e. a uniform measure over a ball) in $\mathbb{R}^{n}$ is absolutely continuous with respect to the Lebesgue measure.
My question is - how to ...
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Which random variables can be written as the difference of two independent positive random variables?
Can we characterize random variables $X$ that satisfy
$$
X\sim Y - Z
$$
for two independent positive random variables $Y$ and $Z$?
Are $Y$ and $Z$ unique in some sense?
Can (one possible choice of) $Y$...
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Convolution, Fourier transforms, and area preservation [closed]
Consider the convolution of two functions, f * g. And let us assume, for practicality, some example case where an integral of f or g can be interpreted as the "area under the curve" (or the ...
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Existence of unique convolution semigroups of probability measures on more general spaces then $\mathbb R^d$
Let $E$ be a $\mathbb R$-Banach space, $\mathcal M_1(E)$ (resp. $\mathcal M_1^\infty(E)$) denote the set of probability measures (resp. infinitely divisible probability measures) on $E$, $\varphi_\mu$ ...
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Convolution of functionals on compact quantum group
Let $\mathbb{G}= (A, \Delta)$ be a ($C^*$-algebraic) compact quantum group. In a paper I'm reading, the space $A^*= B(A, \mathbb{C})$ obtains a product
$$\omega_1*\omega_2:= (\omega_1\otimes \omega_2) ...
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Characterization of convolution operators via the Fourier transform
Let $\mathcal{L}$ be a linear and continuous operator from the space of tempered distributions $\mathcal{S}'(\mathbb{R})$ to itself. The Fourier transform of a tempered distribution $f$ is denoted by $...
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Inverting convolutions over finite intervals
There are well-known techniques for inverting convolutions over the whole or half real line with Fourier and Laplace transformations, but on the face of it they can't be applied to an integral ...
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