# Questions tagged [convolution]

The convolution tag has no usage guidance.

108
questions

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votes

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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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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 ...

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votes

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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) ...

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votes

**0**answers

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 $...

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votes

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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)...

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votes

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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, ...

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vote

**0**answers

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

**0**answers

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

**3**answers

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 $r>0$ ...

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votes

**1**answer

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

**1**answer

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

**1**answer

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+\...

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votes

**0**answers

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

**3**answers

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

**1**answer

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

**1**answer

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 ...