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Questions tagged [distribution-theory]

Use this tag for questions about distributions (or generalized functions). For questions about "probability distributions", use (probability-distributions). For questions about distributions as sub-bundles of a vector bundle, use (differential-geometry).

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12 views

How are Ancillary and Complete Statistics related?

We also know that X > theta I am told to first find a complete statistic for theta. My understanding is that I have to calculate the density function first. How would the PDF determine completeness, ...
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0answers
8 views

Minimal Sufficient Statistics and Sufficient Statistic

The procedure I have to take is to first identify that both U(X) are unbiased for theta, and that U2(X) = E[U1(X)|T2(X)]. My question is How would that relate to U2 having a smaller variance than U1,...
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0answers
20 views

Interchanging Integration Order involving Fourier Transform

$$f(\omega,u):=\frac1{\omega+iu}$$ where $i$ is the imaginary unit number. We see that the integral of a Fourier transform $$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
2
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1answer
24 views

Weak convergence in $W^{1,p}(\Omega)$

I am a bit confused by weak convergence in Sobolev spaces. I am making this post to hopefully clarify some of my doubts. Recall that in a Banach space, we say a sequence $x_n\in X$ converges weakly ...
2
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1answer
60 views

Show that there exists a linear functional $T:\mathcal{D}(Q)\rightarrow\mathbb{C}$ which is not continuous.

How to construct a linear functional on the space of test functions $\mathcal{D}(Q)$ which is not continuous? In other words, how to find a linear map $T:\mathcal{D}(Q)\rightarrow\mathbb{C}$ such that ...
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1answer
21 views

Calculating the Cramer-Rao lower bound for the variance of unbiased estimators of $θ$

Assume the standard situation, that is, let X1, . . . , Xn be independent and identically distributed with $X_k ∼ P_θ$, where$P_θ(x) = 1/(2θ^3).x^2.exp(−x/θ) , 0 < x < ∞$. This is a special case ...
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1answer
17 views

Extension of Sokhotski-Plemelj rule for computing integral

I am trying to calculate the following integral, $$ \int_{0}^\infty dk e^{i(\alpha k^2 + \beta k +\gamma)} $$ where $\alpha,\beta,\gamma$ are real numbers and $\alpha>0$. I am aware that the ...
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0answers
32 views

Are all the radial, positive homogeneous of degree $a\in(-N,0)$ tempered distributions of $\mathbb{R}^N$ proportional to $x\mapsto |x|^a$?

Let $\mathcal{S}$ be the space of Schwartz test functions on $\mathbb{R}^N$ and let $\mathcal{S'}$ be the space of tempered distributions on $\mathbb{R}^N$. Let $\mathcal{O}$ be the space of unitary ...
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1answer
41 views

What is $F_\mu(\mu)$?

Given a measure $\mu$ and the corresponding distribution function $F_\mu$. What happens if one looks at $$F_\mu(\mu)~? $$ One might as well assume that $\mu <<\lambda$. Thanks in advance!
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1answer
32 views

Probability for cumulative density function [on hold]

I have no idea how to do this since I don't have the specific function, and prof says that it's the point in p.d.f. as it is $f(x)$. This is the problem: The random variable $x$ is distributed ...
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1answer
29 views

Find the support of $T(\phi) = \sum_{j = 1}^{+\infty} \frac{1}{j}(\phi(\frac{1}{j}) - \phi(0))$

the distribution not being defined through a locally integrable function then the support is the complement open set on which $T$ vanishes. if we consider test function with support not including $[0,...
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0answers
16 views

Find the support of the distribution $T(\phi) = \int_\limits{0}^{+\infty}\phi'(x) \log x \, \mathrm dx$

$$T : \mathcal{D(\mathbb{R})} \to \mathbb{R}\text{ and }T(\phi) =\int_{0}^{+\infty}\phi'(x) \log x \, \mathrm dx$$ I have proved in a previous post that this is a distribution of order $1$. Now I ...
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0answers
27 views

How to find the exact order of a distribution?

