Questions tagged [convolution]
The convolution tag has no usage guidance.
157
questions
2
votes
1
answer
316
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 \...
5
votes
0
answers
242
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
$$F(x)^\lambda F(y)^{1-\lambda}\leq ...
2
votes
0
answers
109
views
Anticommutation of convolution products on trace class operators of quantum groups
This question was originally posted to MathStackExchange.
Let $\mathbb{G}$ be a locally compact quantum group and let $W$ and $V$ be the left and right fundamental unitaries, i.e., they implement the ...
3
votes
1
answer
318
views
Fast computation of convolution integral of a gaussian function
Given a convolution integral
$$
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
$$
where
$\varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\...
6
votes
2
answers
494
views
When is $\lVert f*g\rVert_\infty=\lVert f\rVert_1\lVert g\rVert_\infty$?
If $1\leq p<\infty$, it is easy to find nice necessary and sufficient equality conditions for the convolution inequality $$\lVert f*g\rVert_p\leq\lVert f\rVert_1\lVert g\rVert_p\qquad (f\in L^1(\...
3
votes
1
answer
567
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, ...
5
votes
1
answer
315
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:
$$
\begin{bmatrix}
a&b&c\\
d&0&-d\\
-c&-b&-a
\end{...
1
vote
1
answer
450
views
Convolution of an Airy function with a Gaussian
I wonder if the convolution
\begin{equation}
f(y)=\int_{-\infty}^{+\infty} \mathrm{Airy}(a\cdot x)\cdot e^{-b(y-x)^2} dx
\end{equation}
can be solved analytically. Or in case not, if there is an ...
3
votes
2
answers
407
views
Vanishing convolution between density and compactly supported function
Find a pair of functions $f,g:\mathbb{R}\to\mathbb{R}$ such that:
$f$ is smooth and compactly supported (say, on $[0,1]$ but this isn't crucial),
$g(x)>0$ for all $x\in\mathbb{R}$, $\int g(x)\,dx=...
1
vote
1
answer
896
views
Young's convolution inequality for weighted norms
Young's convolution inequality states that, for $1/p+1/q=1/r+1$ ($1\leq p,\, q, r\leq \infty$), we have $$\lVert f * g \rVert_r \leq \lVert f\rVert_p \lVert g\rVert_q.$$
It is implicit here that the ...
1
vote
0
answers
113
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 ...
4
votes
1
answer
239
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 ...
0
votes
1
answer
124
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
302
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
272
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]$
...
3
votes
1
answer
2k
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 ...
0
votes
1
answer
133
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
305
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
741
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
$$
for two independent positive random variables $Y$ and $Z$?
Are $Y$ and $Z$ unique in some sense?
Can (one possible choice of) $Y$...
1
vote
1
answer
438
views
Convolution, Fourier transforms, and area preservation [closed]
Consider the convolution of two functions, f * g. And let us assume, for practicality, some example case where an integral of f or g can be interpreted as the "area under the curve" (or the ...
1
vote
1
answer
151
views
Existence of unique convolution semigroups of probability measures on more general spaces then $\mathbb R^d$
Let $E$ be a $\mathbb R$-Banach space, $\mathcal M_1(E)$ (resp. $\mathcal M_1^\infty(E)$) denote the set of probability measures (resp. infinitely divisible probability measures) on $E$, $\varphi_\mu$ ...
3
votes
1
answer
233
views
Convolution of ball measures
It is well known that convolution of two ball measures (i.e. a uniform measure over a ball) in $\mathbb{R}^{n}$ is absolutely continuous with respect to the Lebesgue measure.
My question is - how to ...
3
votes
2
answers
248
views
Convolution of functionals on compact quantum group
Let $\mathbb{G}= (A, \Delta)$ be a ($C^*$-algebraic) compact quantum group. In a paper I'm reading, the space $A^*= B(A, \mathbb{C})$ obtains a product
$$\omega_1*\omega_2:= (\omega_1\otimes \omega_2) ...
0
votes
0
answers
116
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 $...
4
votes
1
answer
273
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 ...
0
votes
0
answers
56
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 ...
1
vote
1
answer
1k
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,
$$
f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right)
$$
$$
g(y)=\...
2
votes
0
answers
73
views
Particular Ehrenpreis factorization for covariance function
Let $f:\mathbb{R}^d\to\mathbb{R}$ be a smooth compactly supported covariance function of a stationary random fields (hence positive definite).
Is there a compactly supported function $g:\mathbb{R}^d\...
1
vote
0
answers
70
views
Convolution Integral Equation on a compact subset of the real line
I am dealing with the following equation: $$ 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.$$ where $X$...
1
vote
1
answer
68
views
Limit of an integral / Boundary behaviour of a Gaussian convolution / single layer potential
Let $k(t,x)$ be the transition density of Brownian motion $$ k(t,x) := \frac{1}{\sqrt{2 \pi t}} \exp \left\{ \frac{-x^2}{2t} \right\} , \quad t \geq 0, x \in {\mathbb R.}$$
Question
Let $0 < x &...
5
votes
3
answers
1k
views
Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?
Consider the following formula which defines a piece-wise function which I believe corresponds to a series representation for the Dirac delta function $\delta(x)$. The parameter $f$ is the evaluation ...
