Questions tagged [convolution]
The convolution tag has no usage guidance.
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Convolution vanishes on an interval
Fix a "test" function $f(x)=x\exp(-x^2)$, which is nonzero except $x=0$. Suppose that $g$ is a function with some necessary regularity. Consider the convolution.
$$
(f\ast g )(x)=\int_{-\infty}^{+\...
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Which classes of functions are "convolution ideals"?
If $g$ is continuous then $f*g$ is continuous.
If $g$ is smooth then $f*g$ is smooth.
If $g$ is a polynomial then $f*g$ is a polynomial.
If just one of the two functions belongs to the class of well-...
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Is the set of the convolutions of two-point measures dense in the set of all measures?
A measure supported in two points is a measure of the form
$$
\mu=\alpha\delta_a+(1-\alpha)\delta_b,
$$
where $a<b$ and $\alpha\in (0,1)$.
The question is:
Given a finite non-negative measure ...
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Convolution of measures - entropy growth
Imagine you have two shift-invariant measures $\mu, \nu$ in the Bernoulli space $\{0,1\}^{\mathbb{N}}$ with positive entropy and both are not the Bernoulli measure $(\frac{1}{2},\frac{1}{2})$. I know ...
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convolution integral involving modified Bessel functions of the first kind
I'm stuck with this convolution integral ($z \geq 0$)...
\begin{equation}
f_{Z}(z)=\int^{\infty}_{-\infty}f_{1}(x)f_{2}(z-x)dx = \mbox{ } ???
\end{equation}
which represents the pdf of the sum $Z = ...
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Complete solution set of a Convolutional Equation?
Here is a problem that am I stuck and I appreciate any help. In essence, I am trying to show that the only solutions for the described problem are the ones provided below. Best..
Setup: In what ...
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Do they have the same limit?
Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$
$$
\frac{1}{T}\int_{\mathbb{R}}dx\int_{[-T,T]^2}d\mathbf{v}\int_{[-T,T]^2}...
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Maximum of a mollified/convolution function
I have a function $f:{\mathbb R}\rightarrow {\mathbb R}_+$ which has a unique maximum at $x=0$. $f$ can be symmetric or asymmetric. I am interested on the mollified-f function
$$\tilde{f}(x)=\int_{-\...
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Existence and smoothness of convolutions of distributions in Sobolev spaces
Let $f\in H^{s_1}(\mathbb{R}^n)$ and $g\in H^{s_2}(\mathbb{R}^n)$, where $s_1, s_2 \in \mathbb{R}$ and can be positive or negative.
It is easy to show that $f *g$ is defined pointwise when $s_1+s_2\...
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Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)
Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as
$$...
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Deconvolution of sum of two random variables
Let $Z = X + c \cdot Y$ where $X$ and $Y$ are independent random variables drawn form the same distribution given by the pdf $g()$ and $0 < c < 1$
I have observations of $Z_i$'s and thus can ...
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Bounded convolutions with binomial coefficients
I need to figure out a nice family of decaying functions such that
$\sum_{d=2}^k {k \choose d} f_k(d) \leq 1/k$ and $f_k(d)\geq f_k(d+1)$
How can I figure out what good candidates could be?
Any ...
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Banach algebra for measures induced by Haar measures
It is classical that $L^1(G, m)$ is a Banach algebra when $G$ is a locally compact group with Haar measure $m$ by using the operation of convolution via the integral
$$(f*g)(y)=\int_Xf(x)g(yx^{-1})\,...
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What is convolution intuitively?
If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...