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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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1answer
32 views

Fourier transform of $\varphi_m(u)=\int |x|^mu(x)dx$

I'm stuck with the following problem Let $\varphi_m \in \mathcal{S}'(\mathbb{R}^{n})$, $n \in \mathbb{N}$, $m\in \mathbb{C}$, $0 >\text{Re}(m)>-n$ the distribution defined by $$\varphi_m(u)...
2
votes
1answer
34 views

Does the Lebesgue measure on the segment $y=x$ represent this distribution?

Set $\Omega=(-1,1)^2$. Consider the following measure on $\Omega$: $\mu(A)=m(A \cap L)$, where $L=\{ (x,x) \, | \, -1 < x < 1\}$ (the segment of the line $y=x$ in $\Omega$) , and $m$ is the ...
0
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1answer
32 views

Linear ODE Cauchy problem: issue in the distributional solution

I am considering the following Initial Value problem: $$ \begin{aligned} &u'+\alpha u = \cos \omega t\\ & u(0)=u_0 \end{aligned} $$ The solution is: $$u(t) ={\rm e}^{-\alpha t} \left(u_0-{\...
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0answers
14 views

Relation between adjoint of trace operator and Dirac delta

Assume $u \in \mathscr{D}(\mathbb{R}^n)$ is a distribution of order $k$, with compact support on a smooth manifold $\Gamma \subset \mathbb{R}^n$. We know that we can write this distribution (Thm 2.3.5,...
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0answers
71 views

If $F_1$ and $F_2$ are distributions with compact support , how to well define the convolution of them , and how to show $F_1*F_2=F_2*F_1$

Suppose $F_1$ and $F_2$ are given distributions with $F_2$ having compact support , then we define the convolution $F_1*F_2$ as the distributions $(F_1*F_2)(\varphi)=F_1(F_2^{a}*\varphi)$ where $F_2^a(...
0
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1answer
18 views

Line integral of multi dimensional Dirac Delta

How do we compute the line integral of a Dirac Delta? e.g. $\int_a^b \delta(x-x(t),y-y(t))dt$ ? Consequently, is $\int_{-\infty}^{\infty}\delta(x-t)\delta(y-t)=\delta(x-y)$ or $\delta((x-y)/\sqrt2)$? ...
1
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1answer
37 views

System of Distributional Differential Equations

Here are the problem and my attempt to the solution. Is it correct?
2
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1answer
37 views

Distribution Derivative

Question: Find the second derivative of the function $f(x) = e^{-|x|}$ using the sense of Theory of distribution. My Attempt: $$ \left< f',\phi \right> = -\left< f,\phi' \right> = -\...
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votes
2answers
30 views

limiting distribution of an exponential function

Question: Find the limit of the function $f(x) = \lim_{\epsilon \rightarrow 0^+}\frac{e^\frac{-x^2}{\epsilon}}{\sqrt \epsilon}$ in the distribution sense: My Attempt : $\left< f_\epsilon,\phi \...
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2answers
92 views

Are there test functions $\in \mathcal{C}_{\text{c}}^{\infty}$ which aren't in the algebra generated by $f_{a,b}(x) := e^{\frac{C}{(x-a)(x-b)}}$

The standard test ($\in \mathcal{C}_{\text{c}}^{\infty}(\mathbb{R})$) function is the following \begin{equation*} f_{a,b}^{(c)}: \mathbb{R} \to \mathbb{R}_{\ge 0}, \ x \mapsto \begin{cases} \exp\left(\...
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0answers
42 views

Integrabilty of distibutions

Let $T : D(\Omega) \to \mathbb R$, be a generalised function. What can we say about its integrability? For example, does it belong to $L^p(\mathbb R)$? I am interested in the case of non-regular ...
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0answers
26 views

Term for Dirac delta-like functions

Is there a common term for functions that are like the Dirac delta function but perhaps shifted sideways, i.e., functions of the form $\rho(x)=\delta(x-a)$, where $\delta$ is the Dirac delta function ...
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0answers
19 views