0
votes
1
answer
191
views
$\frac{\partial}{\partial x}\int_{\mathbb{R}}\frac{1}{\sqrt{2 \pi \varepsilon}}e^{-\frac{(x-y)^2}{2\varepsilon}}l(y)dy\leq C\frac{1}{x}$
Let $l$ be a continuous bounded function ($l$ is not differentiable). I want to prove for $x$ large enough that
$$\frac{\partial}{\partial x}\int_{\mathbb{R}}\frac{1}{\sqrt{2 \pi \varepsilon}}e^{-\...
3
votes
1
answer
1k
views
If the convolution of two functions $f\star g$ is equal to $g$, $f$ is even with compact support and $g$ is bounded, implies that $g$ is constant?
Let $f$ be an even continuous function with compact support such that
$$
\int f(t)\,\mathrm{d}t=1,
$$
and let $g$ be a bounded continuous function such that the convolution $f\star g$ satisfies the ...
3
votes
1
answer
241
views
Prove or disprove the linearity of expectiles
For expectation (mean), there are many useful properties such as Linearity of Expectation:
$\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]$
$\mathbb{E}[\alpha X]=\alpha\mathbb{E}[X]$
(The 2 equations ...
4
votes
2
answers
675
views
Earliest use of deconvolution by Fourier transforms
From a previous discussion here Origin of the convolution theorem, it was shown that the property of convolution $y(t)$=$a$*$b$ becoming a multiplication after Fourier transform: $F$$(y(t))$= $F(a)F(b)...
4
votes
1
answer
918
views
Convergence of semi convex functions
Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...
1
vote
1
answer
221
views
The derivative of a filter with respect to a output signal [closed]
I have two signals, $d(t)$ and $p(t)$, respectively the input and the output of the matching filter $w(t)$, i.e.
$$
d(t)*w(t)=p(t)
$$
where $*$ denotes convolution.The impulse response $w(t)$ may be ...
1
vote
1
answer
158
views
Wavelet momentum identity
I am reading an article on wavelet connection coefficients (G. Beylkin, "On the representation of operators in bases of compactly supported wavelets", 1992 (MSN)) and I came across Equation (3.31):
\...
2
votes
1
answer
2k
views
History- calculating convolution by tabular method
I often see a trick for calculating convolution of discrete data by a so-called Tabular method. There are a lot of Youtube videos and many Indian textbooks on Signal Processing [Books].1
Basically, ...
1
vote
0
answers
71
views
Convolve a 4D Gaussian function along a plane?
There is a 4D Gaussian function $G(u,s)=G(x|c,\mu,\Sigma )$ where $x=\begin{bmatrix}u\\ s\end{bmatrix}$,$u$ and $s$ is all 2D vector.
Now I want to blur (convolve) it along with $u$ by another 2D ...
0
votes
0
answers
219
views
Derivative of a convolution integral of the following type?
I'm looking to find the derivative of a convolution integral of the following form:
\begin{equation}
\frac{d}{dr}((G(r,t)*f(t)) = \frac{d}{dr} (\int_{-\infty}^{\infty} G(r,t-\tau)f(\tau) d\tau)
\end{...
3
votes
3
answers
250
views
When does convolution erase non-monotonicities?
Suppose $\phi:\Bbb R\to[0,\beta]$ is a bounded continuous function such that $\phi(-\infty)=0$ and $\phi(\infty)=\beta$.
Assume $\phi$ is non-decreasing except near zero, i.e. there exists $r>0$ ...
3
votes
1
answer
404
views
Is there (fast) fourier transform for vector convolution?
Given a list of variables $u_1,\dots,u_m\in\mathbb R$ and $v_1,\dots,v_n\in\mathbb R$ the standard convolution is defined
$$U*V(t)={\sum_{i}} u_iv_{t-i}.$$
Given a list of vectors $u_1,\dots,u_m\in\...
8
votes
1
answer
539
views
Convolution in K-Theory via an Example (From StackExchange)
I've spent lots of time in Chriss and Ginzburg's "Complex Geometry and Representation Theory" and despite convolution (in Borel-Moore homology or K-theory) being very central, I feel like I'm still ...
4
votes
1
answer
2k
views
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+\...
11
votes
0
answers
159
views
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{...
5
votes
3
answers
789
views
Mathematical Techniques to Reduce the Width of a Gaussian Peak
In the chemical analysis by instruments, the signals of several molecules are overlapped which makes it difficult to determine the true area of each peak, such as those shown in red. I simulated this ...
3
votes
1
answer
626
views
Exponential deconvolution using the first derivative
There is an interesting observation using the first derivative to deconvolve an exponentially modified Gaussian:
The animation is here at terpconnect.umd.edu.
The main idea is that if we have an ...
8
votes
1
answer
1k
views
Origin of the convolution theorem
I am a chemist, with some interest in signal processing. Sometimes, we use the deconvolution process to remove the instruments response from the desired signals. I am looking for the earliest ...
10
votes
1
answer
1k
views
Gelfand's trick (Gelfand's lemma) in positive characteristic?
I came across this preprint that claims in Lemma 1.1 that Gelfand's trick (also known as Gelfand's lemma) only works in characteristic zero:
Let $H < G$ be finite groups. Suppose we have an anti-...