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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Prove that the derivative of the mollification approaches the strong $L^p$ derivative
Let $g \in C_c(\mathbb{R})$ be such that $\int_\mathbb{R} g dx = 1$. Suppose that $f \in L^p(\mathbb{R}$ has a strong $L^p$ derivative, i.e. there is some $h$ such that $\displaystyle\lim_{y \to 0} \...
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Calculate $\sum_{n=1}^\infty (1-\alpha) \alpha^nP^{*n}(x)$
$$
\mbox{Let}\quad
P'(x)=\sum_{j=1}^n a_j\frac{a^jx^{j-1}}{(j-1)!}e^{-ax},x\geq 0
$$
be the density function of a mixture of Erlangs and let $\alpha\in(0,1)$:
Is is possible to determine an analytic ...
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Does convolution preserve local integrability in the sense that if $f \in L^p_{\text{loc}}$ and $g \in L^1$, then $f \ast g \in L^p_{\text{loc}}$?
Consider the Euclidian space $\mathbb R^n$, where $n \in \mathbb N$ is a fixed integer, equipped with the Lebesgue measure. Let $L^p_{\text{loc}}(\mathbb R^n)$($1 \leqslant p < \infty$) denote the ...
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Density of similiar Sobolev space
Consider the space of functions defined as,
$$
D = \{f \in L^p(0,\infty): f \in AC_{loc}(0,\infty) \text{ and } xf'(x) \in L^p(0,\infty)\},
$$
where $AC_{loc}(0,\infty)$ is the set of locally ...
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Find a sequence polynomials that converges to $e^{x-y}$ on $S^1$ but diverge anywhere else
I'm trying to do Exercise 3.3.1 in scv.pdf
Let $z=x+i y$ as usual in $\mathbb{C}$. Find a sequence of polynomials in $x$ and $y$ that converge uniformly to $e^{x-y}$ on $S^1$, but diverge everywhere ...
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Fourier transform with and without convolution theorem not equivalent
This is a problem involving Fourier transforming an integral relevant to the computation of Feynman diagrams, which is of the form:
$S(r_1,r_2)=\int d^3 r_3 \space v(r_1,r_3)f(r_3,r_2),$ where $v(r_1,...
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Preservation of strict log-concavity under convolution
I have spent an embarrassing amount of time trying to prove or disprove any of this. I am aware that a similar question was posted in 2014, but since I couldn't make anything out of the two sources ...
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Pixel sensitivity in a convolution operation
So a Gaussian blur convolution looks like this:
formula for Gaussian Blur
Now, it seems rather common to take the partial derivatives with respect to a change in x or y. And in CNNs you get the ...
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Haar measure and convolution on a Lie subgroup
Let $G$ be a Lie group and $H \leq G$ a closed Lie subgroup.
Take Haar measures $dg$ and $dh$ on $G$ and $H$. Can we get $dh$ from $dg$ (up to some constant)? I would also like to understand it in ...
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A 3D integral (Hylleraas wave function)
In quantum mechanical context I would like to evaluate this function (Hylleraas type wave function for Helium atom ground state):
$$ I(\boldsymbol{r}) = \int_{R^3} d^3 r' e^{- a_1 r'} \Vert \...
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Complex exponential Fourier coefficients of a convolution involving the exponetial function
In the book "Elementary Classical Analysis", by Marsden, the following is proposed as a worked example:
Let $f:[0,2\pi]\to\mathbb{R},g:[0,2\pi]\to\mathbb{R}$ and extend by periodicity. ...
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Is there a good method to recover the original functions from their convolution?
Occasionally, I failed to detect the function which is just the convolution of two simple functions. For instance
$$
2a\sin{at}*\sin{at}=\sin{at}-at\cos{at}
$$
One possible way is to observe the ...
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$g: R \to R$ is integrable and has compact support. $f(y) =sin^2(y) ln|y|$. Show that $(f ∗ g)(x)$ is well defined for $x \in R$ and of class $C^1$?
$g : \mathbb{R} \to \mathbb{R}$ is integrable over $\mathbb{R}$ and has a compact support.
$f(y) = \sin^2(y) \ln|y|$
Show that the convolution $(f ∗ g)(x)$ is well defined for $x \in \mathbb{R}$ and ...
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Positive integral appears negative after applying the convolution theorem
I am Fourier transforming momentum representation of quantum mechanical wave functions to the position representation. In the weak sense (tempered distribution) we have
$$\int_{\mathbb{R}^3} d^3k \ e^{...
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Fourier Transform of a Triangular function (using the formula and the convolution theorem)
I am studying physics at a degree level, and I was given a seemingly simple question: FT of a triangular function. $\\f(x)=\frac{1}{a}(a-|x|)$ and I needed to show $\DeclareMathOperator{\sinc}{sinc} \...
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