Eigenvalue problems for elliptic operators on unbounded domains

Assuming $L$ is symmetric, elliptic second order differential operator, I want to to know about solutions to $$ -Lu = \lambda u \quad \text{in } \mathbf{R}^n.$$ Due to the unboundedness of the ...
1
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1answer
31 views

Rigorous explanation of integration involving delta distribution

In a physics class, I saw the following: The charge density of a uniformly charged circle (charge $Q$) of radius $R$ can be described in cylindrical coordinates using the delta distribution as $$ \...
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0answers
16 views

Distributional limit of $m\sin(m|x^2+y^2-1|)$

Consider the sequence $$ f_m(x,y)=m\sin(m|x^2+y^2-1|) $$ as sequence of distributions in $\mathcal{D}^\prime(\mathbb{R}^2\setminus\{0\})$. What is the limit as $m\to\infty$? In other words, given a ...
1
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1answer
42 views

Dirac Distribution

I'm trying to solve the following homework problem: Show that, \begin{equation} 1 + 2\sum_{n=1}^{\infty} \cos(2\pi nx) = \sum_{k= - \infty}^{\infty}\delta(x-k), \end{equation} in the sense of ...
0
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1answer
26 views

$u(x,y)=H(x-y)$ is a solution of a partial differential equation

I am trying to prove that $u(x,y)=H(x-y)$ ($H$ denotes the heaviside function) is a solution of the partial differential equation $$\frac{ \partial^2 u}{\partial x^2} -\frac{ \partial^2 u}{\partial y^...
2
votes
1answer
46 views

computation of the norm

I try to understand the notions of weak derivative and Sobolev space I take this example: $f(x)= |x|\quad $ for $ \quad x\in [-1, 1]$ The derivative in the sense of distributions is $(T_f)^{'}= ...
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2answers
25 views

Equivalence between Dirac Delta definition as a measure and as a distribution.

I always thought of Dirac Delta as the distribution $\delta_ {x_0}$ which performs $\phi\ \mapsto \phi(x_0)$. With respect to this definition we can think the Delta as the weak limit of some $L^1_{loc}...
1
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1answer
30 views

Example a distribution of finite order which is not tempered

I am looking for an example of a distribution of finite order that is not a tempered distribution. Could anyone help me with an example. It is known that any tempered distribution is of finite order. ...
0
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1answer
20 views

Converges in distribution sense : what does it really mean?

I'm always a bit confuse with this convergence in distribution sense. For example, take $f_n(x)=n\boldsymbol 1_{[0,1/n]}$. We have that $f_n(x)\to 0$ a.e., so why do we say that $f_n\to \delta _0$ in ...
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0answers
32 views

'Half-sifting' of Dirac Delta

Is the solution of the following integral $0$ or $1$ or something else? And why so? $$ \int_{0}^{t} f(\tau) \;\delta(\tau) \;d\tau$$ with $\;\;\;t \in \mathbb{R}\,,\; f: \mathbb{R}\rightarrow\...
2
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1answer
46 views

Prove that the distributional derivative of the Cantor function is the the $\log_3 2$-Hausdorff measure restricted to the Cantor set

The Cantor function has weak derivative equal to $0$ a.e. Its distributional derivative should be the $\log_3 2$-Hausdorff measure restricted to the Cantor set, but I'm having troubles doing the ...
2
votes
1answer
47 views

Distributional derivative of Weierstrass function

How can we compute the distributional derivative of the Weierstrass function $$W(x) =\sum_{k=1}^\infty \lambda^{(s-2)k}\sin(\lambda^k x)$$ where $s \in (0,2)$ and $\lambda$ are fixed parameters? We ...
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votes
1answer
36 views

Existence of fundamental solutions to general linear PDE with const. coefficients

Let $p(D)$ be a nontrivial polynomial with $D=-i\partial$ such that $$ p(D) = \sum_{a} c_aD^a $$ for a multiindex $a$ and constants $c_a$. I was wondering how and if we can prove that a fundamental ...
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votes
0answers
26 views

