Use this tag for questions about Schwartz distributions, also known as Generalised 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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3
votes
1answer
130 views

Convergence to the Dirac Delta Function

Let $h\colon[0,1]\to \mathbb{R}^+$ be any bounded measurable non-negative function with a unique maximum at $a$ and $h$ is continuous at $a$. For $\lambda>0$ define $h_\lambda(x)=C_\lambda ...
0
votes
0answers
31 views

Differential Equation of a Distribution

How to solve the differential equation in $\mathcal{D'}(\mathbb{R})$: \begin{equation} T''=\delta_a \Leftrightarrow \langle T'',\phi\rangle=\langle \delta_a,\phi''\rangle, \forall \phi \in ...
0
votes
0answers
41 views

Tensor product of the Heaviside distribution

I would like to prove that: \begin{equation} H_{(a,b)}=H_a \otimes H_b \end{equation} So far I have: \begin{equation} \langle H_a(x) \otimes H_b(y), \phi\rangle=\langle H_a(x),\langle ...
4
votes
2answers
152 views

Are there any differences between distributions (generalized functions) and probability distributions?

A distribution/generalized function is an element of the dual space of $$S=\{f\in C^{\infty}(\mathbb{R})\colon \|f\|_{\alpha,\beta}<\infty \text{ for all } \alpha ,\beta\}$$ Where ...
0
votes
1answer
43 views

Prove if $T\in\mathcal{D}'(\mathbb{R})$ and $\mathrm{d}T/\mathrm{d}x=0$ then $T$ is the constant distribution.

Prove if $T\in\mathcal{D}(\mathbb{R})$ and $\mathrm{d}T/\mathrm{d}x=0$ then $T$ is the constant distribution. I am having some problems both proving this problem, as well as understanding ...
1
vote
0answers
21 views

Deriving existence of classical Fourier transform via the space of temperate distributions

If for some measurable function $f:{\bf R}^n\rightarrow{\bf R}$ the functional $h\mapsto\int fh$ is in ${\scr S}'$ (space of temperate distributions) and there exists some measurable $g$ such that the ...
3
votes
2answers
42 views

Pointwise convergence and the convergence of a distribution sequence.

The sequence of functions $f_n(x)=\{tanh(nx)\}_{n=0}^{\infty}$ does not converge uniformly to $f(x)=sgn(x)$ but only pointwise. Is it then, still possible that the sequence of distributions ...
-1
votes
1answer
26 views

Proof that the distibution sequence $\int_{0}^{\infty}(tanh(nx)-1)\phi(x)dx \to 0$

Let $T:\phi \mapsto \langle T_f,\phi\rangle =\int _{-\infty}^{\infty}f(x)\phi(x)dx$, where $\phi(x)$ is a test function, be a distribution. I would like to prove that the sequence of distributions: ...
0
votes
0answers
23 views

A question involving weak convergence in $\sigma(L^1, L^\infty)$

Exercise 4.15 from H. Brezis - "Functional analysis..." Let I = (0, 1) and $f_n = n e^{-n x}$ a sequence of functions. Show that $$ f_n \;\; \text{does not converge weakly to} \;\; 0 \;\; \text{in} ...
0
votes
0answers
24 views

How to check a linear map between topological space is continuous?

I am reading something about distributions, and I have a question. I think it is not hard, but I don't know how to explain it rigorously. Suppose $M$ is a smooth manifold, $V$ is a Fréchet space, and ...
0
votes
1answer
36 views

Order of a distribution and its derivatives

For $\varphi\in C_{0}^{\infty}(\mathbb{R}^{3})$ , define $u(\varphi):=\int\partial^{\alpha}\varphi(x,0,0)dx$ for some multiindex $\alpha$ . It's pretty clear to me that $u$ is a distribution. ...
1
vote
2answers
37 views

A sequence of distributions converges to a certain distribution.

