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).

learn more… | top users | synonyms

3
votes
1answer
111 views

Why $Pu\in C^{\infty}(I)$ implies $u\in C^{\infty}(I)$?

I got stuck in the second question of the following problem. I cannot prove anything significant other than the base case. So I think I need some help. Here is what I did: Let $I\in \mathbb{R}$ be ...
6
votes
1answer
246 views

What is wrong with my `proof'?(solved)

The question is: Let $k\in C^{0}(\mathbb{R}^{n}-\{0\})$ be a function such that $$k(xt)=t^{-n}k(x)$$ for $0\not=x\in\mathbb{R}^{n},t>0$. Show that the principal value $$\int ...
3
votes
1answer
143 views

Why $\langle \frac{1}{x^{2}},\phi\rangle =\int^{\infty}_{0}\frac{\phi(x)+\phi(-x)-2\phi(0)}{x^{2}}dx$?

I cannot get this identity by using the condition: $\frac{1}{x^{2}}=-\partial(\frac{1}{x})$, and the integrands are defined by continuity at $x=0$. My reasoning goes as $$\langle ...
1
vote
0answers
90 views

Symbol of a partial differential operator.

We have $$P(fu)=\sum_{|\alpha|\le m}a_{\alpha}(x)\sum_{\beta+\gamma=\alpha}\frac{\alpha!}{\beta!\gamma!}\partial^{\beta}f\partial^{\gamma}u$$ The author claimed it is `obvious' we can put it into the ...
4
votes
0answers
140 views

Distributional derivative of bounded functions

Let $f$ be a bounded measurable function on $\mathbb{R}$. We may consider its derivative $f'$ as a distribution on $\mathbb{R}$. Is there a reasonable description of those distributions $\psi$ ...
1
vote
1answer
75 views

Question about distribution

Let $(f_k)_{1\le k\le \infty}\in L_{1}^\mathrm{loc}(\mathbb{R}^n)$ be a sequence of real valued functions such that $\operatorname{supp} f_k \subset \{|x|\le k^{-1}\}$, $$\int f_k (x)\,dx=1,k\in ...
1
vote
1answer
722 views

Fourier transform of convolution of sinusoidal signals, or product of distributions (generalized functions)

I will unashamedly say that this was at least spurred by homework. However I have gone far beyond the syllabus of the course and still can't find an authoritative answer. And it seems an interesting ...
3
votes
0answers
169 views

Does $\sqrt{\delta^2}$ make more sense than $\delta^2$?

What is the Product of $\delta$ functions with itself? was already asked some time ago. In a comment the OP states: I want to create $\delta(t)$ such that the product with itself is also ...
0
votes
1answer
95 views

Limit of the sequence $f_n(t) = \frac{1}{t + n + i/n}$ of smooth functions

Let $\mathcal{E}$ be the space $C ^{\infty}(\mathbb R)$ with the system of seminorms: $$ p_{N,n}(f) := \max{\lbrace |f^{(k)}(t)| : k = 0, 1, \dots , n; t \in [-N, N] \rbrace},\quad n = 0, 1, 2, ...
3
votes
1answer
146 views

Function of $C_0(\mathbb{R})$

I need to prove that $$g(x) = \text{p.v.} \int\limits_{-1/2}^{1/2}\frac{e^{-itx}}{t\cdot \ln{|t|}}dt $$ is function of $C_{0}(\mathbb{R})$. So, I need to prove that $$ ...
1
vote
1answer
307 views

Identify distribution by a constant function [duplicate]

Possible Duplicate: On distributions over $\mathbb R$ whose derivatives vanishes Why can I identify a distribution $G \in \mathcal{D}'((a,b))$, $\partial G = 0$ by a constant function?
5
votes
4answers
535 views

What is the relationship between generalized functions and things like the Riesz representation theorem?

I just watched this video of Prof. Osgood's lecture on Fourier Transforms, and it seems to me that there's some connection between his talk of distributions (generalized functions) and the usual ...
2
votes
0answers
56 views

Action of linear functional on integral depending of parameter

Let $K(x,\omega) \in C^{\infty}(\Omega \times \Omega)$, where $\Omega$ is a domain in $\mathbb{R}^{n}$. Let $\mu$ be a probability measure on $\Omega$. My question is under which conditions an ...
2
votes
1answer
158 views

Principal Valued Distributions

I am currently studying applied functional analysis and I see a proof about principal valued distributions. It is easy to prove that $x \times P/x =1$, where $P/x$ is the principal valued ...
2
votes
0answers
230 views

Representation of compactly supported distribution

Is this true? Any compactly supported distribution $T\in \cal D'$ can be represented as finite sum of partial derivatives of functions.
3
votes
4answers
561 views

