Questions tagged [convolution]
The convolution tag has no usage guidance.
64 questions with no upvoted or accepted answers
11
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0
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161
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Bijections $\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ with vanishing local means
This is just a summer-time curiosity arisen after a recent question by Dominic van der Zypen.
For a finite subset $S$ of $\mathbb{Z}\times\mathbb{Z}$ and a function $f$ on $\mathbb{Z}\times\mathbb{...
- 60.6k
8
votes
0
answers
294
views
Which classes of functions are "convolution ideals"?
If $g$ is continuous then $f*g$ is continuous.
If $g$ is smooth then $f*g$ is smooth.
If $g$ is a polynomial then $f*g$ is a polynomial.
If just one of the two functions belongs to the class of well-...
6
votes
0
answers
305
views
Distribution class closed under convolution counterexample?
Define the class of probability density functions $\mathcal{C}$: $\,p \in \mathcal{C}$ iff $p(x)=p(|x|)$, and $\log p(\!\sqrt{x})$ is convex on $[0,\infty)$.
Conjecture: if $p,q \in \mathcal{C}$, then ...
- 391
5
votes
0
answers
167
views
Computing sums with linear conditions quickly
Let $f:\{1,\dotsc,N\}\to \mathbb{C}$, $\beta:\{1,\dotsc,N\}\to [0,1]$ be given by tables (or, what is basically the same, assume their values can be computed in constant time). For $0\leq \gamma_0\leq ...
- 20.2k
5
votes
0
answers
276
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
$$F(x)^\lambda F(y)^{1-\lambda}\leq ...
- 93
4
votes
0
answers
105
views
Convolution algebra of a simplicial set
Consider a simplicial set $X^\bullet$ with face maps $d_i$ (assume the set is finite in each degree so there are no measure issues). Then given two functions $f,g:X^1\to \mathbb{C}$ one can form their ...
- 1,198
4
votes
0
answers
181
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Shifted convolution problem for Coefficients of automorphic forms
The shifted convolution problem for coefficients of modular forms is well studied and many estimates were established for the shifted convolution sums of Hecke eigenvalues. So, one may ask about the ...
- 1,257
4
votes
1
answer
2k
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Norm of convolution operator
By Young's inequality for any $f\in L^p(\mathbf{R})$ the map $T_f:g\mapsto f\star g$ is a continuous operator from $L^q(\mathbf{R})$ to $L^r(\mathbf{R})$ where $1\leq p,q,r\leq \infty$ satisfy $1+\...
- 3,425
3
votes
0
answers
95
views
Is it true that p-integrable function can be written as a convolution of an integrable function and p-integrable function?
We know that convolution of an integrable function with an $p$-integrable is an $p$-integrable function. This follows from Young's inequality.
My question: Is it true that $L^p(\mathbb{R}^n)\subseteq ...
- 31
3
votes
0
answers
154
views
Inequality involving convolution roots
I am struggling with the following problem. Let $f$ be a real smooth function. Let assume that $f$ is:
increasing
strictly convex on $(-\infty,0)$
strictly concave on $(0,+\infty)$
Let $\sigma>0$ ...
- 393
3
votes
0
answers
143
views
Extrapolated Integral operator (compactness)
I am studying the compactness of some convolution operators. Let the convolution with extrapolation
$$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$
Here $T(\cdot)$ is a ...
- 299
3
votes
0
answers
320
views
Does convolution by a Schwartz function preserve symbol classes?
I am working on a problem involving pseudodifferential operators, and I need a property of the operator "convolution by a Schwartz function". I apologize in advance if the question is ...
- 395
3
votes
0
answers
238
views
Is there a closed form for the discrete convolution of $\sigma_1$ and $\sigma_2$?
I am trying to find a closed form for the following sum:
$$\sum_{k = 1}^{n-1}\sigma_1(k) \sigma_2(n-k)$$
where $\sigma_i$ is the sum of divisors and $\sigma_2$ is the sum of squares of divisors.
...
- 211
3
votes
0
answers
119
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Any chance to get the moments of this exotic distribution?
Let us define the following cumulative distribution:
\begin{align}
\Pr (Y(t)\geq a)=\sum_{n=0}^\infty \frac{e^{-kt}(kt)^n }{n!}\int_a^\infty[\circledast_{f}\circ H_a ]^n\delta (x) dx
\end{align}
where ...
- 634
3
votes
0
answers
81
views
Computing distribution of non-identical coin flips
Suppose I have $N$ coins, where coin $i$ has probability $p_i$ of coming up heads. I flip all $N$ coins and let $S_N$ be the number of heads. How can I compute the distribution of $S_N$ efficiently?