Given the sequence of functions: \begin{equation} f_n(x)=tanh(nx) \end{equation} and knowing that: \begin{equation} \lim_{x \to \pm \infty}f_n(x)=f(x)=\begin{cases} -1, & x<0 \\ 1, & ...
1
vote
2answers
64 views

Dirac delta and non-test functions

Normalization of the delta function (distribution) is often informally written as an integral $$\int_{-\infty}^{+\infty} \delta(x) \, dx = 1$$ An attempt to write this formally would be expression ...
0
votes
2answers
42 views

Has any $L^1(\mathbb{R})$ function a distributional derivative?

If I remember correctly, $L^1(\mathbb{R})$ functions can be identified with distributions via $$L^1(\mathbb{R}) \hookrightarrow D'(\mathbb{R})$$ defined as $f\mapsto T_f\in D'(\mathbb{R})$ and then ...
0
votes
0answers
10 views

If solution of NLS is smooth and decaying at infinity; how to justify it satisfy the conservation law?

We consider the cubic nonlinear Shr\"odinger equation(NLS) $iu_{t}+\frac{1}{2} \Delta u = |u|^{2}u, \ u(x, 0)= u_0(x), (x\in \mathbb R^{d}, t\in \mathbb R)$ I have been trying to understand the ...
2
votes
1answer
49 views

Confusion about inclusions of dual spaces

We have the triple inclusions $H^1_0(\Omega)\subset L^2(\Omega)\subset H^{-1}(\Omega)$, where the second inclusion is not literal but in the sense of distributions. Related to this answer, why is the ...
1
vote
1answer
47 views

Introducing an operator by a bilinear form

Let $I = (0, 1)$ and $H_0^2 (I)$ the closure of $C_c^\infty (I)$ in $W^{2, 2} (I)$. Consider $$a : H_0^2 (I) \times H_0^2 (I) \to \mathbb{R}$$ defined by $$a(u, v) = \int\limits_I u''(x) v''(x) \, ...
0
votes
2answers
28 views

Problem on multiplication of distributions with test functions

This question comes from Grubb's Distributions and Operators, question 3.8(b): Consider $u\in \mathcal{D}'(\mathbb{R}^n)$ and $\phi\in C^{\infty}_0(\mathbb{R}^n)$. Find out whether one of the ...
0
votes
0answers
31 views

Elementary properties of Fourier transform on the space of tempered distributions.

I'm trying to prove that some of the basic facts of Fourier transform on $L^1(\mathbb{R}^n)$ also holds on the space of tempered distributions. For example: Suppose that $F\in \mathcal{S}^\prime$, ...
1
vote
2answers
28 views

Can $\sum_{a = -\infty}^{\infty} e^{i\omega aT_0}$ be represented by dirac delta functions?

The usual definition of dirac delta function says that $\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ip(x-\alpha)}\ dp $. The appearance similarity makes me think that it may be possible ...
0
votes
1answer
52 views

Non-standard examples of Distributions

Can somebody point me to examples of distributions which are not sums of delta functions or derivatives thereof. Also of course not integrals with local integrable functions. Additional: is there a ...
1
vote
0answers
25 views

Weak convergence in $D^{1,p}(\mathbb{R}^n)$ and in $\mathcal{M}(\mathbb{R}^n)$

What it means: $$u_n\rightharpoonup u ~\text{weakly in }~ D^{1,p}(\mathbb{R}^n)\\ |\nabla(u_n-u)|^p\rightharpoonup\mu ~\text{weakly in}~ \mathcal{M}(\mathbb{R}^n)\\|u_n-u|^{p^{*}}\rightharpoonup\nu ...
2
votes
1answer
29 views

$k^2 e^{ikx} \rightharpoonup 0$ in the Sense of Distributions

So I'm concerned showing $k^2 e^{ikx} \rightharpoonup 0$ in the sense of distributions or in other words for any $\phi \in C_c^{\infty}(\mathbb{R})$ we have for any $\epsilon > 0$ $$ \left\lvert ...
1
vote
1answer
45 views

Differentiating the Dirac Delta distribution

More generally, let $\psi (D)$ denote a pseudodifferential operator on $\mathbb{R}^n$ given by the function $\psi \in S^m_{\rho, \eta}$, the usual symbol class. My question is: can we interpret $\psi ...
0
votes
0answers
12 views

Fourier transformation of Principal value distribution [duplicate]

I have the principal value distribution defined as $pv(\frac{1}{x})(\phi)=\int^\infty_0\frac{1}{x}(\phi(x)-\phi(-x))dx$ and I want to show that the fourier transform is given by $-\pi i\cdot ...
1
vote
1answer
50 views

Does pointwise convergence imply convergence in distribution? Counterexample?

