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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7
votes
1answer
381 views

What's the Fourier transform of these functions?

The Fourier transform of $|x|^{\alpha}$. This is the Fourier transform of a homogeneous function, and there are several cases of various $\alpha$: when $a\leq -n$, it's not a temperate distribution; ...
4
votes
1answer
191 views

A question about convolution of two distributions

Generally,when taking convolution of two distributions,at least one of which is supposed to be of compact support. But when u,$v\in S'(\mathbb{R})$ ( temperate distributions) have suports on the ...
2
votes
1answer
121 views

Some questions about distribution theorem

Given an equation $P(D)u=0$, where $P$ is a polynomial (not equal to a constant). Here are some basic information about the distributional solution $u$: If $P$ has at least one real root, then there ...
1
vote
1answer
773 views

Proving the mean value property of harmonic functions using distributions?

A professor I talked to showed me a proof of the mean value property. (He actually showed it for functions solving the heat equation instead of Laplace's equation, but it seems like the argument is ...
3
votes
1answer
289 views

The distribution $\Delta u$ (where $u = \ln|\vec{x}|$)

Problem Consider the function $u(\vec{x})=\ln|\vec{x}|$ as a distribution on $\mathbb{R}^3$ and $\mathbb{R}^2$. We want to determine $\Delta u$ in the distribution sense. First calculate $\Delta u$ ...
1
vote
1answer
636 views

What is the sum of only half the exponential terms that give the Dirac comb?

The following infinite sum of exponential terms gives a Dirac comb: $$ \sum_{n=-\infty}^\infty e^{i n x} = 2 \pi \sum_{n=-\infty}^\infty \delta(x - 2 \pi n) $$ Of course the sum doesn't strictly ...
3
votes
1answer
835 views

Normalization parameter, properties of Dirac delta functions

Suppose $\psi_E (x)=N(E)\exp (ikx)$ where $\psi_E (x)$ is a momentum eigenfunction, $N(E)$ is the normalization constant on the energy scale such that $\langle E'|E\rangle=\int_{-\infty}^\infty ...
1
vote
0answers
177 views

Integration methods for functions with Delta distributions

Which Monte-Carlo methods are available for computing a multidimensional integral with Delta distributions (in case one cannot sample them explicitly)? PS: I also asked a similar question at ...
3
votes
0answers
100 views

Colombeau product of distributions

How can use the Colombeau generalized function method to evaluate the product of distributions $ \delta (x) \times \delta (x) $ or $ \delta ^{n} (x) \times \delta ^{m} (x) $ (derivatives of dirac ...
5
votes
2answers
602 views

limit of an integral with a Lorentzian function

We want to calculate the $\lim_{\epsilon \to 0} \int_{-\infty}^{\infty} \frac{f(x)}{x^2 + \epsilon^2} dx $ for a function $f(x)$ such that $f(0)=0$. We are physicist, so the function $f(x)$ is smooth ...
3
votes
1answer
55 views

Uniform convergence and convergence in $S'(\mathbb{R}^n)$

Let $$\hat{f_\epsilon}: \xi \mapsto \exp(-\epsilon |\xi|) \frac{\sin(|\xi|t)}{|\xi| t}$$ denote to the Fourier transform of $f$. How do I see $\hat{f_\epsilon}$ converges uniformly on ...
-1
votes
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
727 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
vote
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
vote
0answers
298 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
235 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
votes
0answers
109 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
935 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
74 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
vote
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
933 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
181 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
vote
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
152 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
176 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
vote
1answer
258 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
136 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
147 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
62 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
253 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
151 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
99 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
votes
0answers
157 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
810 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
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
votes
1answer
390 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
670 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
59 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
177 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 ...