Solving a distributional equation sin(x)F=0

Can you help me please to solve the equation in $ \mathbf{D'}$: $$sin(x)F=0 $$ The correct answer is $\sum_{n=-\infty}^{\infty} c_n \delta_0(x-\pi n),$ where ${c_n} $ any numbers. But i have no ...
1
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2answers
62 views

Boundary condition inserted in the local PDE

Let us consider the following problem: $$ \begin{align} -&u_{xx}=0&&\forall x\in(0,L)&&\tag{1}\\ &u(0)=0\tag{2}\\ &u_x(L)=\alpha\tag{3} \end{align} $$ It is possible to ...
0
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2answers
22 views

Finding convolution of two distributions.

I am trying to show $\delta'*H =\delta $ and $1*\delta' = 0$,where $H$ denotes heavyside function. I tried this but after some simplication I am not getting anything interesting.
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0answers
16 views

Integration of a function - log-normal distribution

I am very inexperienced on this type of problems. Please help me step by step to solve this problem. $$ log \int A_i W(Z_i, Y) di=? \\ log \int W(Z_i, Y) di=? $$ We could assume log-normal ...
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0answers
22 views

Finite order distributions

Let $F$ be a linear functional on the space of test functions defined by $F(f)=\sum_{n\ge 0} f^{(n)}(n)$. show that $F$ is of infinite order. Attempt: for any $f$ we have that $supp(f)$ is compact. ...
0
votes
1answer
45 views

How to solve distributional equations?

I'm trying to solve this exercise If $T \cdot x=1$ in the sense of distributions then $T=\textrm{p.v.}\left(\frac{1}{x}\right)+c \delta_{0}$ Where $\delta_{0}$ is the Dirac delta distribution ...
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0answers
25 views

convergence of the integral of sin(nx) and a test function

and I am trying to show that the integral $\int_{\mathbb{R}}\sin(nx)fdx$ converges to $0$ for any test function $f$ while $\int_{\mathbb{R}}\sin^2(nx)f(x)dx$ does not go to 0. For the first function,...
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0answers
24 views

Convergence in the space of distributions

If $g_1,g_2,g_3$ are each continuous functions and converge to a continuous function $g$. I am trying to show that for any test function $T$, we have that $\langle g_n, T\rangle$ converges to $\langle ...
0
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1answer
43 views

Solving $f'=0$ in distribution sense

When we solve $f'=0$, we can solve it by using Fourier transform in the space of tempered distribution to get $$i\xi\hat{f}(\xi)=0,$$ which gives us the distribution $\hat{f}(\xi)=0$, thus $f=0$. ...
8
votes
0answers
127 views

Is there a notion of a continuous basis of a Banach space?

If $X$ is a Banach space, then a Hamel basis of $X$ is a subset $B$ of $X$ such that every element of $X$ can be written uniquely as a linear combination of elements of $B$. And a Schauder basis of $...
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0answers
26 views

Comparing (Normalised) Distribution of Two Small-Sample Datasets

I'd like to compare the distribution of a number of datasets that have few values within them, with the results ideally on a 0-1 normalised scale so that it is clear that a distribution approaching 1 ...
2
votes
2answers
64 views

Derivative of Dirac delta function

Is the relation of the Dirac delta function correct? $$ \frac{\partial}{\partial x''}\delta(x''-x') = -\frac{\partial}{\partial x'}\delta(x'-x'').\tag{1} $$ If it is, how to derive the above ...
0
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1answer
32 views

Condition to the convolution product well defined

i'm lost in the following question: why $(1*\delta')*H$ and why $1*(\delta' * H)$ are well defined? Where $*$ is the product of convolution and $H$ is the function of Heaviside. Thank you in advance
0
votes
1answer
17 views

Equality in distribution sense implies almost everywhere sense. (In one dimension)?

Let $\Omega \subset \mathbb{R}$, bounded open set. And $f\in L^2(\Omega)$, if we have that for $g\in D'(\Omega)$. $$\int_\Omega f.\phi dx=\int_\Omega g.\phi dx \; \text{for all } \phi \in D(\Omega)$$...
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0answers
29 views

What is the Fourier transform of the identity function?

