All Questions
Tagged with convolution real-analysis
437
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Integration of the product of a compact supported convolution [closed]
I know that in general case we have $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} f(s)g(t-s) ds dt = \left( \int_{-\infty}^{+\infty} f(t) dt \right) \left( \int_{-\infty}^{+\infty}g(t) dt \right) ...
- 13
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275
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How to prove $(F\ast\sin)(x)=-\sin(x)$, where $F(x)=\frac{1}{2}|x|$?
Wikipedia states in this article about fundamental solutions that if $F\left( x \right) = \tfrac{1}{2} \left| x \right|$, then
$$\left( F \ast \sin \right)\left( x \right) := \int\limits_{-\infty}^{\...
- 139
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134
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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 $E$ ...
- 139
2
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42
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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-...
- 506
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1
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94
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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 ...
- 133
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32
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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 ...
- 41
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33
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Cross-correlation of a function with itself
I came up with the following question while writing on my thesis. We assume $f:\mathbb{R}\rightarrow\mathbb{R}$ to be a real valued $\mathcal{L}^1$-function. Then the cross-correlation of $f$ with ...
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38
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Convolution of $\mathcal{C}^\infty$ is analytic
Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$. Is the convolution (assumed to be well defined) defined as:
$$(f*g)(x) = \int_\mathbb{...
- 506
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1
answer
74
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Seeking Algorithm to Solve a Convolution Integral or Directly Convolve Two CDFs
Hi everyone,
I'm working on a project where I need to find a way to directly convolve two cumulative distribution functions (CDFs) given in polynomial form, and solve the following convolution ...
- 19
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59
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Is the convolution between two CDF always well defined?
Given the integral convolution:
$$(F_X * G_X)(x)=\int_{-\infty}^x F_X(t)G_X(x-t)dt $$
and also that the CDF of random variables are bounded between the interval $(0,1)$ and the flipping of one of them ...
- 495
1
vote
1
answer
65
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Circular shift of a function.
Consider a function $f$ that maps real numbers to real numbers with domain $[-a,a]$. I would like to describe the circular shift of this function by an amount $\delta$ such that, if I shift the ...
- 7,042
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20
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Derivative of convolution of a continuous function with a continuously differentiable function.
Suppose $f \in C^1(\mathbb R)$ has compact support and $g \in
> C(\mathbb R)$ is bounded and $\lVert g \rVert _1 < \infty$. Prove that
the convolution $f * g$ is continuously differentiable and
$...
- 485
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1
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35
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The operator norm regarding to the difference between a mollified function and the function itself
Let $\rho_\epsilon$ be a mollifier that has support in $B(0,\epsilon)$, define the operator
$$T_\epsilon (f)=\rho_\epsilon*f-f,$$
for every $f\in L^2(\mathbb R^d)$, can we prove or disprove that the ...
5
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2
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248
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Calculate $\sum_{n=1}^\infty (1-\alpha) \alpha^nP^{*n}(x)$
$$
\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 ...
- 91
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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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