For example $$T : \mathcal{D(\mathbb{R})} \to \mathbb{R}\text{ and }T(\phi) =\int_{0}^{+\infty}\phi'(x) \log x \, dx $$ The order of $T$ is at most $1$ it is asked to prove that it is exactly $1$ ...
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0answers
16 views

Solve $(1-x^2)^2 u'-2xu=(1-x^2)^2$ in $\mathcal{D}'(I)$

J. J. Duistermaat and J. A. C. Kolk's book proposes the following problem: Prove that the differential equation $$(1-x^2)^2 u' -2xu=(1-x^2)^2$$ has no solution $u\in \mathcal{D}'(I)$ if $I$ is an ...
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0answers
43 views

Fourier transform of $e^{i\sqrt{1+x^2}}$

As the title says: I want to compute the Fourier transform (in the distributional sense) of $f(x)=e^{i\sqrt{1+x^2}}$, $x\in \mathbb{R}^n$ - say $n=1$ for the moment. I have no idea how to get it done: ...
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1answer
162 views

What does the symbol ∟ mean?

I encountered this symbol $\lefthalfcup$ ∟ in the statement of Besicovitch derivation theorem. It says that the Radon-Nykodym decomposition of the given measure $\nu$ is $\nu=f\mu+\nu^s$, $\nu^s$ is ...
2
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0answers
40 views

Convergence in $\mathcal{D}^\prime(\mathbb{R})$

I'm trying to solve this problem about distributions: "Let $f\in\mathcal{D}(\mathbb{R})$ a test function with $\mathrm{supp}(f)\subseteq(-1,1)$ and $\||f|\|_1=1$. For each $k\in\mathbb{N}$, let $f_k$ ...
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0answers
16 views

How evenly are the nodes with the highest degree distributed among communities in a graph?

Suppose I have two network graphs, each has 5 communities, each community in every graph has a distinct number of nodes within with a different degree. I need to estimate how evenly the nodes with ...
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0answers
35 views

Is my reasoning about existence of a distributional solution correct?

Does the equation $x''+\lambda x' - x = \phi$, where $\lambda\in\mathbb{R}$ and $\phi\in\mathcal{D}(\mathbb{R}^N)$ is a given arbitrary test function, always have a solution in a distributional sense? ...
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0answers
24 views

calculate log-normal parameters using the mean and var of the given distribution and obtain the mean and var from the generated random variable

Following this question, I am trying to estimate the parameters of lognormal dist based on Var and mean, and generate some rand numbers. Then for checking the accuracy, I calculate the mean and var ...
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1answer
30 views

About convolution of a distribution with test function

I am fighting with the following problem: Given a a distribution T, I want to prove that $F_T:D(R^n)→C^\infty(R^n)$ defined by $$F_T(ϕ)=T∗ϕ$$ (convolution of a distribution with a test function) is ...
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1answer
40 views

Density of $C_c^\infty([0,1])$ in $H_0^1(0,1)$

I'm trying to prove that $C_c^\infty([0,1])$ is dense in $H_0^1(0,1) = \{ f \in H^1(0,1) : f(0) = f(1) =0\}$ for the usual Sobolev norm $\Vert f \Vert = \Vert f \Vert_{L^2(0,1))} + \Vert f' \Vert_{L^2(...
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0answers
8 views

Problem with continuity of a convolution of a distribution and test

I am fighting with the following problem: Given a a distribution T, I want to prove that $F_T:\mathcal{D}(R^n)\rightarrow\mathcal{C}^\infty(R^n)$ defined by $$F_T(\phi)=T\ast\phi$$ (convolution of a ...
2
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1answer
32 views

Distributional derivative of absolutely continuous function

In $\textit{Rudin, Functional Analysis, p. 148} $, the example 6.14 says that if $\Omega \subset \mathbb{R}$ is an interval and $f$ is a function of bounded variation which is left continuous at every ...
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0answers
33 views

Upper bound for a division of two integrals

Consider two integrable functions $a(x)>b(x)>0$ for all $x$ and $$f(y)=\int_0^{\infty}a(x){\rm sinc}^2[(x_0-x)y]+b(x){\rm sinc}^2[(x_0+x)y] dx,$$ $$g(y)=\int_0^{\infty}b(x){\rm sinc}^2[(x_0-x)y]+...
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0answers
69 views