$f_1,f_2,\dots ,$ and $f$ are in $L_{loc}^1(U)$. I'm trying to give a counterexample where $f_n\to f$ pointwise, but not $f_n\to f$ in $\mathcal{D}^\prime (U)$, where ...
0
votes
2answers
46 views

Does convergence in $L^p$ imply convergence in distribution?

$f_1,f_2,\dots ,$ and $f$ are in $L_{loc}^1(U)$. Prove that if $f_n\in L^p(U)(1\leq p\leq \infty)$ and $f_n\to f$ in the $L^p$ norm or weakly in $L^p$, then $f_n\to f$ in ...
2
votes
1answer
36 views

Distribution agreeing with function

I'm trying to figure out how to show distributions agree with a given function on some domain. For instance, let $f \in C(\mathbb{R^n}\setminus\{0\})$ such that $f(rx) = r^{-n}f(x)$ and $\int f ...
8
votes
1answer
115 views

Why do we give $C_c^\infty(\mathbb{R}^d)$ the topology induced by all good seminorms?

Briefly, my question boils down to the following: What benefits do we gain from considering the space of test functions in the topology induced by all "good" seminorms, as opposed to other topologies ...
1
vote
0answers
48 views

Negative index Sobolev spaces, properties

I was hoping to find some references for the following facts: Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^n$. (i) If $\nabla p$ is a distribution in $H^{-1}(\Omega)$ then ...
1
vote
1answer
35 views

Tempered distributions

Let P be a vector whose componentes are polynomials in $\mathbb{R}^n$ and harmonics. its true that exists a polinomial T that $\nabla T = P$? I think this has something to be with fourie transform, ...
2
votes
0answers
61 views

Applications of Banach-Alaoglu theorem in the theory of distributions?

Are there some interesting applications of Banach-Alaoglu theorem in the theory of distributions? The theorem provides compact subsets in the $w^*$-topology, so distributions seem a great place for ...
0
votes
1answer
49 views

Entire function of exponential type $1$ bounded by $1/(1+|x|)$

Let $f$ be an entire function of exponential type $1$ such that $|f(x)| \leq \frac{1}{1+|x|}$ for all $x\in \mathbb{R}$. First, I have to show that $|f(z)| \leq \frac{Ce^{|Im(z)|}}{1+|z|} z\in ...
0
votes
1answer
39 views

fourier transform and principal values

Fourier transform and principal values Can anyone tell me from how can i get the fouries transformation of prinicipal value of (1/x) $$p.v\int \frac{1}{x}\Bigg(\int e^{-wix}\varphi(w)dw\Bigg)dx$$
2
votes
1answer
42 views

distributions whose derivative is zero?

I just learned about the notion of tempered distributions $\mathcal{S}'(\mathbb{R})$. But it is unclear that if such a distribution has a 0 derivative (of course in the distribution sense) then it ...
1
vote
2answers
25 views

Distributional solution of this equation

I am having trouble finding the distributional solutions $u$ of: $x^2u = \delta$. Could somebody help? Thanks in advance
0
votes
0answers
23 views

Proving the absolute value of a smooth function is $W^{1,p}$ [duplicate]

How could one prove the following: Take $u \in C^1_c(\mathbb{R}^n)$ Then, $|u|$ is in $W^{1,p}(\mathbb{R}^n)$, $p \in [1;\infty)$. The problem is to show that the derivative in the distribution sens ...
0
votes
1answer
37 views

