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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1answer
132 views

Approximating an integral

Might be simple, but i don't get it. Why is the integral in the last line approximately equal to $n(\varphi(\frac{-1}{2n}) - \varphi(\frac{1}{2n}))$?
4
votes
1answer
709 views

Application of closed graph theorem.

I'm having a problem applying the closed graph theorem, which I think stems from distributions still being very new to me. I am reading a proof in Stein and Weiss, Introduction to Fourier Analysis ...
3
votes
0answers
57 views

Why $\partial_{i}(A^{*}u)=A^{*}(\sum^{n}_{j=1}a_{ji}\partial_{j} u)$?

We define the affine transformation on distributions by $$\langle A^{*}u, \phi \rangle=\frac{1}{\det(A)}\langle u,\phi(A^{-1}x)\rangle$$ Assume this we should have $$\langle \partial_{i}(A^{*}u), ...
1
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0answers
84 views

wavefront set of a distribution

If $(x_0,\xi_0)\in\mathbb{R}^{2n}$ is a given point in phase space, how do I construct a compactly supported distribution $u$ which has WF$(u)=\{(x_0,t\xi_0) | t>0\}$ ?
0
votes
1answer
1k views

Fourier transformation of sin, cos, sinh and cosh

I am trying to solve the following exercise Use $\mathcal{F}(e^{xb}) = 2\pi \delta_{ib}$ to calculate the Fourier-Transformation of $\sin x$, $\cos x$, $\sinh x$ and $\cosh x$ Now I am a little ...
1
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0answers
293 views

Poisson equation on half-space

Let $H$ be the open half-space of $\Bbb R^n$ defined by $x_n > 0$. Let $f : \overline H \to \Bbb R$ continuous and harmonic on $H$. Define the function $F : \Bbb R^n \to \Bbb R$ by $$ ...
0
votes
1answer
199 views

How to use the Malgrange-Ehrenpreis-Theorem

In the Wikipedia article of this theorem http://en.wikipedia.org/wiki/Malgrange%E2%80%93Ehrenpreis_theorem it is said that i could be used to prove that $P(\partial/\partial x_i)u(x)=f(x)$ has a ...
4
votes
1answer
233 views

Questions concerning a proof that $\mathcal{D}$ is dense in $\mathcal{S}$.

I am currently working through this lecture notes and on page 164, there it is said The space of $\mathcal{D}(\mathbb{R}^n)$ of smooth complex-valued functions with compact support is contained ...
2
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0answers
105 views

Why does the following define a distribution and of which order?

I want to show that $$\phi\mapsto\underset{\varepsilon\searrow 0}{lim}\int_{-\infty}^{\infty}\frac{\phi(x)}{x+i\varepsilon}dx$$ defines a distribution on $\mathcal{D}(\mathbb{R})$ but I just don't ...
3
votes
1answer
916 views

Does zero distributional derivative imply constant function?

If a real function $f\colon[a,b]\to\mathbb{R}$ is differentiable and its derivative $f'$ is zero, then $f$ is constant. Does this result still hold when $f$ has a weak derivative? Explicitly, suppose ...
1
vote
1answer
73 views

whats the order of a distributional derivate?

I have to calculate the derivatives of order $\le 2$ of for example $f(x) = |x|$, is it the same as the second derivate, what does this "of order $\le 2$" mean? calculating distributionell derivatives ...
1
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1answer
50 views

connection between the support and the representation of a distribution

I want to show, that for $u' \in \mathcal{D}'(\mathbb{R}^n)$ supp $u$ = $\{ 0 \}$ iff there exist numbers $m \in \mathbb{N}, c_{\alpha} \in \mathbb{K}$ such that $u = \sum_{|\alpha| \le m} c_{\alpha} ...
0
votes
2answers
70 views

why is $\frac{\phi(x)-\phi(-x)}{x}$ for smooth $\phi$ bounded at $x=0$

Why is $\frac{\phi(x)-\phi(-x)}{x}$ for smooth $\phi$ bounded at $x=0$? If i set $\phi(x) = \sqrt{|x|}$, it definitely not bounded. I saw this on page 293 of ...
4
votes
2answers
916 views

principal value as distribution, written as integral over singularity

Let $C_0^\infty(\mathbb{R})$ be the set of smooth functions with compact support on the real line $\mathbb{R}.$ Then, the map $$\operatorname{p.\!v.}\left(\frac{1}{x}\right)\,: ...
2
votes
1answer
177 views

What does it mean to say an integral exists 'in the distributional sense'?

What exactly does it mean to say that an integral exists 'just in the distributional sense'? For example, the Fourier transform of $x^2 e^{-\lambda x}$ or of $H(R-|x|)$ where $R > 0$ and $H$ is the ...
0
votes
2answers
1k 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 = ...
1
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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
votes
1answer
114 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
votes
2answers
150 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 $f''$ ...
0
votes
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
votes
2answers
175 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
255 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
vote
0answers
146 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
61 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
votes
1answer
114 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
250 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 ...
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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 ...
5
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0answers
154 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
80 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
804 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
382 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
657 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
58 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
173 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
246 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
678 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
193 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
336 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
392 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
575 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 ...
4
votes
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
464 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 ...