Questions tagged [convolution]

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2
votes
1answer
137 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 \...
4
votes
0answers
119 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^\lambda(x)F^{1-\lambda}\leq F(\...
2
votes
0answers
86 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
1answer
73 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
2answers
469 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(\...
0
votes
0answers
17 views

Weierstrass transform of chirped Gauss-Hermite function

I would like to understand if there is any hope of having an analytic expression for the following convolution integral: $$ \int_{-\infty}^{+\infty}\mathrm{d}y\; \mathrm{e}^{-\frac{(x-y)^2}{2\sigma_1^...
3
votes
1answer
241 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
1answer
285 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
1answer
112 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
2answers
184 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
1answer
162 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
0answers
98 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
1answer
105 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
1answer
108 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
1answer
240 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(...
0
votes
1answer
87 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
1answer
477 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
1answer
80 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
2answers
203 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
2answers
308 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) $Y$...
0
votes
1answer
173 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
1answer
79 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
1answer
123 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
2answers
211 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
0answers
77 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
1answer
71 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
0answers
45 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
1answer
264 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
0answers
67 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
0answers
50 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
1answer
58 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 &...
4
votes
2answers
647 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
1answer
172 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
1answer
587 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
1answer
169 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
2answers
487 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)...
2
votes
1answer
174 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
1answer
170 views

The derivative of a filter with respect to a output singal [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
1answer
127 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
1answer
747 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
0answers
32 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
0answers
103 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
3answers
219 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 $r>0$ ...
2
votes
1answer
256 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
1answer
391 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
1answer
955 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
0answers
142 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
3answers
438 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
1answer
366 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 ...
7
votes
1answer
744 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 ...