An application of the Dirac delta function

I do apologize if this question is a bit vague, but I shall try to be as clear as possible. We were introduced to the Dirac delta function $\delta(x)$. I have seen examples in applied courses where ...
2
votes
1answer
1k views

weak derivative of a nondifferentiable function

I am reading a book on Sobolev and having trouble understanding a notion of weak derivative. I consider a function $(x-1)^+=\max(x-1,0),x\in[0,2]$, I have a problem at $x=1$, so it is continuous and ...
1
vote
1answer
180 views

Homogeneous and rotational invariant distribution

If $u \in \mathcal D'(\mathbb R^n)$, $u$ is homogeneous of degree $0$ and rotational invariant, it is necessarily that $u$ is a constant? (Since if $u \in C^\infty$, the conclusion obviously hold.)
2
votes
0answers
277 views

The topology of distributions

I have been wondering about the following concerning the spaces $\mathcal D$ of test functions (say on $\mathbf R$). It is my understanding that the topology on this space is inductive limit ...
2
votes
0answers
88 views

The topology of $C_0^\infty(M) $

I have read definitions in my PDE book as follows: If $M$ is a smooth paracompact manifold, the space of all linear functional on $C^\infty(M) $ is denoted by $\mathcal E'$ and the space of all linear ...
5
votes
1answer
343 views

How should I understand a PDE that contains distribution or measure mathematically?

We know that the Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, $$ \delta(x)=\begin{cases}+\infty, ...
5
votes
2answers
4k views

Dirac Delta function

I know that $\int_{-\epsilon}^\infty f(x)\delta(x)dx=f(0)$ but what about $\int_0^\infty f(x)\delta(x)dx$? I suppose we have to do this by definition since the lower limit is bang on $0$?
3
votes
2answers
497 views

Verifying the 2-dimensional fundamental solution of the wave equation

I'm trying to verify that $$u(t,x)=H(t-|x|)(t^2-|x|^2)^{-1/2}$$ is the fundamental solution of the 2-dimensional wave equation; that is, $\Box u = u_{tt}-\Delta u = \delta_{0}$. I know there are ...
2
votes
1answer
2k views

Clear explanation of heaviside function fourier transform

I know that fourier transform of Heaviside function is : $\hat{H}(x) = \pi \delta(\omega) + i (v.p. \frac{1}{\omega})$ How can i proof this result?
4
votes
1answer
406 views

How to make this heuristic extension of Itô-Tanaka's formula valid

Here is my story, I have the following function : $$ g(x)=(1+x)\cdot\exp\left(-\frac{(\log(x+a)+c)^2}{2\sigma^2}\right)1[x\ge y]=f(x)\cdot1[x\ge y] $$ with $a,c,\sigma$ being "good" reals so that ...
2
votes
1answer
108 views

Differential equation for distribution

Consider a distribution $T \in D'(\mathbb{R})$ such as (E) : $T' + gT = 0$ with $g \in D(\mathbb{R})$. Could you prove that $T$ is a strong solution of (E) ? I know that we must use the ...
4
votes
2answers
347 views

A sufficient condition for a function to be of class $C^2$ in the weak sense.

Let $f\colon\mathbb{R}\to\mathbb{R}$ be a continuous function with weak derivative (i.e. the derivative in the sense of distribution) in $C^1(\mathbb{R})$. Does this condition imply that $f$ is two ...
1
vote
1answer
160 views

Verify this distribution convolution: $E(t,x)\ast (g(x)\delta(t)) = t\int_{\omega\in S^2}{\frac{g(x-t\omega)}{4\pi}dS(\omega)}$

In our class notes we are asked to verify the following equality: $$E(t,x)\ast (g(x)\delta(t)) = t\int_{\omega\in S^2}{\frac{g(x-t\omega)}{4\pi}dS(\omega)}$$ where ...
2
votes
2answers
395 views

Convolution of different objects

I am studying an article in which convolution is used very much and it is used between all sorts of objects: functions, distributions and measures. I know that convolving a $L^1$ function (for ...
1
vote
1answer
272 views

Problem regarding to Dirac Delta function

I'm reading a paper about the fundamental solution of the wave operator in $\mathbb{R}^3$. The author said that the fundamental solution equals $$cH(t)\delta(t^2-|x|^2)=c/2tH(t)\delta(t-|x|)$$ where c ...
3
votes
2answers
105 views

Understanding an example of a Distribution

Whilst reading the article about restrictions of distributions (generalized functions) on Wikipedia (here) I had trouble understanding the example of a distribution defined on the subset $V = (0,2) ...
2
votes
1answer
355 views

Question about support of distributions

I'm reading "Pseudo-differential Operators and the Nash-Moser Theorem" and at the top of on page 8 they write: "Finally, we note that if $u \in C^0(\Omega)$, then the support of $u$ defined above ...
2
votes
1answer
545 views

Dirac delta and derivative inside an integral

In development of a calculus (in GR but it doesn't matter here) I have seen one dubious substitution: in an integral of the form: $$\int dx ~\delta(x-x_0)~\partial_x F(x) $$ The author substitutes ...
3
votes
1answer
375 views