Show that the limit of distributions is Dirac delta

I would like to show that the following statement is true $ \lim_{a\searrow 0} \theta(x)\frac{x^{1-a}}{\Gamma(a)} = \delta(x). $ $\Gamma$ is the gamma function. The above limit is in the sense of ...
1
vote
1answer
33 views

Convergence of $f_n(x)=n^2f(nx)$ in the sense of distributionas

Let $f$ be a test function such that $\int_{-\infty}^\infty f(x)dx=0$ and $f_n(x)=n^2f(nx)$. Find the distributional limit $\lim_{n\to\infty}f_n$. How can I use the Dominated Convergence Theorem ...
1
vote
1answer
27 views

Distribution equation (x-a)T=0

I have to solve the equation $(x-a)T=0$ , T is a distribution. By definition : $(x-a)\int T(x)\varnothing (x)=0$ I know if I pose $X=x-a$ I find $XT(X)=0$ and $T(X)=\delta(X)$. But I stuck to find ...
3
votes
1answer
33 views

Seminorms in distribution theory are norms, right?

In distribution theory the seminorms that you use there $p_m( \phi) := \max_{|\alpha| \le m} \sup_{x \in \Omega} |(\partial^{\alpha}(\phi) (x)|, \phi \in C_c^{\infty}(\Omega)$. Those guys are norms ...
1
vote
1answer
23 views

Bound on the set of compactly supported distributions with support in the same compact set

Consider the set of all compactly supported distributions $v\in\mathcal{\mathcal{E}^{\prime}}(\mathbb{R}^{n})=\left(C^{\infty}\right)^{*}$ with compact support in a fixed compact set $\Omega$ . ...
1
vote
1answer
42 views

A variant of the fundamental lemma of calculus of variation

If $F$ is a distribution and its distributional derivative is equal to 0, how can I show that $F$ is (represented by) a constant function i.e. there exists a constant $c$ such that $F(\phi)=c\int\phi$ ...
5
votes
1answer
114 views

Distributional linear differential equations

What are the most general distributional solutions $u \in \mathcal{D}'(\mathbb{R})$ to $-\frac{d^n u}{dx^n} + c_{n-1}\frac{d^{n-1}u}{dx^{n-1}} + ... + c_0 u = 0$; $-x\frac{d^n u}{dx^n} + ...
1
vote
0answers
59 views

Complex distributions - what are the appropriate test functions?

In the theoretical physics literature on conformal field theory, one encounters distributional formulas like $$ \frac{1}{\pi}\partial_{\bar z}\frac{1}{z} = \delta(z), $$ where $\partial_{\bar z}$ is ...
7
votes
3answers
254 views

Distribution theory and differential equations.

How does distribution theory plays role in solving differential equations? This question might seem to be very general. I will try to explain, please bear with me. I understand, distributions make it ...
2
votes
1answer
47 views

Sign mistake in Fourier transform of $\frac{x}{1+x^2}$.

I want to calculate the distributional Fourier transform of $u(x) = \frac{x}{1+x^2}$ in one dimension in the distributional sense as $u\notin L^1$. I use the distributional definition of the Fourier ...
5
votes
2answers
42 views

How to prove this simple fact without using distribution theory?

Suppose function $f(x) $ is normalized to unity, i.e., $$ \int dx |f(x)|^2 =1 . $$ Now consider the Fourier transform of $f$, i.e., $$ F(k) = \int d x f(x) e^{-i k x} . $$ Here we assume that $f $ ...
3
votes
1answer
54 views

Show that the distribution is of the form $C \delta + f$

I'm trying to solve this problem: Let $ u = p.v.(1/x)$, $\phi$, $\psi \in C^{\infty}_c$. I want to show that the distribution $(\phi u )* (\psi u)$ is of the form $C \delta + f$ for some constant C ...
2
votes
1answer
146 views

Generalized Functions (Distributions) over Manifolds

What is the right way of making sense of generalized functions over manifolds? For concreteness, let me restrict my question to the dirac delta function. The article on Wikipedia on Dirac delta ...