How to find the Fourier transform of $x \mapsto x$ using distribution $\delta$? Since $FT(1)=\sqrt{2\pi} \delta(k)$ then $FT(x \cdot 1)=\sqrt{2\pi} i \delta'(k)$ But also since $1=d/dx (x)$ then $...
3
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1answer
46 views

Derivate of distribution, how does it work?

Let $f\in L^1$. Then $f$ define a distribution $$L_f(\varphi )=\int_{\mathbb R}f\varphi=:\left<f,\varphi \right>,\quad \varphi \in \mathcal S(\mathbb R^n).$$ The derivative of $f$ (in ...
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votes
0answers
26 views

Convergence in distribution to delta

Let us assume $f_n\in S'(\mathbb{R}^k)$ (Schwartz space) such that $f_n\to\delta_t$ in distribution sense. That is, $$\int_{\mathbb{R}^k}f_n(s)F(s)\mathrm{d}s\to F(t)$$ for all $F\in S(\mathbb{R}^k)$. ...
1
vote
1answer
46 views

Why do we have $\int_{t_0}^t \delta(t-t') c(t')=\frac{1}{2} c(t)$?

I am reading a physics paper in which they claim that: $$\int_{t_0}^t \delta(t-t') c(t')=\frac{1}{2} c(t).$$ They just say about this equation: "The result will always hold when $\int_{-\infty}^...
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votes
0answers
21 views

Novice question about power laws

Say I have a cumulative distribution that is described by a power law, $$N = ax^b$$ I.e., N(x) is the number of objects greater or equal than x and $b<0$ so that there are more objects with small ...
1
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0answers
45 views

Solutions of linear PDE in the sense of distributions

Let us consider the wave equation written in the form $$ \partial_{tt}\phi-\Delta\phi+\phi=0. $$ By a Fourier transform, this takes the form $$ (k_0^2-{\bf k}^2-1)\phi(k)=0. $$ This has as a solution, ...
1
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1answer
42 views

$\forall\varepsilon > 0,\exists\ a >0 : |f(x)|\,\le\, a\|f\|_2 + \varepsilon\|f'\|_2$ for $f\in H^1(0,1)$

I know that function evaluation in $H^1(0,1)$ is continuous (see, e.g., Is the Delta distribution a continuous functional on $H^1(\mathbb R)$). So, $\delta_x : H^1(0,1)\to\mathbb C$ is a continuous ...
1
vote
1answer
27 views

Conditional probability of two dependent continuous random variables

I have two continuous random variables $V_1$ and $V_2$ defined as $$\begin{aligned}V_1 &:= a_1 \cdot W_1 + a_2 \cdot W_2 + a_3 \cdot W_3 + a_4 \cdot W_4 + a_5 \cdot W_5 \\ V_2 &:= b_1 \cdot Y ...
2
votes
1answer
41 views

If $ f*g$ is a polynomial of degree at most $m$ for all $g \in C_{0}^{\infty} $. Show $f$ is a polynomial of degree at most $m$ almost everywhere

$C_{0}^{\infty}$ denotes the set of smooth functions with compact support. In attempt of this, I've evaluated the convolution at the (m+1)th derivative to obtain $$ \frac{d}{d^{m+1}}(f*g) =\int f(x-y)...
2
votes
0answers
21 views

Does convergence of operators imply convergence of kernels?

Schwarz Kernel Theorem states that: If $A: C_0^\infty(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n)$ is a linear continuous operator, then there exists a unique kernel $K\in \mathcal{D}'(\mathbb{R}^n\...
5
votes
1answer
48 views

Density of test functions in the space of distributions — a clarification

Let $U \subseteq \mathbb{R}^n$ be open and denote by $\mathcal{D}(U)$ the space of all compactly supported smooth functions $U \to \mathbb{R}$. Let $\mathcal{D}^\prime(U)$ be the space of all ...