Questions tagged [convolution]
Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schwartz's sense) or measures.
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Evaluating the convolution of $e^{-at^2}$ and $e^{-bt^2}$ via Fourier transforms
Problem Statement: Use the convolution theorem on the function $ f(t) = e^{-at^2} $ and $ f(t) = e^{-bt^2} $, $ a, b \in \mathbb{R} $. Calculate $ (f \ast g)(t) $.
I got a hint that I should first ...
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Convolution between $L^1$ function and a singular integral kernel
I meet a problem when reading Modern Fourier Analysis(3rd. Edition) written by L.Grafakos. On pg.82 he writes:
Fix $L\in\mathbb{Z}^+$. Suppose that $\{K_j(x)\}_{j=1}^L$ is a family of functions ...
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I can't prove a wiki statement about convolutins
Here is the link:
https://en.wikipedia.org/wiki/Fundamental_solution#Application_to_the_example
It states that :
$$∫_{-∞}^∞\frac{1}{2} |x-y| \sin(y) dy=-sin(x)$$ as distributions
The best I can come ...
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convolution of the fundamental solution with the homogeneous solution
I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero?
Let $U$ and $E$ ...
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Analycity of $f*g$ with $f$ and $g$ smooth on $\mathbb{R}$ and analytic on $\mathbb{R}^*$
Posted also on MO with a bounty
Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ analytic on $\mathbb{R}\backslash\{0\}$ but non-...
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Algorithm to compute a convolution recursively
Let
$$
f(t) = \int_0^t k(t-s)g(s) \, ds.
$$
Assume that $g$ is only given in a grid $t_j = j\delta_t$, and that we wish to compute similarly $f$ on the same grid.
What's an efficient algorithm to ...
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Clarification Needed on 1D Convolution and Kernel Purpose
I am confused about the definition of 1D convolution.
Given $ a = [-\frac{1}{2}, \frac{1}{2}] $ and $b = [1, 1, 1, -1, -1, -1] $, what will be the result of the convolution $( a * b )$?
From my ...
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Convolution of slightly multivariate Gaussians slightly modified
Starting with
$ p(a) = \int p(a|b) p(b) db$
replace $p(b)$ with $\tilde{p}(b) = \mathcal{N}(b; \mu_b, \Sigma_b + \tilde{D})$
where $\tilde{D}$ is an additive diagonal covariance.
Assuming ...
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Convolution preserve the boundary condition
Here, I want to know if convolution will preserve the Neumann condition or not. Suppose $K$ is a continuous function on some interval $[-L,L]$, and $u$ is some 'good enouth' function on $[0,L]$ that ...
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What's the necessary and sufficient condition for a real sequence to be written as the self-convolution of another real sequence?
Definition For a sequence $a_0,a_1,\cdots,a_n$, the corresponding self-convolution is another sequence $\displaystyle b_m=\sum\limits_{i+j=m}a_ia_j$ where $0\leq m\leq 2n$.
Calculating the self-...
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Associativity of Convolutions
In Folland's real analysis textbook, there are the following propositions:
Assuming that all integrals in question exist, we have
$$
(f*g)*h=f*(g*h) $$
The proof is based on the Fubini's theorem.But ...
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Help with understanding combination of probability distributions
I have two probability mass functions (PMFs) across the surface of a sphere. They are localised Gaussians (a few degrees in expanse), whose centres have arbitrary positions though they are quite close ...
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Inverse Fourier Transform - convolution of exponential and rectangular window
I'm trying to get the response in the time domain of the convolution between the exponential $u(t)e^{-at}$ and the rectangular window ($u(t+1)-u(t-1)$). I had already obtained its result by ...
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Is convolution theorem on $l^2(\mathbb{Z})$ valid?
I have a doubt about Fourier transform $F:L^2([0,2\pi])\to l^2(\mathbb{Z})$. If $f,g\in l^2(\mathbb{Z})$ then $f*g\in l^2(\mathbb{Z})$, then, $\mathcal{F}^{-1}(f*g)\in L^2([0,2\pi])$.
Question $\...
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The Fourier transform of product of derivatives
I have the task to compute the Fourier transform of the product in matlab:
$$ \left( \frac{\partial u(t, x)}{\partial x} \right)^2 \left( \frac{\partial^2 u(t, x)}{\partial x^2 } \right)$$
I was ...
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