If $\int_{\mathbb{R}}f_1(x)g(x) dx=\int_{\mathbb{R}}f_2(x)g(x) dx, \forall g \in \mathcal{C}^{\infty}_{c}(\mathbb{R}),$ then $f_1=f_2$ as functions

Is the proof below correct? Any feedback is much appreciated. Let $f_1,f_2 \in \mathcal{C}^{\infty}_{c}(\mathbb{R}),$ the set of infinitely differentiable functions with compact support in $\mathbb{...
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1answer
30 views

Equivalent characterization of zero-sum test function

I am trying to prove an offhand claim from Meyer's Wavelets & Operators, and am stuck on the following. Let $\mathcal{D}(\mathbb{R})$ be the space of compactly supported real-valued test functions ...
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1answer
31 views

Support of the Principal value of $\frac {1}{x}$

I was studying a proof and encountered a part where they used the fact that the support of the principal value of $\frac {1}{x}$ that is $Pr (\frac {1}{x})$ is $\mathbb R$. Please what is the proof ...
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0answers
6 views

How to select a small amount of numbers such that a distribution matches another as closely as possible?

I have two sets of numbers (1000 in each set, integers), one of those distributions remains constant (reference), but the other changes slightly and some numbers are dropped. When a number is dropped, ...
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0answers
27 views

Show that $\psi$ is a distribution

How would one show that $$T: \mathcal D \to\mathbb R $$ $$\psi\to \newcommand\inner[2]{\langle #1,#2 \rangle}\inner{T}{\psi} = \sum_{n\in\mathbb Z} \psi (n) $$ is a distribution. I know that $T$ is a ...
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0answers
24 views

If $(a_n)_n\subseteq \Bbb{R}$ and $\left\langle T,\varphi \right\rangle=\sum^{\infty}_{n=0}a_n\varphi(n)$, then $T\in D'(\Bbb{R}).$

Let $(a_n)_n\subseteq \Bbb{R}$ such that $$\left\langle T,\varphi \right\rangle=\sum^{\infty}_{n=0}a_n\varphi(n).$$ I want to prove that $T\in D'(\Bbb{R}).$ My trial It suffices to prove that $T\...
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1answer
22 views

show that the formula $\langle S,\phi\rangle= \sum_{n=1}^{+\infty} \phi^{(n)}(1/n)$ defines a distribution over $\mathbb{R^{*}}$

I'm stuck and I can't see the necessity of $\mathbb{R^{*}}$ instead of $\mathbb{R}$ what I noticed so far : partial sum defines a distribution because for every $\phi \in \mathcal{D}(\mathbb{R})$ (...
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0answers
22 views

“Derivative” of piecewise Lipschitz continuous function

Suppose that we are given a family of piecewise Lipschitz continuous functions $f^{\varepsilon}$ with one single jump at $x^{\varepsilon}$. We suppose that $x^{\varepsilon}$ is differentiable with ...
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0answers
16 views

fourier inversion formula and compactly support distributions

I know that the Fourier transform of a compactly support distribution $u\in \mathscr{E}'(\mathbb{R}^{n})$ is smooth and also satisfies $$ |\hat{u}(\xi)|\leqslant C_{N}(1+|\xi|)^N,\label{1}\tag{1} $$ ...
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1answer
27 views

Example of a compactly supported $\delta$-sequence.

Recall that a $\delta$-sequence can be defined as a sequence, $(\phi_n)_{n\in\mathbb N}$, of continuously differentiable, non-negative, real-valued functions for which $\int_\mathbb R\phi_ndx=1$ for ...
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1answer
67 views

Fourier transform of $\frac{1}{t} \theta(t)$

I'm looking for the Fourier transform of $f(t)=\frac{1}{t}\theta(t)$ ($\theta$ is the step function), I know how to do both factors separately but not if they are multiplying. Can someone help me ? I ...
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0answers
14 views

Fourier Restriction: extension operator identity

Consider the extension operator: $$ Eg(x)=\int_S g(\xi)e^{2\pi i x\cdot \xi}d\sigma(\xi). $$ For simplicity we consider the 2D-case, where $S$ is the paraboloid $\xi\mapsto \xi^2$, $\xi\in [-1,1]$. (...
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1answer
64 views

All the derivatives of distributions are also distributions, but what about the converse?

