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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14 views

Properties of Certain Example of Nonuniqueness to Heat equation

We say that a function $g\in C^{\infty}(\mathbb{R}^{n})$ is of Gevrey class $G^{\sigma}(\mathbb{R}^{n})$ if for each compact $K\subset\mathbb{R}^{n}$, there exist $C,R>0$ such that $$\sup_{x\in ...
2
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1answer
18 views

Existence of Certain Locally Integrable Function Defining a Tempered Distribution

We say that a distribution $T$ is tempered if for every sequence $\left\{\phi_{n}\right\}$ in $C_{c}^{\infty}(\mathbb{R}^{n})$ tending to $0$ in the topology of the Schwartz space ...
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0answers
34 views

Is the following property of a Fourier Transform valid?

We know that $$\mathscr{F}\left\{f*g\right\}=\mathscr{F}\left\{f\right\}\mathscr{F}\{g\}$$ so I was wondering whether the inverse is true: ...
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0answers
27 views

Square of (derivative of Dirac Delta) [on hold]

Let $f(x)$ is test function. I have to compute $\int_{\mathbb{R}}|\delta'(x)|^2~f(x)$. I tried usual by parts (though I have a confusion of how well difined is $\delta'\circ \delta'$). I tried as ...
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0answers
25 views

Rigorous Justification of Infinitesimal Techniques

As you may know that there are a bunch of heuristic techniques in physics to make integrals converge. For example, when we define a following Fourier transform, we add a positive infinitesimal and let ...
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1answer
52 views

$f, \hat{f} \in L^{p}(\mathbb R) \cap C(\mathbb R) \implies |f(x)| \to 0$ as $|x| \to \infty$?

Suppose $f, \hat{f} \in L^{p}(\mathbb R) \cap C(\mathbb R)\cap L^{\infty}(\mathbb R), (1<p<\infty).$ My Question: Can we expect $\lim_{|x|\to \infty} |f(x)|=0$ ? (In other words, If $f$ and its ...
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2answers
46 views

Writing integral in terms of distributions

EDIT (now asking how to write $F$ as distributions, instead of writing the integral in terms of distributions): Let $F$ be the distribution defined by its action on a test function $\phi$ as ...
2
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1answer
38 views

Prove that $\lim_{n\to \infty}\langle \operatorname{erfc}(-nx), \phi\rangle =\langle H_0, \phi\rangle $

Define the error function $\operatorname{erf}(x)$ as: \begin{equation} \operatorname{erf}(x):=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-y^2}dy \end{equation} and ...
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1answer
28 views

What's the difference between $C_c^{\infty}(\mathbb{R})$ and Schwartz Space for test spaces of distributions? Are there other spaces?

A distribution is an element of either $\left(C_c^{\infty}(\mathbb{R})\right)^{\ast}$ or of $S^{\ast}$, depending on literature. Why would we use one vs the other? What other spaces are there and ...
2
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1answer
33 views

Uniform convergence for the sequence $\psi_n(x)=n^{-1}e^{-n^2/(n^2-x^2)}$

How can I prove that the sequence of functions: \begin{equation} \psi_n(x)=\begin{cases} n^{-1}e^{-n^2/(n^2-x^2)}, & |x|\leq n \\ 0, & |x|\geq n \end{cases} \end{equation} convergences ...
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1answer
25 views

Problem with second distributional derivative

I have the following function: $ f(x) = \begin{cases} \sqrt{x}, & \text{if $x>0$} \\ \sqrt[3]{|x|}, & \text{if $x<0$ } \end{cases} $. I have to find $f'(x)$, $f''(x)$ as ...
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0answers
19 views

Example concearning the Schwartz Kernel Theorem?

Let $\Omega_i\subseteq \mathbb R^{n_i}$ ($i=1, 2$) be open subsets. Given $K\in\mathscr{D}^\prime(\Omega_1\times \Omega_2)$ one can obtain a continuous linear operator ...
9
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1answer
167 views

Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is ...
4
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0answers
67 views

How to prove that this solution of heat equation is not a tempered distribution?

A theorem of PDEs sais that the following Cauchy problem for the heat equation \begin{align*} & \partial_t u = \partial_{x}^2 u, \quad (t,x) \in \mathbb{R_+} \times \mathbb{R}, \\ & u|_{ t = ...
3
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1answer
108 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 ...
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0answers
30 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 ...
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0answers
39 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
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2answers
116 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 ...
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1answer
23 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 ...
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0answers
18 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 ...
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0answers
42 views

The Analysis of Linear Partial Differential Operators I Prerequisites

I am a graduate level student in Mathematics and I would like to study the books titled "the analysis of linear partial differential operators I-IV" by Hörmander. As I have been away from mathematics ...
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2answers
37 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 ...
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1answer
24 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: ...
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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} ...
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0answers
19 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 ...
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1answer
22 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. ...
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2answers
30 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, & ...
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2answers
51 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 ...
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2answers
36 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 ...
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0answers
9 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 ...
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1answer
28 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 ...
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1answer
44 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) \, ...
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0answers
24 views

Support of Distribution Function

Suppose I have a distribution function $$C(u,v)$$ with domain $I^2$. Let us define the support of this function as the complement of the union of all open subsets of $I^2$ with C-measure zero. Based ...
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2answers
25 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 ...
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0answers
23 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 $F\in \mathcal{S}^\prime$, ...
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2answers
24 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
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1answer
47 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
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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
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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 ...
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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 ...
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0answers
11 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 ...
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1answer
44 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
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2answers
44 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
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1answer
31 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 ...
5
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0answers
55 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 ...
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0answers
19 views

Expand a distribution to a linear differential form

A linear differential operator $P$ is defined as: $$ PT=\sum_{\alpha=0}^\infty p_{\alpha} \partial^\alpha T $$ A distribution $Q$ which satisfies $P \delta = Q$ is given, where $\delta$ is the Dirac ...
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45 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 ...
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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
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0answers
55 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
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1answer
39 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 ...