...
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3
votes
0
answers
75
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Is there a name for the general type of operation that sweeps a kernel over a function (e.g. like convolution, morph. dilation, registration, etc)
There is a certain family of 'sweeping' operators / functions $S(y; f,k,g)$, where:
$f$ is a function $f : x \mapsto \mathbb{R}^N$
$k$ is a 'kernel' function $k : x \mapsto \mathbb{R}^N$
$y$ ...
3
votes
0
answers
267
views
Link between standard convolution and Day convolution
There is a notion of convolution product between two functors called "Day convolution". (See here nlab for instance) I know that the definition of this notion is inspired by the discrete convolution $$...
- 1,017
3
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0
answers
543
views
Existence and smoothness of convolutions of distributions in Sobolev spaces
Let $f\in H^{s_1}(\mathbb{R}^n)$ and $g\in H^{s_2}(\mathbb{R}^n)$, where $s_1, s_2 \in \mathbb{R}$ and can be positive or negative.
It is easy to show that $f *g$ is defined pointwise when $s_1+s_2\...
- 256
3
votes
0
answers
741
views
Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)
Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as
$$...
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3
votes
1
answer
77
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Level sums, displacements: how to determine them efficiently?
Let $R =\mathbb{Z}/N \mathbb{Z}$. Let $f:R\to \mathbb{R}$,
$\rho:R\to \lbrack 0,1\rbrack$. We assume that it takes trivial time to compute any given value $f(m)$ or $\rho(m)$.
Define $$S(\delta,m) = ...
- 20.2k
2
votes
0
answers
191
views
Smoothing property of the heat kernel on the one-dimensional torus
Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$ $G$ is given by \begin{equation}
G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \...
- 185
2
votes
0
answers
73
views
On a possible generalization of heat kernel semigroups on Lie groups
Let $G$ be a compact matrix Lie group with Haar measure $\mu$. Then the heat kernel $\rho: G\times (0,\infty) \rightarrow \mathbb{R}$ satisfies
(1) $\rho(g_1g_2,t)=\rho(g_2g_1,t)$ for all positive $t$,...
- 577
2
votes
0
answers
116
views
A technical question concerning convolution product
Let $v\in L^p(\Bbb R^d)$, $1\leq p<\infty$ be nonzero function, i.e., $v\not\equiv 0$.
Define $$u(x)= |v|*\phi(x)= \int_{\Bbb R^d} |v(y)|\phi(x-y)d y$$ with $\phi(x)= ce^{-|x|^2}$ and $c>0$ so ...
- 1,101
2
votes
0
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139
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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.
...
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2
votes
0
answers
86
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}
$...
2
votes
0
answers
112
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 ...
- 59
2
votes
0
answers
74
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\...
- 321
2
votes
0
answers
80
views
Are the roots of an infinitely divisible probability infinitely divisible themselves?
Let $\mu$ be an infinitely divisible probability on a topological group $G$. If $\nu ^{* n} = \mu$ for some $n$, is $\nu$ an infinitely divisible probability too?
A sufficient criterion would be to ...
- 5,407
2
votes
0
answers
66
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Classifying Algebras of Convolution
$L^1(\mathbb R^n)$, $L^1(\mathbb R_+)$, $C^0_c(\mathbb R_+)$, $C^\infty_c(\mathbb R_+)$ are algebras of convolution.
Question 1: is there a classification of subalgebras of convolution of $L^1(\...
- 16.2k
2
votes
0
answers
200
views
Some detail in Fefferman's thesis
Recently, I am reading "Inequalities for strongly singular convolution operator" written by Fefferman. I have some question on the detail of proof of Theorem 2'.
Let $\theta \in (0,1)$.
Let $f \in ...
- 121
2
votes
0
answers
2k
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convolution integral involving modified Bessel functions of the first kind
I'm stuck with this convolution integral ($z \geq 0$)...
\begin{equation}
f_{Z}(z)=\int^{\infty}_{-\infty}f_{1}(x)f_{2}(z-x)dx = \mbox{ } ???
\end{equation}
which represents the pdf of the sum $Z = ...
- 163
1
vote
0
answers
68
views
Breakdown of the fourier series identity $e_n e_m = e_{n+m}$ on a perturbed torus
I would like to apologize for leaving things a bit vague. I think my question could be stated much more precisely but right now that is difficult for me to do. I nevertheless think it is an ...
