Questions tagged [convolution]
The convolution tag has no usage guidance.
164
questions
3
votes
1
answer
350
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}\...
- 250
6
votes
2
answers
497
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 $$\lVert f*g\rVert_p\leq\lVert f\rVert_1\lVert g\rVert_p\qquad (f\in L^1(\...
- 199
3
votes
1
answer
593
views
Can we show that the characteristic function of an infinitely divisible probability measure has no zeros
Let $E$ be a normed $\mathbb R$-vector space, $\mu$ be a probability measure on $\mathcal B(E)$ and $\varphi_\mu$ denote the characteristic function$^1$ of $\mu$.
Assume $\mu$ is infinitely divisible, ...
- 157
5
votes
1
answer
317
views
Is there a name for this type of matrix?
For my thesis in neural networks, I was trying to find a way to generalize a Sobel operator. I quickly thought of this:
$$
\begin{bmatrix}
a&b&c\\
d&0&-d\\
-c&-b&-a
\end{...
- 153
1
vote
1
answer
479
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 ...
- 113
3
votes
2
answers
437
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=...
- 75
1
vote
1
answer
1k
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 ...
- 75
1
vote
0
answers
121
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
4
votes
1
answer
254
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 ...
- 1,021
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
312
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(...
- 71
2
votes
1
answer
303
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 $[0,\infty]$
...
- 31
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 ...
- 791
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}\...
- 157
2
votes
2
answers
310
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 ...
- 157