Delta function integral

I encountered an issue when doing some problems in solid state physics, and I spent a whole day trying to clear this up, unsuccessfully. I'm posting it here because my issue is purely mathematical. ...
1
vote
0answers
94 views

Introductory book on Distribution theory [duplicate]

Possible Duplicate: Distribution theory book Is there a good alternative to Friedlander Introduction to the Theory of Distributions ? Many thanks !
1
vote
0answers
64 views

$C^{\infty}$ function represented by the diverging integral

There is a theorem (see Treves: "Introduction to Pseudodifferential and Fourier integral operators") that states that the kernel of any pseudodifferential operator, i.e. the distribution $$ K(x,y) ...
2
votes
0answers
173 views

What does it mean to carry out calculations modulo $S^{- \infty}$

I have trouble making sense of the following remark that is given in a set of lecture notes I am currently going through on my own: " Oscillatory integrals are used for the study of singularities of ...
7
votes
1answer
711 views

Meaning of polynomially bounded

I am currently reading through some lecture notes on Fourier Transform and Distributions on my own, and came upon the notion of a polynomially bounded function. I am not sure I understand this ...
1
vote
1answer
413 views

how could i prove this (delta function representation)

how could i prove that the sucession $ \frac{k^{k+1}}{k!}t^{k}e^{-kt} $ tends in the limit $ k \to \infty$ to the delta function $ \delta (t-1) $ this is used inside the post 's inversion formula ...
3
votes
2answers
140 views

Is the following derivative defined?

I am new to this site so I am not sure if this is the right place to be asking this question but I will try anyway. I am reading an economics paper for my research and the author does the following: ...
2
votes
2answers
201 views

Simplifying the generalized function $x^{\lambda}_+$ in the strip $-n - 1 < \mbox{Re}\lambda < -n$

Note: this post is a follow up to an earlier question. The (divergent) integral of $x^{\lambda}_+$ can be analytically continued into the region Re $\lambda > -n - 1$, $\lambda \ne -1, -2 , \ldots ...
4
votes
4answers
3k views

Which of these two ways to take the derivative of a delta function times another function is correct?

A well known identity of the Dirac delta function is that for any function $f(x)$: $$ \delta(x) f(x) = \delta(x) f(0). $$ If we take the derivative of the right hand side we get: $$ ...
2
votes
2answers
195 views

Tempered distributions of finite order?

Is every tempered distribution of finite order? It seems that yes with the definition.
4
votes
4answers
266 views

Borel Measure such that integrating a polynomial yields the derivative at a point

Does there exist a signed regular Borel measure such that $$ \int_0^1 p(x) d\mu(x) = p'(0) $$ for all polynomials of at most degree $N$ for some fixed $N$. This seems similar to a Dirac measure ...
6
votes
1answer
187 views

Showing that $\int_0^1 x^{\lambda} [ \: \phi(x) - \phi(0)\: ] dx$ is convergent for $\lambda > -2$

Id' appreciate help understanding why the integral $$ \int_0^1 x^{\lambda} [ \: \phi(x) - \phi(0)\: ] dx $$ is convergent provided $\lambda > -2$, where $\phi \in \mathcal{D}(\mathbb{R})$. To ...
5
votes
1answer
499 views

Sequential and topological duals of test function spaces

Given a test function space, in particular $\mathcal{S}=\mathcal{S}(\mathbb{R}^n)$ (the Schwartz space) or $\mathcal{D}=\mathcal{D}(\mathbb{R}^n)$ (the space of compactly supported smooth test ...
1
vote
1answer
323 views

Delta function representation

Suppose I have a set of functions $(f_\epsilon)$ such that as $\epsilon\to 0$, $f_\epsilon\to F$ s.t. $F(x)=0$ for $x\neq 0$ and $F(x)=\infty$ for $x=0$; $\int_{-\infty}^\infty f_\epsilon(x) dx=1$ ...
1
vote
2answers
381 views

When is the convolution with a tempered distribution again a tempered distribution?

If $f$ is a Schwartz function on $\mathbb R^n$ and $g \in L^1(\mathbb R^n)$, then if $g$ is the Poisson kernel, is $f\ast g$ a Schwartz function? are there any known sufficient conditions on $g$ to ...
2
votes
0answers
82 views

Deduce the global differential equation from the pointwisely defined equation in Fourier space

Let $G\in \mathcal{F}(\mathbb{R}^{n+1})'$ be a distribution on the space of spatial Fourier transform'able function, ie an $L^1_{\mathrm{loc}}(\mathbb{R^{n+1}})$ function, $G = G(t,\xi)$. Assume ...
9
votes
1answer
175 views

Oscillatory integral giving me the willies

So now that my term's over, I've been brushing up on my quantum field theory, and I came across the following line in my textbook without any justification: ...