Questions tagged [convolution]

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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 ...
1 vote
1 answer
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+50

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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1 vote
0 answers
43 views

Apply gaussian blur to get original image

Suppose I have an image A. Is it possible to construct an image A' from A so I can get the ...
2 votes
0 answers
70 views

Local Rankin-Selberg Zeta-function and Coates' p-adic L-Functions

$\DeclareMathOperator\Kern{Kern}\DeclareMathOperator\diag{diag}$ Let $F$ be a non-archimedean local field, $\mathcal{O}$ its ring of integers, $\mathfrak{p}$ its maximal ideal and $\pi$ a uniformizer. ...
0 votes
1 answer
222 views

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 ...
0 votes
1 answer
134 views

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 $...
5 votes
1 answer
482 views

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 ...
7 votes
2 answers
984 views

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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1 vote
0 answers
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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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0 votes
0 answers
28 views

Regularity of Volterra convolution integral

I have a question about the regularity of the convolution integral. Let $f: [0,\infty) \to \mathcal{L}(\mathbb{R}^n)$ given by $f(t) = e^{tA}$. Let $g: (0,T) \to \mathbb{R}^n$ such that $g \in L^2(0,T;...
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4 votes
1 answer
125 views

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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3 votes
2 answers
383 views

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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3 votes
1 answer
568 views

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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2 votes
1 answer
79 views

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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4 votes
1 answer
189 views

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 ...
0 votes
0 answers
155 views

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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2 votes
0 answers
73 views

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} $...
3 votes
0 answers
134 views

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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2 votes
1 answer
92 views

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 $f*g$ ...
1 vote
0 answers
39 views

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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4 votes
1 answer
186 views

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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2 votes
1 answer
222 views

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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5 votes
0 answers
160 views

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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2 votes
0 answers
105 views

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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3 votes
1 answer
176 views

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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6 votes
2 answers
486 views

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(\...
3 votes
1 answer
403 views

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, ...
4 votes
1 answer
293 views

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{...
1 vote
1 answer
254 views

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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3 votes
2 answers
293 views

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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1 vote
1 answer
377 views

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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1 vote
0 answers
102 views

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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4 votes
1 answer
143 views

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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0 votes
1 answer
111 views

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 ...
3 votes
1 answer
258 views

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(...
1 vote
1 answer
158 views

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 $[0,\infty]$ ...
2 votes
1 answer
1k views

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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0 votes
1 answer
101 views

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}\...
2 votes
2 answers
239 views

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 ...
6 votes
2 answers
525 views

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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0 votes
1 answer
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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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1 vote
1 answer
114 views

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 $E$, $\varphi_\mu$ ...
3 votes
1 answer
159 views

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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3 votes
2 answers
228 views

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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0 votes
0 answers
91 views

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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4 votes
1 answer
135 views

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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0 votes
0 answers
46 views

What is the term for convoluting but scaling the time domain instead of shifting?

Given that the convolution definition as far as I am aware is: $(f*g)(t) = \int_{-\infty}^\infty f(\tau)g(t-\tau)d\tau$ Here I see that the functions f and ...
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1 vote
1 answer
676 views

Convolution of two Gaussian mixture model

Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is, $$ f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right) $$ $$ g(y)=\...
2 votes
0 answers
72 views

Particular Ehrenpreis factorization for covariance function

Let $f:\mathbb{R}^d\to\mathbb{R}$ be a smooth compactly supported covariance function of a stationary random fields (hence positive definite). Is there a compactly supported function $g:\mathbb{R}^d\...
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1 vote
0 answers
57 views

Convolution Integral Equation on a compact subset of the real line

I am dealing with the following equation: $$ f(x) = g(x) + \intop_{X} dt K(x-t)f(t) \;,\qquad \left\lbrace \begin{array}{c}f(x)>0\;,\;x\in X \\ f(x)<0\;,\;x\notin X \end{array}\right.$$ where $X$...
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