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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2answers
979 views

distributional derivative

I want to calculate the first and second distributional derivate of the $2\pi$-periodic function $f(t) = \frac{\pi}{4} |t|$, it is $$ \langle f', \phi \rangle = - \langle f, \phi' \rangle = ...
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1answer
82 views

How can an ordinary function be a distribution?

I think distributions are linear and continuous functionals on the set of testfunctions. In a textbook I saw this question: Let $f$ be a $2\pi$-periodic function with $f(t) = \frac{\pi}{4}|t|$ ...
2
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1answer
112 views

Show that a functional is a distribution

Consider the following functional $$ \langle u , \phi \rangle = \int_0^{\infty} \phi(t) \frac{\mathrm{d}t}{t^{\alpha}} $$ I want to show that it is a functional in $\mathcal{D'}{(\mathbb{R})}$. ...
4
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2answers
149 views

Function $f$ such that Fourier-series converges uniformly, but the series of the derivatives are divergent

I am studying Fourier-transformation right now, and I am asking if there exists a function $f$ such that is Fourier-series converges uniformly, the Fourier-series of $f'$ only in $L_2$ and that $f''$ ...
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2answers
38 views

what could be said about the series of test-functions?

in another thread an offtopic question came to my mind. Consider a test/bump function $\phi$, then consider all its derivatives $\phi^{(k)}$, of course they are bounded by constants $M_k$, next form ...
2
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2answers
174 views

What is wrong in this counter-example?

In reading my textbook, the author give a lemma as follows: Let $X\subset \mathbb{R}^{n}$ be an open set, and let $u\in \mathcal{E}'(X)$ have order $N$. Then $\langle u,\phi \rangle=0$ for all $\phi$ ...
1
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1answer
246 views

Are test/bump functions always bounded?

a bump function is a infinitely often differentiable function with compact support. I guess that such functions are always bounded, especially because the set where they are not zero is compact and ...
3
votes
1answer
135 views

What is the mistake in my reasoning?

I am working on the following problem trying to use strategy in this problem. I am trying to simplify the proof by working with $v_{i}=0,i\not=1$ case. But the result looks very different from what I ...
1
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0answers
145 views

Fundamental solution of a vector field

Consider a smooth real vector field $X(x,\partial _x)=\sum a_i(x) \frac{\partial}{\partial x_i}$ on an open subset $\Omega \subset \mathbb{R}^n$ (which is assumed to be sufficiently regular) as a ...
2
votes
1answer
58 views

Confusion on continuous linear forms

In Friedlander's book "introduction to the theory of distributions" he claimed(on page 35): "Now the equation $$|\langle u,\phi\rangle| \le C\sum_{|a|\le N|}\sup\{|\partial^{\alpha}\phi|:x\in K\}$$ ...
3
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1answer
113 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
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1answer
249 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
149 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
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0answers
98 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
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0answers
149 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
79 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
797 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
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0answers
172 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
96 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
147 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 $$ ...
2
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1answer
379 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
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4answers
640 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
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0answers
57 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
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1answer
170 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
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0answers
242 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
667 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
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1answer
189 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
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0answers
321 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
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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
380 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
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2answers
569 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 ...
3
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1answer
3k 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
451 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
153 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
387 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
163 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
432 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
278 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
109 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
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1answer
440 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
586 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
388 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
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0answers
98 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
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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
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0answers
181 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
749 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
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1answer
441 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
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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: ...