# Questions tagged [convolution]

The convolution tag has no usage guidance.

149
questions

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### Convolution of two modified Bessel functions

Does a closed formula (or power series expansion) for the following convolution exist?
$$I_{\nu}(x)=\int_{0}^{\infty} K_{\nu}(x-\tau)K_{\nu}(\tau)d\tau$$
Here $K$ stand for the modified Bessel ...

0
votes

1
answer

99
views

### Does convolution with $(1+|x|)^{-n}$ define an operator $L^p(\mathbb R^n) \to L^p(\mathbb R^n)$

Suppose that $f : \mathbb R^n \to \mathbb R^n$ is a locally integrable function. I am interested in the integral
$$ x \to \int_{\mathbb R^n} ( 1 + |y| )^{-n} f(x-y) \;dy $$
If the decay of the ...

1
vote

0
answers

15
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 ...

0
votes

0
answers

61
views

### Lower bound of the derivative $(f*g_\sigma)'$ at the zero-crossing point

I am stuck with the following problem. Let consider $f$ a smooth real function such that:
$f$ is negative before 0,
$f$ is positive after 0,
we have $|f'(0)|>0$.
Let $\sigma>0$ and $g_\sigma$ ...

2
votes

0
answers

233
views

### Recent progress restriction conjecture - Problem 2.7 (Terence Tao lecture notes)

I've been tackling the following problem for some time,
Problem 2.7. (a) Let $S:=\left\{(x, y) \in \mathbf{R}_{+} \times \mathbf{R}_{+}: x^2+y^2=1\right\}$ be a quarter-circle. Let $R \geq 1$, and ...

3
votes

0
answers

138
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 $\sigma>0$ ...

2
votes

1
answer

99
views

### Uniqueness of the zero of $f-f*G_\sigma$ with $f$ convex/concave

I am struggling with the following problem. Let $f$ be a real smooth function:
strictly convex on $(-\infty,0)$,
strictly concave on $(0,\infty)$,
strictly increasing.
For $\sigma>0$, how can one ...

2
votes

1
answer

206
views

### Distance between root of $f$ and its Gaussian convolution

Let $f$ be a :
$f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$,
for all $x> 0,~f(x)>0$,
for all $x< 0,~f(x)<0$,
I am struggling to find a bound for the distance between the root of $f$ ...

0
votes

0
answers

80
views

### Does the tensor product of mollifiers work for $L^{p,q}$ spaces?

Let $X$ and $Y$ be compact regions of $n$- and $m$-dimensional Euclidean spaces respectively.
For any $p,q \in [1,\infty)$, define $L^{p,q}(X \times Y)$ be the space of real valued functions $f :X \...

1
vote

1
answer

106
views

### Convolution with the Jacobi Theta-function on "both the space and time variables" - still jointly smooth?

Let $\Theta(x,t)$ be the Jacobi-Theta function:
\begin{equation}
\Theta(x,t):=1+\sum_{n=1}^\infty e^{-\pi n^2 t} \cos(2\pi n x)
\end{equation}
Usually, the heat equation with the periodic boundary ...

3
votes

1
answer

160
views

### Is there a real/functional analytic proof of Cramér–Lévy theorem?

In the book Gaussian Measures in Finite and Infinite Dimensions by Stroock, there is a theorem with a comment
The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, ...

6
votes

3
answers

681
views

### Convolution of $L^2$ functions

Let $u\in L^2(\mathbb R^n)$: then $u\ast u$ is a bounded continuous function. Let me assume now that $u\ast u$ is compactly supported. Is there anything relevant that could be said on the support of $...

4
votes

1
answer

182
views

### Just how regular are the sample paths of 1D white noise smoothed with a Gaussian kernel?

Adapted from math stack exchange.
Background: the prototypical example of---and way to generate---smooth noise is by convolving a one-dimensional white noise process with a Gaussian kernel.
My ...

4
votes

0
answers

94
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 ...

0
votes

1
answer

119
views

### Does convolution commute with Lebesgue–Stieltjes integration?

Let $g: \mathbb R \to \mathbb R$ be a function of locally bounded variation, and $f$ a locally integrable function with respect to $dg$, the Lebesgue–Stieltjes measure associated with $g$.
Let $\eta$ ...

1
vote

1
answer

177
views

### Extracting eigenvalues of a circulant matrix using discrete Fourier matrix

The eigenvalues of a circulant matrix $C$ can be extracted as $$
\Lambda=F^{-1} C F
$$
where the $F$ matrix is a discrete Fourier transform matrix and $\Lambda$ is a diagonal matrix of eigenvalues.
...

5
votes

0
answers

163
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 ...

2
votes

0
answers

113
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 ...

3
votes

1
answer

106
views

### Convolution between normal distribution and the maximum over $m$ Gaussian draws

$\DeclareMathOperator\erf{erf}$
Let's consider the Gaussian distribution $P_X(x)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{x^2}{2 \sigma^2}}$. Now consider the random variable $W \equiv \max \{ X_1, ...

3
votes

0
answers

137
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 ...

1
vote

0
answers

56
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 ...

2
votes

1
answer

162
views

### Approximating a function by a convolution of given function?

Let $g:\mathbb{R}\to \mathbb{R}$ be a given differentiable function of exponential decay on both sides. Now let us be given a function $f:\mathbb{R}\to \mathbb{R}$, also of exponential decay, if you ...

4
votes

1
answer

173
views

### 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

0
answers

174
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, ...

1
vote

0
answers

66
views

### Apply gaussian blur to get original image [closed]

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

133
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

233
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

244
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

570
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

1k
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 :)

1
vote

0
answers

106
views

### 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....

4
votes

1
answer

233
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 ...

3
votes

2
answers

399
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{...

3
votes

1
answer

639
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)...

2
votes

1
answer

99
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-...

4
votes

1
answer

269
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

214
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 ...

2
votes

0
answers

79
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

234
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 ...

2
votes

1
answer

100
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

77
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 ...

4
votes

1
answer

204
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;...

2
votes

1
answer

306
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

210
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

108
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

267
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

493
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

537
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

313
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

393
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 ...