- 243
1
vote
0
answers
86
views
Fourier transform relation for spherical convolution
Let $f$ and $g$ be two functions defined over the 2d sphere $\mathbb{S}^2$.
The convolution between $f$ and $g$ is defined as a function $f * g$ over the space $SO(3)$ of 3d rotations as
$$(f*g)(R) = \...
- 2,306
1
vote
0
answers
69
views
Is there an generalisation of convolution theorem to integral transforms
Basic convolutions can be computed efficiently by taking fourier transforms and applying the convolution theorem. Is there something analogous for a more general transform, where we have a varying ...
1
vote
0
answers
27
views
Spectrum of the convolution of the Maxwell collision kernel with a distribution
Given the Maxwell collision kernel $A(z) = |z|^2I_d - z \otimes z$, where $I$ denotes the $d\times d$ identity matrix and $z\otimes z = zz^T$ is the outer product, it is easy to see that $A(z)$ has ...
- 43
1
vote
0
answers
72
views
Solve linear matrix equation involving convolution
I am facing following equation:
$$
A * X + C \cdot X = D
$$
with:
$A, C, D \in \mathbb{R}^{n \times n}$ some known matrices without any particular structure,
$X \in \mathbb{R}^{n \times n}$ the ...
- 11
1
vote
0
answers
212
views
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, ...
- 879
1
vote
0
answers
118
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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....
- 879
1
vote
0
answers
279
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 ...
- 273
1
vote
0
answers
58
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$\mathbb{P}_1$-finite element as convolution of $\mathbb{P}_0$-finite element
For a vector $\vec{u}\in\mathbb{R}^N$ let's denote $\pi_N\left(\vec{u}\right)$ the unique piecwise linear and $1$-periodic function matching the components of $\vec{u}$ on the discretization $x_k = \...
- 3,425
1
vote
0
answers
99
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 ...
- 101
1
vote
0
answers
127
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 ...
- 105
1
vote
0
answers
76
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 $X$...
- 105
1
vote
0
answers
78
views
Convolve a 4D Gaussian function along a plane?
There is a 4D Gaussian function $G(u,s)=G(x|c,\mu,\Sigma )$ where $x=\begin{bmatrix}u\\ s\end{bmatrix}$,$u$ and $s$ is all 2D vector.
Now I want to blur (convolve) it along with $u$ by another 2D ...
- 11
1
vote
0
answers
47
views
Show a convolution of distributions ε-close to min-entropy k is ε^t-close to min-entropy k
Assume $X_1,...,X_t$ are independent distributions on $\mathbb{Z}_2^n$ s.t. each $X_i$ is $\epsilon$-close to min-entropy $k$; i.e. there exist distributions $Y_1,...,Y_t$ on $\mathbb{Z}_2^n$ s.t:
$$ \...
- 21
1
vote
0
answers
151
views
Convolution of $\ell$-adic sheaves and group homomorphisms
This question follows this one , where I defined convolution of $\ell$-adic/perverse sheaves.
Here I am working with a perfect field $k$ ($char(k)\neq l$) and with a smooth separated groupscheme $G$ ...
- 329
1
vote
0
answers
134
views
Convolution integral of series involving the non-trivial zeros of $\zeta(s)$
Let us consider the convolution $$f\left(x\right):=\int_{2}^{x-2}\sum_{\rho_{1}}\frac{u^{\rho_{1}}}{\rho_{1}}\sum_{\rho_{2}}\frac{\left(x-u\right)^{\rho_{2}}}{\rho_{2}}du,\,x>4$$ where $\rho_{i},\,...
- 219
1
vote
0
answers
116
views
Existence theorems Volterra Equation of second kind on unbounded domains
The general Volterra Equation of the second kind in convolution form can be described by:
$$
\phi(x) = \int_a^x K(x-t)\phi(t)\, \mathrm{d}t + f(x), \text{ for } x\geq a
$$
Suppose we wish to ...
- 31
1
vote
0
answers
40
views
Envelope of a parametrized family of convolutions
For a certain application I need to compute a pointwise supremum of this family of gaussian convolutions:
$$\sup_s f(x)\otimes e^{-\frac{x^2}{s^2}}$$
where $f(x),x\in \mathbb{R}^2$ is known and $\...
- 2,205
1
vote
0
answers
154
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Convolution in Hardy spaces
Question Are there non-trivial restrictions on the coefficients of functions in Hardy spaces ($H_p(\mathbb{D})$, $p<1$) that make a subspace that is closed under convolution?
Definition The Hardy ...
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