Say you have some linear functional $f$ well defined on $\mathscr{D}(\mathbb{R})$: then what if for some test function $\phi$ you have $$ -f(\phi') = g(\phi)? $$ If that $g$ defines a distribution,...
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0answers
15 views

show that positive distributions are of order $0$?

$T$ is a postive distribution if $ 0\leq\phi \in \mathcal{D}(\mathbb{R}) \implies T(\phi) \geq 0$ we have to show the following, let $K = [-m,m]$ such that $supp \phi \subset K$ then there must exist ...
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0answers
22 views

show that $fp(\frac{1}{x^2})$ defines a distribution?

let $\phi$ be a test function, let $\epsilon > 0$, then : $$\langle fp(\frac{1}{x^2}),\phi\rangle = \lim_{\epsilon \to 0} [\int_{|x| \geq \epsilon} \frac{\phi(x)}{x^2}dx - 2\frac{\phi(0)}{\epsilon}...
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0answers
41 views

show that $\delta''(\frac13x^3 +x) = \delta''(x) - 2\delta(x)$

I tried to go about about this using the definition : let $\phi$ be a test function $$\begin{align}\langle \delta''(\frac13x^3 +x) ,\phi\rangle &= \langle \delta''(\frac13x^3 +x) ,\frac{x^2+1}{x^...
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1answer
58 views

Which theorem is this? If $\phi(x)$ is differentiable at $0$, then there exists $\psi(x)$, continuous at $0$, such that $\phi(x) = \phi(0) + x\psi(x)$

I'm self studying generalized functions, and it's been the third time or so that I come across the following statement : If $\phi(x)$ is differentiable at $0$, then there exists a function $\psi(x)...
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2answers
45 views

How to properly estimate a Schwartz function

For every Schwartz function $\varphi \in S(\mathbb{R}^{n})$ there exists a constant $c_{\beta, k}$ such that one can estimate the Schwartz function by $$|\partial^{\beta}\varphi(x)| \leq \frac{c_{\...
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1answer
23 views

Question on vector valued distributions: weak time derivative calculation

Although I read all the similar posts here, I still can't find the answer in my question so I 'll try to pose it as clear as I can. DEFINITION $1$: Let $f\in L^1_{loc}(I, X)$. Then $\;\;\langle T_f, \...
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1answer
47 views

Solving a distribution differential equation

The exercise is to solve $$x u'(x) = \delta(x).$$ By using the definitions $$\begin{cases}(u'|\varphi) = (-u|\varphi') \\ (fu|\varphi) = (u|f\varphi) \end{cases}$$ we get to solve $$(-u|\varphi + ...
3
votes
1answer
17 views

Is this estimate true for functionals on Frechet spaces?

In class the other day my professor made the following claim about the Schwartz class $\mathcal S$: Let $u: \mathcal S \to \mathbb C$ be linear. $u$ is continuous iff there exist $C,N>0$ such ...
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1answer
56 views

Computing (distributional) gradient of a singular function

This question could well belong better to the physics stackexchange, but I'm hoping that posting it here could give me a more mathematical perspective. I am trying to find the expression for the ...
2
votes
1answer
62 views

Continuous distribution-valued function induces distribution

Suppose that the map $\mathbb{R}^n \to \mathcal{D}'(\mathbb{R}^n), \hspace{3mm}\eta\mapsto E_\eta$ is continuous. Furthermore let $\mu$ be a Radon-measure with compact support. I'm having trouble ...
1
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1answer
25 views

Are these distributions the same?

Consider the following distribution, where $\delta$ is the Dirac delta: $$f(x,y)=\delta(x)+\delta(y).\tag1$$ This can be viewed as a limit of the following sequence of smooth functions: $$\...
0
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0answers
22 views

Is Wiener's Generalized Harmonic Analysis still relevant now that we have Distribution theory?

I only just now discovered Wiener's work "Generalized Harmonic Analysis", which from what I understand was a generalization of Fourier Transform for functions that are not square-integrable. However, ...